Knapsack problem/0-1
You are encouraged to solve this task according to the task description, using any language you may know.
A tourist wants to make a good trip at the weekend with his friends.
They will go to the mountains to see the wonders of nature, so he needs to pack well for the trip.
He has a good knapsack for carrying things, but knows that he can carry a maximum of only 4kg in it, and it will have to last the whole day.
He creates a list of what he wants to bring for the trip but the total weight of all items is too much.
He then decides to add columns to his initial list detailing their weights and a numerical value representing how important the item is for the trip.
Here is the list:
Table of potential knapsack items item weight (dag) value map 9 150 compass 13 35 water 153 200 sandwich 50 160 glucose 15 60 tin 68 45 banana 27 60 apple 39 40 cheese 23 30 beer 52 10 suntan cream 11 70 camera 32 30 T-shirt 24 15 trousers 48 10 umbrella 73 40 waterproof trousers 42 70 waterproof overclothes 43 75 note-case 22 80 sunglasses 7 20 towel 18 12 socks 4 50 book 30 10 knapsack ≤400 dag ?
The tourist can choose to take any combination of items from the list,
but only one of each item is available.
He may not cut or diminish the items, so he can only take whole units of any item.
- Task
Show which items the tourist can carry in his knapsack so that their total weight does not exceed 400 dag [4 kg], and their total value is maximized.
[dag = decagram = 10 grams]
- Related tasks
11l
<lang 11l>F totalvalue(comb)
V totwt = 0
V totval = 0
L(item, wt, val) comb
totwt += wt
totval += val
R I totwt <= 400 {(totval, -totwt)} E (0, 0)
V items = [
(‘map’, 9, 150), (‘compass’, 13, 35), (‘water’, 153, 200), (‘sandwich’, 50, 160), (‘glucose’, 15, 60), (‘tin’, 68, 45), (‘banana’, 27, 60), (‘apple’, 39, 40), (‘cheese’, 23, 30), (‘beer’, 52, 10), (‘suntan cream’, 11, 70), (‘camera’, 32, 30), (‘t-shirt’, 24, 15), (‘trousers’, 48, 10), (‘umbrella’, 73, 40), (‘waterproof trousers’, 42, 70), (‘waterproof overclothes’, 43, 75), (‘note-case’, 22, 80), (‘sunglasses’, 7, 20), (‘towel’, 18, 12), (‘socks’, 4, 50), (‘book’, 30, 10)
]
F knapsack01_dp(items, limit)
V table = [[0] * (limit + 1)] * (items.len + 1)
L(j) 1 .. items.len
V (item, wt, val) = items[j - 1]
L(w) 1 .. limit
I wt > w
table[j][w] = table[j - 1][w]
E
table[j][w] = max(table[j - 1][w], table[j - 1][w - wt] + val)
[(String, Int, Int)] result
V w = limit
L(j) (items.len .< 0).step(-1)
I table[j][w] != table[j - 1][w]
V (item, wt, val) = items[j - 1]
result.append(items[j - 1])
w -= wt
R result
V bagged = knapsack01_dp(items, 400) print("Bagged the following items\n "sorted(bagged.map((item, _, _2) -> item)).join("\n ")) V (val, wt) = totalvalue(bagged) print(‘for a total value of #. and a total weight of #.’.format(val, -wt))</lang>
- Output:
Bagged the following items banana compass glucose map note-case sandwich socks sunglasses suntan cream water waterproof overclothes waterproof trousers for a total value of 1030 and a total weight of 396
360 Assembly
Non recurvive brute force version. <lang 360asm>* Knapsack problem/0-1 16/02/2017 KNAPSA01 CSECT
USING KNAPSA01,R13
B 72(R15)
DC 17F'0'
STM R14,R12,12(R13)
ST R13,4(R15)
ST R15,8(R13)
LR R13,R15 end of prolog
L R0,N n
LA R1,1
POWER MH R1,=H'2' *2
BCT R0,POWER
BCTR R1,0 -1
ST R1,IMAX imax=2**n-1
SR R6,R6 i=0
DO WHILE=(C,R6,LE,IMAX) do i=0 to imax
SR R10,R10 im=0
SR R8,R8 iw=0
SR R9,R9 iv=0
LA R7,1 j=1
DO WHILE=(C,R7,LE,N) do j=1 to n
LR R1,R6 i
LR R2,R7 j
BAL R14,TSTBIT call tstbit(i,j)
IF C,R0,EQ,=F'1' THEN if tstbit(i,j)=1 then
LA R10,1(R10) im=im+1
LR R3,R7 j
BCTR R3,0
SLA R3,5
LA R1,24(R3)
A R8,DATA(R1) iw=iw+data(j).w
LA R1,28(R3)
A R9,DATA(R1) iv=iv+data(j).v
ENDIF , endif
LA R7,1(R7) j=j+1
ENDDO , enddo j
IF C,R8,LE,MAXW,AND,C,R9,GT,XV THEN if w<=maxw and iv>xv then
ST R6,XB xb=i
ST R10,XM xm=im
ST R8,XW xw=iw
ST R9,XV xv=iv
ENDIF , endif
LA R6,1(R6) i=i+1
ENDDO , enddo i
MVC PG(2),=C'n='
L R1,N n
XDECO R1,XDEC edit n
MVC PG+2(2),XDEC+10
XPRNT PG,L'PG print buffer
LA R6,1
DO WHILE=(C,R6,LE,N) do i=1 to n
L R1,XB xb
LR R2,R6 i
BAL R14,TSTBIT call tstbit(xb,i)
IF C,R0,EQ,=F'1' THEN if tstbit(xb,i)=1 then
LR R1,R6 i
BCTR R1,0
SLA R1,5
LA R2,DATA(R1) @data(i).n
MVC PG(24),0(R2)
XPRNT PG,24 print item
ENDIF , endif
LA R6,1(R6) i=i+1
ENDDO , enddo i
L R1,XM xm
XDECO R1,XDEC edit xm
MVC PGT+6(2),XDEC+10
L R1,XW xw
XDECO R1,XDEC edit xw
MVC PGT+16(3),XDEC+9
L R1,XV xv
XDECO R1,XDEC edit xv
MVC PGT+26(4),XDEC+8
XPRNT PGT,L'PGT print buffer
L R13,4(0,R13) epilog
LM R14,R12,12(R13)
XR R15,R15
BR R14 exit
TSTBIT EQU * R1 value to test the R2 bit
LA R3,32 32
SR R3,R2 (32-i)
STC R3,XSLL+3
LR R0,R1 n
EX 0,XSLL SLL R0,(32-i)
SRL R0,31
BR R14 return R0
XSLL SLL R0,0 shift left logical
MAXW DC F'400' maximum weight N DC A((DATAE-DATA)/32) IMAX DS F number of combinations XB DS F max vector XM DS F max items XW DS F max weight XV DS F max value PG DC CL80' ' PGT DC CL32'items=.. weight=... value=....' XDEC DS CL12 DATA DC CL24'map',F'9',F'150'
DC CL24'compass',F'13',F'35'
DC CL24'water',F'153',F'200'
DC CL24'sandwich',F'50',F'160'
DC CL24'glucose',F'15',F'60'
DC CL24'tin',F'68',F'45'
DC CL24'banana',F'27',F'60'
DC CL24'apple',F'39',F'40'
DC CL24'cheese',F'23',F'30'
DC CL24'beer',F'52',F'10'
DC CL24'suntan cream',F'11',F'70'
DC CL24'camera',F'32',F'30'
DC CL24'T-shirt',F'24',F'15'
DC CL24'trousers',F'48',F'10'
DC CL24'umbrella',F'73',F'40'
DC CL24'book',F'30',F'10'
DC CL24'waterproof trousers',F'42',F'70'
DC CL24'waterproof overclothes',F'43',F'75'
DC CL24'note-case',F'22',F'80'
DC CL24'sunglasses',F'7',F'20'
DC CL24'towel',F'18',F'12'
DC CL24'socks',F'4',F'50'
DATAE DC 0C
YREGS
END KNAPSA01</lang>
- Output:
n=22 map compass water sandwich glucose banana suntan cream waterproof trousers waterproof overclothes note-case sunglasses socks items=12 weight=396 value=1030
Ada
<lang Ada>with Ada.Text_IO; with Ada.Strings.Unbounded;
procedure Knapsack_01 is
package US renames Ada.Strings.Unbounded;
type Item is record
Name : US.Unbounded_String;
Weight : Positive;
Value : Positive;
Taken : Boolean;
end record;
type Item_Array is array (Positive range <>) of Item;
function Total_Weight (Items : Item_Array; Untaken : Boolean := False) return Natural is
Sum : Natural := 0;
begin
for I in Items'Range loop
if Untaken or else Items (I).Taken then
Sum := Sum + Items (I).Weight;
end if;
end loop;
return Sum;
end Total_Weight;
function Total_Value (Items : Item_Array; Untaken : Boolean := False) return Natural is
Sum : Natural := 0;
begin
for I in Items'Range loop
if Untaken or else Items (I).Taken then
Sum := Sum + Items (I).Value;
end if;
end loop;
return Sum;
end Total_Value;
function Max (Left, Right : Natural) return Natural is
begin
if Right > Left then
return Right;
else
return Left;
end if;
end Max;
procedure Solve_Knapsack_01 (Items : in out Item_Array;
Weight_Limit : Positive := 400) is
type W_Array is array (0..Items'Length, 0..Weight_Limit) of Natural;
W : W_Array := (others => (others => 0));
begin
-- fill W
for I in Items'Range loop
for J in 1 .. Weight_Limit loop
if Items (I).Weight > J then
W (I, J) := W (I - 1, J);
else
W (I, J) := Max (W (I - 1, J),
W (I - 1, J - Items (I).Weight) + Items (I).Value);
end if;
end loop;
end loop;
declare
Rest : Natural := Weight_Limit;
begin
for I in reverse Items'Range loop
if W (I, Rest) /= W (I - 1, Rest) then
Items (I).Taken := True;
Rest := Rest - Items (I).Weight;
end if;
end loop;
end;
end Solve_Knapsack_01;
All_Items : Item_Array :=
( (US.To_Unbounded_String ("map"), 9, 150, False),
(US.To_Unbounded_String ("compass"), 13, 35, False),
(US.To_Unbounded_String ("water"), 153, 200, False),
(US.To_Unbounded_String ("sandwich"), 50, 160, False),
(US.To_Unbounded_String ("glucose"), 15, 60, False),
(US.To_Unbounded_String ("tin"), 68, 45, False),
(US.To_Unbounded_String ("banana"), 27, 60, False),
(US.To_Unbounded_String ("apple"), 39, 40, False),
(US.To_Unbounded_String ("cheese"), 23, 30, False),
(US.To_Unbounded_String ("beer"), 52, 10, False),
(US.To_Unbounded_String ("suntan cream"), 11, 70, False),
(US.To_Unbounded_String ("camera"), 32, 30, False),
(US.To_Unbounded_String ("t-shirt"), 24, 15, False),
(US.To_Unbounded_String ("trousers"), 48, 10, False),
(US.To_Unbounded_String ("umbrella"), 73, 40, False),
(US.To_Unbounded_String ("waterproof trousers"), 42, 70, False),
(US.To_Unbounded_String ("waterproof overclothes"), 43, 75, False),
(US.To_Unbounded_String ("note-case"), 22, 80, False),
(US.To_Unbounded_String ("sunglasses"), 7, 20, False),
(US.To_Unbounded_String ("towel"), 18, 12, False),
(US.To_Unbounded_String ("socks"), 4, 50, False),
(US.To_Unbounded_String ("book"), 30, 10, False) );
begin
Solve_Knapsack_01 (All_Items, 400);
Ada.Text_IO.Put_Line ("Total Weight: " & Natural'Image (Total_Weight (All_Items)));
Ada.Text_IO.Put_Line ("Total Value: " & Natural'Image (Total_Value (All_Items)));
Ada.Text_IO.Put_Line ("Items:");
for I in All_Items'Range loop
if All_Items (I).Taken then
Ada.Text_IO.Put_Line (" " & US.To_String (All_Items (I).Name));
end if;
end loop;
end Knapsack_01;</lang>
- Output:
Total Weight: 396 Total Value: 1030 Items: map compass water sandwich glucose banana suntan cream waterproof trousers waterproof overclothes note-case sunglasses socks
APL
<lang APL> ∇ ret←NapSack;sum;b;list;total [1] total←400 [2] list←("map" 9 150)("compass" 13 35)("water" 153 200)("sandwich" 50 160)("glucose" 15 60) ("tin" 68 45)("banana" 27 60)("apple" 39 40)("cheese" 23 30)("beer" 52 10) ("suntan cream" 11 70)("camera" 32 30)("t-shirt" 24 15)("trousers" 48 10) ("umbrella" 73 40)("waterproof trousers" 42 70)("waterproof overclothes" 43 75) ("note-case" 22 80) ("sunglasses" 7 20) ("towel" 18 12) ("socks" 4 50) ("book" 30 10) [3] list←list[⍒3⊃¨list] [4] [5] ret←⍬ [6] :while 0≠⍴list [7] ret←ret,(b←total>sum←+\2⊃¨list)/list [8] list←1↓(~b)/list [9] total←total-sum←¯1↑(total>sum)/sum [10] :end [11] ret←⊃ret,⊂'TOTALS:' (+/2⊃¨ret)(+/3⊃¨ret)
∇</lang>
- Output:
NapSack water 153 200 sandwich 50 160 map 9 150 note-case 22 80 waterproof overclothes 43 75 suntan cream 11 70 waterproof trousers 42 70 glucose 15 60 banana 27 60 socks 4 50 compass 13 35 sunglasses 7 20 TOTALS: 396 1030 Average runtime: 0.000168 seconds
AWK
<lang AWK>
- syntax: GAWK -f KNAPSACK_PROBLEM_0-1.AWK
BEGIN {
- arr["item,weight"] = value
arr["map,9"] = 150
arr["compass,13"] = 35
arr["water,153"] = 200
arr["sandwich,50"] = 160
arr["glucose,15"] = 60
arr["tin,68"] = 45
arr["banana,27"] = 60
arr["apple,39"] = 40
arr["cheese,23"] = 30
arr["beer,52"] = 10
arr["suntan cream,11"] = 70
arr["camera,32"] = 30
arr["T-shirt,24"] = 15
arr["trousers,48"] = 10
arr["umbrella,73"] = 40
arr["waterproof trousers,42"] = 70
arr["waterproof overclothes,43"] = 75
arr["note-case,22"] = 80
arr["sunglasses,7"] = 20
arr["towel,18"] = 12
arr["socks,4"] = 50
arr["book,30"] = 10
sack_size = 400 # dag
PROCINFO["sorted_in"] = "@val_num_desc"
for (i in arr) {
if (total_weight >= sack_size) {
break
}
split(i,tmp,",")
weight = tmp[2]
if (total_weight + weight <= sack_size) {
printf("%s\n",tmp[1])
total_items++
total_value += arr[i]
total_weight += weight
}
}
printf("items=%d (out of %d) weight=%d value=%d\n",total_items,length(arr),total_weight,total_value)
exit(0)
} </lang>
- Output:
water sandwich map note-case waterproof overclothes waterproof trousers suntan cream banana glucose socks compass sunglasses items=12 (out of 22) weight=396 value=1030
BASIC
QBasic
<lang qbasic>N = 7: G = 5: a = 2 ^ (N + 1) RANDOMIZE TIMER DIM L(N), C(N), j(N), q(a), d(a), q$(a)
FOR i = a - 1 TO (a - 1) \ 2 STEP -1
k = i
DO ' cipher 0-1
q$(i) = LTRIM$(STR$(k MOD 2)) + q$(i)
k = INT(k / 2)
LOOP UNTIL k = 0
q$(i) = MID$(q$(i), 2, LEN(q$(i)))
NEXT i
PRINT " # Mass Cost" FOR i = 1 TO N
L(i) = INT(RND * 3 + 1)' mass & cost C(i) = 10 + INT(RND * 9) PRINT i, L(i), C(i)
NEXT i ' origin
PRINT CHR$(10) + "Mass Cost Chifer" FOR h = a - 1 TO (a - 1) / 2 STEP -1
FOR k = 1 TO N
j(k) = VAL(MID$(q$(h), k, 1)) ' j() = cipher
q(h) = q(h) + L(k) * j(k) * C(k) ' 0 or 1
d(h) = d(h) + L(k) * j(k)
NEXT k
IF d(h) <= G THEN PRINT d(h), q(h), q$(h)
NEXT h
PRINT CHR$(10) + "Mass MAX Chifer" max = 0: h = 1 FOR i = 1 TO a
IF d(i) <= G AND q(i) > max THEN max = q(i): h = i
NEXT i PRINT d(h), q(h), q$(h)</lang>
- Output:
Same as QB64 entry.
Yabasic
<lang freebasic>N = 7 : G = 5 : a = 2^(N+1) dim L(N), C(N), j(N), q(a), d(a), q$(a)
for i = a-1 to int((a-1)/2) step -1
k = i
repeat // cipher 0-1
q$(i) = ltrim$(str$(mod(k, 2))) + q$(i)
k = int(k / 2)
until k = 0
q$(i) = mid$(q$(i), 2, len(q$(i)))
next i
print " # Mass Cost" for i = 1 to N
L(i) = int(ran(3)) + 1 // mass & cost C(i) = 10 + int(ran(9)) print i, chr$(9), L(i), chr$(9), C(i)
next i // origin
print chr$(10) + "Mass Cost Chifer" for h = a-1 to (a-1)/2 step -1
for k = 1 to N
j(k) = val(mid$(q$(h), k, 1)) // j() = cipher
q(h) = q(h) + L(k) * j(k) * C(k) // 0 or 1
d(h) = d(h) + L(k) * j(k)
next k
if d(h) <= G print d(h), chr$(9), q(h), chr$(9), q$(h)
next h
print chr$(10) + "Mass MAX Chifer" maxx = 0 : h = 1 for i = 1 to a
if d(i) <= G and q(i) > maxx maxx = q(i) : h = i
next i print d(h), chr$(9), q(h), chr$(9), q$(h) end</lang>
- Output:
Same as QB64 entry.
Batch File
<lang dos>
- Initiate command line environment
@echo off setlocal enabledelayedexpansion
- Establish arrays we'll be using
set items=map compass water sandwich glucose tin banana apple cheese beer suntancream camera tshirt trousers umbrella waterprooftrousers waterproofoverclothes notecase sunglasses towel socks book set weight=9 13 153 50 15 68 27 39 23 52 11 32 24 48 73 42 43 22 7 18 4 30 set importance=150 35 200 160 60 45 60 40 30 10 70 30 15 10 40 70 75 80 20 12 50 10
- Put the above 3 arrays into their own variables with the form of "item[]", "w[]" and "i[]"
set tempnum=0 for %%i in (%items%) do (
set /a tempnum+=1 set item!tempnum!=%%i
) set tempnum=0 for %%i in (%weight%) do (
set /a tempnum+=1 set w!tempnum!=%%i
) set tempnum=0 for %%i in (%importance%) do (
set /a tempnum+=1 set i!tempnum!=%%i
)
- Define the array "r[]" as the ratio between the importance ("i[]") and the weight ("w[]").
for /l %%i in (1,1,22) do set /a r%%i=!i%%i!*100/!w%%i! & rem batch doesn't support decimals, so the numerator is multiplied by 100 to get past this
set totalimportance=0 set totalweight=0 set amount=0
- Find the largest number in "r[]" and define some temp variables based off it
- load
set tempr=0 set tempitem=0 for /l %%i in (1,1,22) do (
if !r%%i! gtr !tempr! ( set tempr=!r%%i! set tempitem=%%i set /a testweight=%totalweight%+!w%%i! if !tempr!==0 goto end if !testweight! geq 400 goto end )
)
- Do basic error checking using the temp variables from above and either output and end the program or send back to load
set /a totaltempweight=%totalweight%+!w%tempitem%!
if %totaltempweight% gtr 400 (
set !r%tempitem%!=0 goto load
)
set totalweight=%totaltempweight% set /a totalimportance+=!i%tempitem%! set taken=%taken% !item%tempitem%! set /a amount+=1 set r%tempitem%=0 & rem set the ratio variable of the item we just added to the knapsack as 0 to stop it repeat
goto load
- end
echo List of things taken [%amount%]: %taken% echo Total Value: %totalimportance% Total Weight: %totalweight% pause>nul </lang>
- Output:
List of things taken [12]: map socks suntancream glucose notecase sandwich sunglasses compass banana waterproofoverclothes waterprooftrousers water Total Value: 1030 Total Weight: 396
BBC BASIC
<lang bbcbasic> HIMEM = PAGE + 8000000
nItems% = 22
maxWeight% = 400
DIM Tag{ivalue%, list%(nItems%-1), lp%}
DIM items{(nItems%-1)name$, weight%, ivalue%}
FOR item% = 0 TO nItems%-1
READ items{(item%)}.name$, items{(item%)}.weight%, items{(item%)}.ivalue%
NEXT
DATA "map", 9, 150, "compass", 13, 35, "water", 153, 200, "sandwich", 50, 160
DATA "glucose", 15, 60, "tin", 68, 45, "banana", 27, 60, "apple", 39, 40
DATA "cheese", 23, 30, "beer", 52, 10, "suntan cream", 11, 70, "camera", 32, 30
DATA "t-shirt", 24, 15, "trousers", 48, 10, "umbrella", 73, 40, "book", 30, 10
DATA "waterproof trousers", 42, 70, "waterproof overclothes", 43, 75
DATA "note-case", 22, 80, "sunglasses", 7, 20, "towel", 18, 12, "socks", 4, 50
carry% = FN_Knapsack(items{()}, nItems% - 1, maxWeight%, cache{()})
FOR i% = 0 TO cache{(carry%)}.lp%-1
n% = cache{(carry%)}.list%(i%)
TotalWeight% += items{(n%)}.weight%
TotalValue% += items{(n%)}.ivalue%
PRINT items{(n%)}.name$ " "
NEXT
PRINT '"Total weight = " ; TotalWeight%
PRINT "Total value = " ; TotalValue%
END
DEF FN_Knapsack(i{()}, i%, w%, RETURN m{()})
LOCAL included{}, excluded{}, tmp%, index%
DIM m{(16384)} = Tag{}, included{} = Tag{}, excluded{} = Tag{}
index% = i% << 9 OR w%
IF m{(index%)}.ivalue% THEN = index%
IF i% = 0 THEN
IF i{(0)}.weight% > w% THEN
m{(index%)}.ivalue% = 0 : REM Item doesn't fit
ELSE
m{(index%)}.ivalue% = i{(0)}.ivalue%
m{(index%)}.list%(m{(index%)}.lp%) = 0
m{(index%)}.lp% += 1
ENDIF
= index%
ENDIF
tmp% = FN_Knapsack(i{()}, i% - 1, w%, m{()})
excluded{} = m{(tmp%)}
IF i{(i%)}.weight% > w% THEN
m{(index%)} = excluded{} : REM Item weighs too much
= index%
ELSE
tmp% = FN_Knapsack(i{()}, i% - 1, w% - i{(i%)}.weight%, m{()})
included{} = m{(tmp%)}
included.ivalue% += i{(i%)}.ivalue%
included.list%(included.lp%) = i%
included.lp% += 1
ENDIF
IF included.ivalue% > excluded.ivalue% THEN
m{(index%)} = included{}
ELSE
m{(index%)} = excluded{}
ENDIF
= index%</lang>
- Output:
map compass water sandwich glucose banana suntan cream waterproof trousers waterproof overclothes note-case sunglasses socks Total weight = 396 Total value = 1030
Bracmat
<lang bracmat>(knapsack=
( things
= (map.9.150)
(compass.13.35)
(water.153.200)
(sandwich.50.160)
(glucose.15.60)
(tin.68.45)
(banana.27.60)
(apple.39.40)
(cheese.23.30)
(beer.52.10)
("suntan cream".11.70)
(camera.32.30)
(T-shirt.24.15)
(trousers.48.10)
(umbrella.73.40)
("waterproof trousers".42.70)
("waterproof overclothes".43.75)
(note-case.22.80)
(sunglasses.7.20)
(towel.18.12)
(socks.4.50)
(book.30.10)
)
& 0:?maxvalue & :?sack & ( add
= cumwght
cumvalue
cumsack
name
wght
val
tings
n
ncumwght
ncumvalue
. !arg
: (?cumwght.?cumvalue.?cumsack.(?name.?wght.?val) ?tings)
& -1:?n
& whl
' ( 1+!n:~>1:?n
& !cumwght+!n*!wght:~>400:?ncumwght
& !cumvalue+!n*!val:?ncumvalue
& ( !tings:
& ( !ncumvalue:>!maxvalue:?maxvalue
& !cumsack
(!n:0&|!name)
: ?sack
|
)
| add
$ ( !ncumwght
. !ncumvalue
. !cumsack
(!n:0&|!name)
. !tings
)
)
)
)
& add$(0.0..!things) & out$(!maxvalue.!sack));
!knapsack;</lang>
- Output:
<lang bracmat> 1030 . map
compass water sandwich glucose banana suntan cream waterproof trousers waterproof overclothes note-case sunglasses socks</lang>
C
<lang c>#include <stdio.h>
- include <stdlib.h>
typedef struct {
char *name; int weight; int value;
} item_t;
item_t items[] = {
{"map", 9, 150},
{"compass", 13, 35},
{"water", 153, 200},
{"sandwich", 50, 160},
{"glucose", 15, 60},
{"tin", 68, 45},
{"banana", 27, 60},
{"apple", 39, 40},
{"cheese", 23, 30},
{"beer", 52, 10},
{"suntan cream", 11, 70},
{"camera", 32, 30},
{"T-shirt", 24, 15},
{"trousers", 48, 10},
{"umbrella", 73, 40},
{"waterproof trousers", 42, 70},
{"waterproof overclothes", 43, 75},
{"note-case", 22, 80},
{"sunglasses", 7, 20},
{"towel", 18, 12},
{"socks", 4, 50},
{"book", 30, 10},
};
int *knapsack (item_t *items, int n, int w) {
int i, j, a, b, *mm, **m, *s;
mm = calloc((n + 1) * (w + 1), sizeof (int));
m = malloc((n + 1) * sizeof (int *));
m[0] = mm;
for (i = 1; i <= n; i++) {
m[i] = &mm[i * (w + 1)];
for (j = 0; j <= w; j++) {
if (items[i - 1].weight > j) {
m[i][j] = m[i - 1][j];
}
else {
a = m[i - 1][j];
b = m[i - 1][j - items[i - 1].weight] + items[i - 1].value;
m[i][j] = a > b ? a : b;
}
}
}
s = calloc(n, sizeof (int));
for (i = n, j = w; i > 0; i--) {
if (m[i][j] > m[i - 1][j]) {
s[i - 1] = 1;
j -= items[i - 1].weight;
}
}
free(mm);
free(m);
return s;
}
int main () {
int i, n, tw = 0, tv = 0, *s;
n = sizeof (items) / sizeof (item_t);
s = knapsack(items, n, 400);
for (i = 0; i < n; i++) {
if (s[i]) {
printf("%-22s %5d %5d\n", items[i].name, items[i].weight, items[i].value);
tw += items[i].weight;
tv += items[i].value;
}
}
printf("%-22s %5d %5d\n", "totals:", tw, tv);
return 0;
} </lang>
- Output:
map 9 150 compass 13 35 water 153 200 sandwich 50 160 glucose 15 60 banana 27 60 suntan cream 11 70 waterproof trousers 42 70 waterproof overclothes 43 75 note-case 22 80 sunglasses 7 20 socks 4 50 totals: 396 1030
C#
<lang csharp>using System; using System.Collections.Generic;
namespace Tests_With_Framework_4 {
class Bag : IEnumerable<Bag.Item>
{
List<Item> items;
const int MaxWeightAllowed = 400;
public Bag()
{
items = new List<Item>();
}
void AddItem(Item i)
{
if ((TotalWeight + i.Weight) <= MaxWeightAllowed)
items.Add(i);
}
public void Calculate(List<Item> items)
{
foreach (Item i in Sorte(items))
{
AddItem(i);
}
}
List<Item> Sorte(List<Item> inputItems)
{
List<Item> choosenItems = new List<Item>();
for (int i = 0; i < inputItems.Count; i++)
{
int j = -1;
if (i == 0)
{
choosenItems.Add(inputItems[i]);
}
if (i > 0)
{
if (!RecursiveF(inputItems, choosenItems, i, choosenItems.Count - 1, false, ref j))
{
choosenItems.Add(inputItems[i]);
}
}
}
return choosenItems;
}
bool RecursiveF(List<Item> knapsackItems, List<Item> choosenItems, int i, int lastBound, bool dec, ref int indxToAdd)
{
if (!(lastBound < 0))
{
if ( knapsackItems[i].ResultWV < choosenItems[lastBound].ResultWV )
{
indxToAdd = lastBound;
}
return RecursiveF(knapsackItems, choosenItems, i, lastBound - 1, true, ref indxToAdd);
}
if (indxToAdd > -1)
{
choosenItems.Insert(indxToAdd, knapsackItems[i]);
return true;
}
return false;
}
#region IEnumerable<Item> Members
IEnumerator<Item> IEnumerable<Item>.GetEnumerator()
{
foreach (Item i in items)
yield return i;
}
#endregion
#region IEnumerable Members
System.Collections.IEnumerator System.Collections.IEnumerable.GetEnumerator()
{
return items.GetEnumerator();
}
#endregion
public int TotalWeight
{
get
{
var sum = 0;
foreach (Item i in this)
{
sum += i.Weight;
}
return sum;
}
}
public class Item
{
public string Name { get; set; } public int Weight { get; set; } public int Value { get; set; } public int ResultWV { get { return Weight-Value; } }
public override string ToString()
{
return "Name : " + Name + " Wieght : " + Weight + " Value : " + Value + " ResultWV : " + ResultWV;
}
}
}
class Program
{
static void Main(string[] args)
{List<Bag.Item> knapsackItems = new List<Bag.Item>();
knapsackItems.Add(new Bag.Item() { Name = "Map", Weight = 9, Value = 150 });
knapsackItems.Add(new Bag.Item() { Name = "Water", Weight = 153, Value = 200 });
knapsackItems.Add(new Bag.Item() { Name = "Compass", Weight = 13, Value = 35 });
knapsackItems.Add(new Bag.Item() { Name = "Sandwitch", Weight = 50, Value = 160 });
knapsackItems.Add(new Bag.Item() { Name = "Glucose", Weight = 15, Value = 60 });
knapsackItems.Add(new Bag.Item() { Name = "Tin", Weight = 68, Value = 45 });
knapsackItems.Add(new Bag.Item() { Name = "Banana", Weight = 27, Value = 60 });
knapsackItems.Add(new Bag.Item() { Name = "Apple", Weight = 39, Value = 40 });
knapsackItems.Add(new Bag.Item() { Name = "Cheese", Weight = 23, Value = 30 });
knapsackItems.Add(new Bag.Item() { Name = "Beer", Weight = 52, Value = 10 });
knapsackItems.Add(new Bag.Item() { Name = "Suntan Cream", Weight = 11, Value = 70 });
knapsackItems.Add(new Bag.Item() { Name = "Camera", Weight = 32, Value = 30 });
knapsackItems.Add(new Bag.Item() { Name = "T-shirt", Weight = 24, Value = 15 });
knapsackItems.Add(new Bag.Item() { Name = "Trousers", Weight = 48, Value = 10 });
knapsackItems.Add(new Bag.Item() { Name = "Umbrella", Weight = 73, Value = 40 });
knapsackItems.Add(new Bag.Item() { Name = "WaterProof Trousers", Weight = 42, Value = 70 });
knapsackItems.Add(new Bag.Item() { Name = "Note-Case", Weight = 22, Value = 80 });
knapsackItems.Add(new Bag.Item() { Name = "Sunglasses", Weight = 7, Value = 20 });
knapsackItems.Add(new Bag.Item() { Name = "Towel", Weight = 18, Value = 12 });
knapsackItems.Add(new Bag.Item() { Name = "Socks", Weight = 4, Value = 50 });
knapsackItems.Add(new Bag.Item() { Name = "Book", Weight = 30, Value = 10 });
knapsackItems.Add(new Bag.Item() { Name = "waterproof overclothes ", Weight = 43, Value = 75 });
Bag b = new Bag();
b.Calculate(knapsackItems);
b.All(x => { Console.WriteLine(x); return true; });
Console.WriteLine(b.Sum(x => x.Weight));
Console.ReadKey();
}
}
}</lang> ("Bag" might not be the best name for the class, since "bag" is sometimes also used to refer to a multiset data structure.)
C#
C# Knapsak 0-1 Russian Binary ciphers
Russian Knapsack 0-1 synthesizes all ciphers from 0 & 1 adding left +1 register and 0 remain on left in cipher
Number of comparisons decreases from N! to 2^N for example N=8 N!=40320 >> 2^N=256
Random values origin are automatically assigned create integral of quantity and quality
<lang C#>using System; // Knapsack C# binary DANILIN using System.Text; // rextester.com/YRFA61366 namespace Knapsack { class Knapsack
{
static void Main()
{
int n = 7;
int Inside = 5;
int all=Convert.ToInt32(Math.Pow(2,(n+1)));
int[] mass = new int[n];
int[] cost = new int[n];
int[] jack = new int[n];
int[] quality = new int[all];
int[] amount = new int[all];
int i; // circle
int k; // circle
int dec;
string[] bin = new string[all];
int list;
int max;
int max_num;
Random rand = new Random();
for (i=0; i<n; i++)
{
mass[i]=1+rand.Next(3);
cost[i]=10+rand.Next(9);
Console.WriteLine("{0} {1} {2}", i+1, mass[i], cost[i]);
}
Console.WriteLine();
for (list = all-1; list>(all-1)/2; list--)
{
dec=list;
while (dec > 0)
{
bin[list] = dec % 2 + bin[list]; // from 10 to 2
dec/=2;
}
if (bin[list] == "")
{
bin[list] = "0";
}
bin[list]=bin[list].Substring(1,bin[list].Length-1);
for (k=0; k<n; k++) // inside 01
{
jack[k]=Convert.ToInt32(bin[list].Substring(k,1));
quality[list]=quality[list]+mass[k]*jack[k]*cost[k]; // integral of costs
amount[list]=amount[list]+mass[k]*jack[k]; // integral of mass
}
if (amount[list]<= Inside) // current mass < Knapsack
{
Console.WriteLine("{0} {1} {2} {3}", Inside, amount[list], quality[list], bin[list]);
}
}
Console.WriteLine();
max=0;
max_num=1;
for (i=0; i < all; i++)
{
if (amount[i]<=Inside && quality[i]>max)
{
max = quality[i]; max_num =i ;
}
}
Console.WriteLine("{0} {1} {2}",amount[max_num],quality[max_num],bin[max_num]);
}
}
}</lang>
- Output:
# Mass Cost 1 2 12 2 3 17 3 1 14 4 3 17 5 1 13 Chifer Mass Cost 11000 5 5 75 01001 5 4 64 00111 5 5 78 !!! 00110 5 4 65 00101 5 2 27 Mass MAX Chifer 5 78 00111
C++
First version
<lang cpp>#include <vector>
- include <string>
- include <iostream>
- include <boost/tuple/tuple.hpp>
- include <set>
int findBestPack( const std::vector<boost::tuple<std::string , int , int> > & ,
std::set<int> & , const int ) ;
int main( ) {
std::vector<boost::tuple<std::string , int , int> > items ;
//===========fill the vector with data====================
items.push_back( boost::make_tuple( "" , 0 , 0 ) ) ;
items.push_back( boost::make_tuple( "map" , 9 , 150 ) ) ;
items.push_back( boost::make_tuple( "compass" , 13 , 35 ) ) ;
items.push_back( boost::make_tuple( "water" , 153 , 200 ) ) ;
items.push_back( boost::make_tuple( "sandwich", 50 , 160 ) ) ;
items.push_back( boost::make_tuple( "glucose" , 15 , 60 ) ) ;
items.push_back( boost::make_tuple( "tin", 68 , 45 ) ) ;
items.push_back( boost::make_tuple( "banana", 27 , 60 ) ) ;
items.push_back( boost::make_tuple( "apple" , 39 , 40 ) ) ;
items.push_back( boost::make_tuple( "cheese" , 23 , 30 ) ) ;
items.push_back( boost::make_tuple( "beer" , 52 , 10 ) ) ;
items.push_back( boost::make_tuple( "suntan creme" , 11 , 70 ) ) ;
items.push_back( boost::make_tuple( "camera" , 32 , 30 ) ) ;
items.push_back( boost::make_tuple( "T-shirt" , 24 , 15 ) ) ;
items.push_back( boost::make_tuple( "trousers" , 48 , 10 ) ) ;
items.push_back( boost::make_tuple( "umbrella" , 73 , 40 ) ) ;
items.push_back( boost::make_tuple( "waterproof trousers" , 42 , 70 ) ) ;
items.push_back( boost::make_tuple( "waterproof overclothes" , 43 , 75 ) ) ;
items.push_back( boost::make_tuple( "note-case" , 22 , 80 ) ) ;
items.push_back( boost::make_tuple( "sunglasses" , 7 , 20 ) ) ;
items.push_back( boost::make_tuple( "towel" , 18 , 12 ) ) ;
items.push_back( boost::make_tuple( "socks" , 4 , 50 ) ) ;
items.push_back( boost::make_tuple( "book" , 30 , 10 ) ) ;
const int maximumWeight = 400 ;
std::set<int> bestItems ; //these items will make up the optimal value
int bestValue = findBestPack( items , bestItems , maximumWeight ) ;
std::cout << "The best value that can be packed in the given knapsack is " <<
bestValue << " !\n" ;
int totalweight = 0 ;
std::cout << "The following items should be packed in the knapsack:\n" ;
for ( std::set<int>::const_iterator si = bestItems.begin( ) ;
si != bestItems.end( ) ; si++ ) {
std::cout << (items.begin( ) + *si)->get<0>( ) << "\n" ;
totalweight += (items.begin( ) + *si)->get<1>( ) ;
}
std::cout << "The total weight of all items is " << totalweight << " !\n" ;
return 0 ;
}
int findBestPack( const std::vector<boost::tuple<std::string , int , int> > & items ,std::set<int> & bestItems , const int weightlimit ) {
//dynamic programming approach sacrificing storage space for execution
//time , creating a table of optimal values for every weight and a
//second table of sets with the items collected so far in the knapsack
//the best value is in the bottom right corner of the values table,
//the set of items in the bottom right corner of the sets' table.
const int n = items.size( ) ;
int bestValues [ n ][ weightlimit ] ;
std::set<int> solutionSets[ n ][ weightlimit ] ;
std::set<int> emptyset ;
for ( int i = 0 ; i < n ; i++ ) {
for ( int j = 0 ; j < weightlimit ; j++ ) {
bestValues[ i ][ j ] = 0 ; solutionSets[ i ][ j ] = emptyset ;
}
}
for ( int i = 0 ; i < n ; i++ ) {
for ( int weight = 0 ; weight < weightlimit ; weight++ ) {
if ( i == 0 ) bestValues[ i ][ weight ] = 0 ; else { int itemweight = (items.begin( ) + i)->get<1>( ) ; if ( weight < itemweight ) { bestValues[ i ][ weight ] = bestValues[ i - 1 ][ weight ] ; solutionSets[ i ][ weight ] = solutionSets[ i - 1 ][ weight ] ; } else { // weight >= itemweight if ( bestValues[ i - 1 ][ weight - itemweight ] + (items.begin( ) + i)->get<2>( ) > bestValues[ i - 1 ][ weight ] ) { bestValues[ i ][ weight ] = bestValues[ i - 1 ][ weight - itemweight ] + (items.begin( ) + i)->get<2>( ) ; solutionSets[ i ][ weight ] = solutionSets[ i - 1 ][ weight - itemweight ] ; solutionSets[ i ][ weight ].insert( i ) ; } else { bestValues[ i ][ weight ] = bestValues[ i - 1 ][ weight ] ; solutionSets[ i ][ weight ] = solutionSets[ i - 1 ][ weight ] ; } }
}
}
}
bestItems.swap( solutionSets[ n - 1][ weightlimit - 1 ] ) ;
return bestValues[ n - 1 ][ weightlimit - 1 ] ;
}</lang>
- Output:
The best value that can be packed in the given knapsack is 1030 ! The following items should be packed in the knapsack: map compass water sandwich glucose banana suntan creme waterproof trousers waterproof overclothes note-case sunglasses socks The total weight of all items is 396 !
Second version
<lang cpp>#include <iomanip>
- include <iostream>
- include <set>
- include <string>
- include <tuple>
- include <vector>
std::tuple<std::set<int>, int> findBestPack(const std::vector<std::tuple<std::string, int, int> > &items, const int weightlimit) {
const auto n = items.size();
int bestValues[n][weightlimit] = { 0 };
std::set<int> solutionSets[n][weightlimit];
std::set<int> bestItems;
for (auto i = 0u; i < n; i++)
for (auto weight = 0; weight < weightlimit; weight++) {
if (i == 0)
bestValues[i][weight] = 0;
else {
auto [_, itemweight, value] = *(items.begin() + i);
if (weight < itemweight) {
bestValues[i][weight] = bestValues[i - 1][weight];
solutionSets[i][weight] = solutionSets[i - 1][weight];
} else {
if (bestValues[i - 1][weight - itemweight] + value > bestValues[i - 1][weight]) {
bestValues[i][weight] = bestValues[i - 1][weight - itemweight] + value;
solutionSets[i][weight] = solutionSets[i - 1][weight - itemweight];
solutionSets[i][weight].insert(i);
} else {
bestValues[i][weight] = bestValues[i - 1][weight];
solutionSets[i][weight] = solutionSets[i - 1][weight];
}
}
}
}
bestItems.swap(solutionSets[n - 1][weightlimit - 1]);
return { bestItems, bestValues[n - 1][weightlimit - 1] };
}
int main() {
const std::vector<std::tuple<std::string, int, int>> items = {
{ "", 0, 0 },
{ "map", 9, 150 },
{ "compass", 13, 35 },
{ "water", 153, 200 },
{ "sandwich", 50, 160 },
{ "glucose", 15, 60 },
{ "tin", 68, 45 },
{ "banana", 27, 60 },
{ "apple", 39, 40 },
{ "cheese", 23, 30 },
{ "beer", 52, 10 },
{ "suntan creme", 11, 70 },
{ "camera", 32, 30 },
{ "T-shirt", 24, 15 },
{ "trousers", 48, 10 },
{ "umbrella", 73, 40 },
{ "waterproof trousers", 42, 70 },
{ "waterproof overclothes", 43, 75 },
{ "note-case", 22, 80 },
{ "sunglasses", 7, 20 },
{ "towel", 18, 12 },
{ "socks", 4, 50 },
{ "book", 30, 10 } };
const int maximumWeight = 400;
const auto &[bestItems, bestValue] = findBestPack(items, maximumWeight);
int totalweight = 0;
std::cout << std::setw(24) << "best knapsack:" << std::endl;
for (auto si = bestItems.begin(); si != bestItems.end(); si++) {
auto [name, weight, value] = *(items.begin() + *si);
std::cout << std::setw(24) << name << std::setw(6) << weight << std::setw(6) << value << std::endl;
totalweight += weight;
}
std::cout << std::endl << std::setw(24) << "total:" << std::setw(6) << totalweight << std::setw(6) << bestValue << std::endl;
return 0;
}</lang>
- Output:
best knapsack:
map 9 150
compass 13 35
water 153 200
sandwich 50 160
glucose 15 60
banana 27 60
suntan creme 11 70
waterproof trousers 42 70
waterproof overclothes 43 75
note-case 22 80
sunglasses 7 20
socks 4 50
total: 396 1030
C_sharp
All combinations, eight threads, break when weight is to large. <lang csharp>using System; // 4790@3.6 using System.Threading.Tasks; class Program {
static void Main()
{
var sw = System.Diagnostics.Stopwatch.StartNew();
Console.Write(knapSack(400) + "\n" + sw.Elapsed); // 60 ms
Console.Read();
}
static string knapSack(uint w1)
{
uint sol = 0, v1 = 0;
Parallel.For(1, 9, t =>
{
uint j, wi, k, vi, i1 = 1u << w.Length;
for (uint i = (uint)t; i < i1; i += 8)
{
k = wi = vi = 0;
for (j = i; j > 0; j >>= 1, k++)
if ((j & 1) > 0)
{
if ((wi += w[k]) > w1) break;
vi += v[k];
}
if (wi <= w1 && v1 < vi)
lock (locker)
if (v1 < vi) { v1 = vi; sol = i; }
}
});
string str = "";
for (uint k = 0; sol > 0; sol >>= 1, k++)
if ((sol & 1) > 0) str += items[k] + "\n";
return str;
}
static readonly object locker = new object();
static byte[] w = { 9, 13, 153, 50, 15, 68, 27, 39, 23, 52, 11,
32, 24, 48, 73, 42, 43, 22, 7, 18, 4, 30 },
v = { 150, 35, 200, 160, 60, 45, 60, 40, 30, 10, 70,
30, 15, 10, 40, 70, 75, 80, 20, 12, 50, 10 };
static string[] items = {"map","compass","water","sandwich","glucose","tin",
"banana","apple","cheese","beer","suntan cream",
"camera","T-shirt","trousers","umbrella",
"waterproof trousers","waterproof overclothes",
"note-case","sunglasses","towel","socks","book"};
}</lang> A dynamic version. <lang csharp>using System class program {
static void Main()
{
knapSack(40);
var sw = System.Diagnostics.Stopwatch.StartNew();
Console.Write(knapSack(400) + "\n" + sw.Elapsed); // 31 µs
Console.Read();
}
static string knapSack(uint w1)
{
uint n = (uint)w.Length; var K = new uint[n + 1, w1 + 1];
for (uint vi, wi, w0, x, i = 0; i < n; i++)
for (vi = v[i], wi = w[i], w0 = 1; w0 <= w1; w0++)
{
x = K[i, w0];
if (wi <= w0) x = max(vi + K[i, w0 - wi], x);
K[i + 1, w0] = x;
}
string str = "";
for (uint v1 = K[n, w1]; v1 > 0; n--)
if (v1 != K[n - 1, w1])
{
v1 -= v[n - 1]; w1 -= w[n - 1]; str += items[n - 1] + "\n";
}
return str;
}
static uint max(uint a, uint b) { return a > b ? a : b; }
static byte[] w = { 9, 13, 153, 50, 15, 68, 27, 39, 23, 52, 11,
32, 24, 48, 73, 42, 43, 22, 7, 18, 4, 30 },
v = { 150, 35, 200, 160, 60, 45, 60, 40, 30, 10, 70,
30, 15, 10, 40, 70, 75, 80, 20, 12, 50, 10 };
static string[] items = {"map","compass","water","sandwich","glucose","tin",
"banana","apple","cheese","beer","suntan cream",
"camera","T-shirt","trousers","umbrella",
"waterproof trousers","waterproof overclothes",
"note-case","sunglasses","towel","socks","book"};
}</lang>
Ceylon
module.ceylon:
<lang ceylon> module knapsack "1.0.0" { } </lang>
run.ceylon:
<lang ceylon> shared void run() {
value knapsack = pack(items, empty(400));
print(knapsack);
}
class Item(name,weight,theValue) {
String name; shared Integer weight; shared Float theValue;
shared actual String string = "item(``name``, ``weight``, ``theValue``)";
}
class Knapsack(items,theValue,weight,available) {
shared Item[] items; shared Float theValue; shared Integer weight; shared Integer available;
shared Boolean canAccept(Item item)
=> item.weight <= available;
String itemsString = items.fold("")((total, remaining) => "``total``\t\n``remaining.string``" );
shared actual String string = "Total value: ``theValue``\nTotal weight: ``weight``\nItems:\n``itemsString``";
}
Knapsack empty(Integer capacity)
=> Knapsack([], 0.0, 0, capacity);
Item[] items =
[
Item("map", 9, 150.0),
Item("compass", 13, 35.0),
Item("water", 153, 200.0),
Item("sandwich", 50, 160.0),
Item("glucose", 15, 60.0),
Item("tin", 68, 45.0),
Item("banana", 27, 60.0),
Item("apple", 39, 40.0),
Item("cheese", 23, 30.0),
Item("beer", 52, 10.0),
Item("cream", 11, 70.0),
Item("camera", 32, 30.0),
Item("tshirt", 24, 15.0),
Item("trousers", 48, 10.0),
Item("umbrella", 73, 40.0),
Item("trousers", 42, 70.0),
Item("overclothes", 43, 75.0),
Item("notecase", 22, 80.0),
Item("sunglasses", 7, 20.0),
Item("towel", 18, 12.0),
Item("socks", 4, 50.0),
Item("book", 30, 10.0)
];
Knapsack add(Item item, Knapsack knapsack)
=> Knapsack { items = knapsack.items.withTrailing(item);
theValue = knapsack.theValue + item.theValue;
weight = knapsack.weight + item.weight;
available = knapsack.available - item.weight; };
Float rating(Item item) => item.theValue / item.weight.float;
Knapsack pack(Item[] items, Knapsack knapsack)
// Sort the items by decreasing rating, that is, value divided by weight
=> let (itemsSorted =
items.group(rating)
.sort(byDecreasing((Float->[Item+] entry) => entry.key))
.map(Entry.item)
.flatMap((element) => element)
.sequence())
packRecursive(itemsSorted,knapsack);
Knapsack packRecursive(Item[] sortedItems, Knapsack knapsack)
=> if (exists firstItem=sortedItems.first, knapsack.canAccept(firstItem))
then packRecursive(sortedItems.rest, add(firstItem,knapsack))
else knapsack;
</lang>
- Output:
Total value: 1030.0 Total weight: 396 Items: item(map, 9, 150.0) item(socks, 4, 50.0) item(cream, 11, 70.0) item(glucose, 15, 60.0) item(notecase, 22, 80.0) item(sandwich, 50, 160.0) item(sunglasses, 7, 20.0) item(compass, 13, 35.0) item(banana, 27, 60.0) item(overclothes, 43, 75.0) item(trousers, 42, 70.0) item(water, 153, 200.0)
Clojure
Uses the dynamic programming solution from Wikipedia. First define the items data: <lang clojure>(def item-data
[ "map" 9 150 "compass" 13 35 "water" 153 200 "sandwich" 50 160 "glucose" 15 60 "tin" 68 45 "banana" 27 60 "apple" 39 40 "cheese" 23 30 "beer" 52 10 "suntan cream" 11 70 "camera" 32 30 "t-shirt" 24 15 "trousers" 48 10 "umbrella" 73 40 "waterproof trousers" 42 70 "waterproof overclothes" 43 75 "note-case" 22 80 "sunglasses" 7 20 "towel" 18 12 "socks" 4 50 "book" 30 10])
(defstruct item :name :weight :value)
(def items (vec (map #(apply struct item %) (partition 3 item-data))))</lang> m is as per the Wikipedia formula, except that it returns a pair [value indexes] where indexes is a vector of index values in items. value is the maximum value attainable using items 0..i whose total weight doesn't exceed w; indexes are the item indexes that produces the value. <lang clojure>(declare mm) ;forward decl for memoization function
(defn m [i w]
(cond
(< i 0) [0 []]
(= w 0) [0 []]
:else
(let [{wi :weight vi :value} (get items i)]
(if (> wi w)
(mm (dec i) w)
(let [[vn sn :as no] (mm (dec i) w)
[vy sy :as yes] (mm (dec i) (- w wi))]
(if (> (+ vy vi) vn)
[(+ vy vi) (conj sy i)]
no))))))
(def mm (memoize m))</lang> Call m and print the result: <lang clojure>(use '[clojure.string :only [join]])
(let [[value indexes] (m (-> items count dec) 400)
names (map (comp :name items) indexes)] (println "items to pack:" (join ", " names)) (println "total value:" value) (println "total weight:" (reduce + (map (comp :weight items) indexes))))</lang>
- Output:
items to pack: map, compass, water, sandwich, glucose, banana, suntan cream, waterproof trousers, waterproof overclothes, note-case, sunglasses, socks total value: 1030 total weight: 396
Common Lisp
Cached method. <lang lisp>;;; memoize (defmacro mm-set (p v) `(if ,p ,p (setf ,p ,v)))
(defun knapsack (max-weight items)
(let ((cache (make-array (list (1+ max-weight) (1+ (length items)))
:initial-element nil)))
(labels ((knapsack1 (spc items)
(if (not items) (return-from knapsack1 (list 0 0 '()))) (mm-set (aref cache spc (length items)) (let* ((i (first items)) (w (second i)) (v (third i)) (x (knapsack1 spc (cdr items)))) (if (> w spc) x (let* ((y (knapsack1 (- spc w) (cdr items))) (v (+ v (first y)))) (if (< v (first x)) x (list v (+ w (second y)) (cons i (third y))))))))))
(knapsack1 max-weight items))))
(knapsack 400
'((map 9 150) (compass 13 35) (water 153 200) (sandwich 50 160) (glucose 15 60) (tin 68 45)(banana 27 60) (apple 39 40) (cheese 23 30) (beer 52 10) (cream 11 70) (camera 32 30) (T-shirt 24 15) (trousers 48 10) (umbrella 73 40) (trousers 42 70) (overclothes 43 75) (notecase 22 80) (glasses 7 20) (towel 18 12) (socks 4 50) (book 30 10))))</lang>
- Output:
(1030 396 ((MAP 9 150) (COMPASS 13 35) (WATER 153 200) (SANDWICH 50 160) (GLUCOSE 15 60) (BANANA 27 60) (CREAM 11 70) (TROUSERS 42 70) (OVERCLOTHES 43 75) (NOTECASE 22 80) (GLASSES 7 20) (SOCKS 4 50)))
Crystal
Branch and bound solution <lang Ruby>require "bit_array"
struct BitArray
def clone
BitArray.new(size).tap { |new| new.to_slice.copy_from (to_slice) }
end
end
record Item, name : String, weight : Int32, value : Int32
record Selection, mask : BitArray, cur_index : Int32, total_value : Int32
class Knapsack
@threshold_value = 0 @threshold_choice : Selection? getter checked_nodes = 0
def knapsack_step(taken, items, remaining_weight)
if taken.total_value > @threshold_value
@threshold_value = taken.total_value
@threshold_choice = taken
end
candidate_index = items.index(taken.cur_index) { |item| item.weight <= remaining_weight }
return nil unless candidate_index
@checked_nodes += 1
candidate = items[candidate_index]
# candidate is a best of available items, so if we fill remaining value with it
# and still don't reach the threshold, the branch is wrong
return nil if taken.total_value + 1.0 * candidate.value / candidate.weight * remaining_weight < @threshold_value
# now recursively check both variants
mask = taken.mask.clone
mask[candidate_index] = true
knapsack_step Selection.new(mask, candidate_index + 1, taken.total_value + candidate.value), items, remaining_weight - candidate.weight
mask = taken.mask.clone
mask[candidate_index] = false
knapsack_step Selection.new(mask, candidate_index + 1, taken.total_value), items, remaining_weight
end
def select(items, max_weight)
@checked_variants = 0
# sort by descending relative value
list = items.sort_by { |item| -1.0 * item.value / item.weight }
# use heuristic of relative value as an initial estimate for branch&bounds
w = max_weight
heur_list = list.take_while { |item| w -= item.weight; w > 0 }
nothing = Selection.new(BitArray.new(items.size), 0, 0)
@threshold_value = heur_list.sum(&.value) - 1
@threshold_choice = nothing
knapsack_step(nothing, list, max_weight)
selected = @threshold_choice.not_nil!
result = [] of Item
selected.mask.each_with_index { |v, i| result << list[i] if v }
result
end
end
possible = [
Item.new("map", 9, 150),
Item.new("compass", 13, 35),
Item.new("water", 153, 200),
Item.new("sandwich", 50, 160),
Item.new("glucose", 15, 60),
Item.new("tin", 68, 45),
Item.new("banana", 27, 60),
Item.new("apple", 39, 40),
Item.new("cheese", 23, 30),
Item.new("beer", 52, 10),
Item.new("suntan cream", 11, 70),
Item.new("camera", 32, 30),
Item.new("T-shirt", 24, 15),
Item.new("trousers", 48, 10),
Item.new("umbrella", 73, 40),
Item.new("waterproof trousers", 42, 70),
Item.new("waterproof overclothes", 43, 75),
Item.new("note-case", 22, 80),
Item.new("sunglasses", 7, 20),
Item.new("towel", 18, 12),
Item.new("socks", 4, 50),
Item.new("book", 30, 10),
]
solver = Knapsack.new used = solver.select(possible, 400) puts "optimal choice: #{used.map(&.name)}" puts "total weight #{used.sum(&.weight)}, total value #{used.sum(&.value)}" puts "checked nodes: #{solver.checked_nodes}" </lang>
- Output:
optimal choice: ["map", "socks", "suntan cream", "glucose", "note-case", "sandwich", "sunglasses", "compass", "banana", "waterproof overclothes", "waterproof trousers", "water"] total weight 396, total value 1030 checked nodes: 992
D
Dynamic Programming Version
<lang d>import std.stdio, std.algorithm, std.typecons, std.array, std.range;
struct Item { string name; int weight, value; }
Item[] knapsack01DinamicProgramming(immutable Item[] items, in int limit) pure nothrow @safe {
auto tab = new int[][](items.length + 1, limit + 1);
foreach (immutable i, immutable it; items)
foreach (immutable w; 1 .. limit + 1)
tab[i + 1][w] = (it.weight > w) ? tab[i][w] :
max(tab[i][w], tab[i][w - it.weight] + it.value);
typeof(return) result;
int w = limit;
foreach_reverse (immutable i, immutable it; items)
if (tab[i + 1][w] != tab[i][w]) {
w -= it.weight;
result ~= it;
}
return result;
}
void main() @safe {
enum int limit = 400;
immutable Item[] items = [
{"apple", 39, 40}, {"banana", 27, 60},
{"beer", 52, 10}, {"book", 30, 10},
{"camera", 32, 30}, {"cheese", 23, 30},
{"compass", 13, 35}, {"glucose", 15, 60},
{"map", 9, 150}, {"note-case", 22, 80},
{"sandwich", 50, 160}, {"socks", 4, 50},
{"sunglasses", 7, 20}, {"suntan cream", 11, 70},
{"t-shirt", 24, 15}, {"tin", 68, 45},
{"towel", 18, 12}, {"trousers", 48, 10},
{"umbrella", 73, 40}, {"water", 153, 200},
{"waterproof overclothes", 43, 75},
{"waterproof trousers", 42, 70}];
immutable bag = knapsack01DinamicProgramming(items, limit);
writefln("Items:\n%-( %s\n%)", bag.map!q{ a.name }.retro);
const t = reduce!q{ a[] += [b.weight, b.value] }([0, 0], bag);
writeln("\nTotal weight and value: ", t[0] <= limit ? t : [0, 0]);
}</lang>
- Output:
Items: banana compass glucose map note-case sandwich socks sunglasses suntan cream water waterproof overclothes waterproof trousers Total weight and value: [396, 1030]
Brute Force Version
<lang d>struct Item { string name; int weight, value; }
immutable Item[] items = [
{"apple", 39, 40}, {"banana", 27, 60},
{"beer", 52, 10}, {"book", 30, 10},
{"camera", 32, 30}, {"cheese", 23, 30},
{"compass", 13, 35}, {"glucose", 15, 60},
{"map", 9, 150}, {"note-case", 22, 80},
{"sandwich", 50, 160}, {"socks", 4, 50},
{"sunglasses", 7, 20}, {"suntan cream", 11, 70},
{"t-shirt", 24, 15}, {"tin", 68, 45},
{"towel", 18, 12}, {"trousers", 48, 10},
{"umbrella", 73, 40}, {"water", 153, 200},
{"waterproof overclothes", 43, 75},
{"waterproof trousers", 42, 70}];
struct Solution { uint bits; int value; } static assert(items.length <= Solution.bits.sizeof * 8);
void solve(in int weight, in int idx, ref Solution s) pure nothrow @nogc @safe {
if (idx < 0) {
s.bits = s.value = 0;
return;
}
if (weight < items[idx].weight) {
solve(weight, idx - 1, s);
return;
}
Solution v1, v2; solve(weight, idx - 1, v1); solve(weight - items[idx].weight, idx - 1, v2);
v2.value += items[idx].value; v2.bits |= (1 << idx);
s = (v1.value >= v2.value) ? v1 : v2;
}
void main() @safe {
import std.stdio;
auto s = Solution(0, 0); solve(400, items.length - 1, s);
writeln("Items:");
int w = 0;
foreach (immutable i, immutable it; items)
if (s.bits & (1 << i)) {
writeln(" ", it.name);
w += it.weight;
}
writefln("\nTotal value: %d; weight: %d", s.value, w);
}</lang> The runtime is about 0.09 seconds.
- Output:
Items: banana compass glucose map note-case sandwich socks sunglasses suntan cream water waterproof overclothes waterproof trousers Total value: 1030; weight: 396
Short Dynamic Programming Version
<lang d>import std.stdio, std.algorithm, std.typecons, std.array, std.range;
struct Item { string name; int w, v; } alias Pair = Tuple!(int,"tot", string[],"names");
immutable Item[] items = [{"apple",39,40}, {"banana", 27, 60},
{"beer", 52, 10}, {"book", 30, 10}, {"camera", 32, 30},
{"cheese", 23, 30}, {"compass", 13, 35}, {"glucose", 15, 60},
{"map", 9, 150}, {"note-case", 22, 80}, {"sandwich", 50, 160},
{"socks", 4, 50}, {"sunglasses", 7, 20}, {"suntan cream", 11, 70},
{"t-shirt", 24, 15}, {"tin", 68, 45}, {"towel", 18, 12},
{"trousers", 48, 10}, {"umbrella", 73, 40}, {"water", 153, 200},
{"overclothes", 43, 75}, {"waterproof trousers", 42, 70}];
auto addItem(Pair[] lst, in Item it) pure /*nothrow*/ {
auto aux = lst.map!(vn => Pair(vn.tot + it.v, vn.names ~ it.name));
return lst[0..it.w] ~ lst[it.w..$].zip(aux).map!q{ a[].max }.array;
}
void main() {
reduce!addItem(Pair().repeat.take(400).array, items).back.writeln;
}</lang> Runtime about 0.04 seconds.
- Output:
Tuple!(int, "tot", string[], "names")(1030, ["banana", "compass", "glucose", "map", "note-case", "sandwich", "socks", "sunglasses", "suntan cream", "water", "overclothes", "waterproof trousers"])
Dart
<lang dart>List solveKnapsack(items, maxWeight) {
int MIN_VALUE=-100;
int N = items.length; // number of items
int W = maxWeight; // maximum weight of knapsack
List profit = new List(N+1);
List weight = new List(N+1);
// generate random instance, items 1..N
for(int n = 1; n<=N; n++) {
profit[n] = items[n-1][2];
weight[n] = items[n-1][1];
}
// opt[n][w] = max profit of packing items 1..n with weight limit w
// sol[n][w] = does opt solution to pack items 1..n with weight limit w include item n?
List<List<int>> opt = new List<List<int>>(N+1);
for (int i=0; i<N+1; i++) {
opt[i] = new List<int>(W+1);
for(int j=0; j<W+1; j++) {
opt[i][j] = MIN_VALUE;
}
}
List<List<bool>> sol = new List<List<bool>>(N+1);
for (int i=0; i<N+1; i++) {
sol[i] = new List<bool>(W+1);
for(int j=0; j<W+1; j++) {
sol[i][j] = false;
}
}
for(int n=1; n<=N; n++) {
for (int w=1; w <= W; w++) {
// don't take item n
int option1 = opt[n-1][w];
// take item n
int option2 = MIN_VALUE;
if (weight[n] <= w) {
option2 = profit[n] + opt[n-1][w - weight[n]];
}
// select better of two options
opt[n][w] = Math.max(option1, option2);
sol[n][w] = (option2 > option1);
}
}
// determine which items to take
List<List> packItems = new List<List>();
List<bool> take = new List(N+1);
for (int n = N, w = W; n > 0; n--) {
if (sol[n][w]) {
take[n] = true;
w = w - weight[n];
packItems.add(items[n-1]);
} else {
take[n] = false;
}
}
return packItems;
}
main() {
List knapsackItems = [];
knapsackItems.add(["map", 9, 150]);
knapsackItems.add(["compass", 13, 35]);
knapsackItems.add(["water", 153, 200]);
knapsackItems.add(["sandwich", 50, 160]);
knapsackItems.add(["glucose", 15, 60]);
knapsackItems.add(["tin", 68, 45]);
knapsackItems.add(["banana", 27, 60]);
knapsackItems.add(["apple", 39, 40]);
knapsackItems.add(["cheese", 23, 30]);
knapsackItems.add(["beer", 52, 10]);
knapsackItems.add(["suntan cream", 11, 70]);
knapsackItems.add(["camera", 32, 30]);
knapsackItems.add(["t-shirt", 24, 15]);
knapsackItems.add(["trousers", 48, 10]);
knapsackItems.add(["umbrella", 73, 40]);
knapsackItems.add(["waterproof trousers", 42, 70]);
knapsackItems.add(["waterproof overclothes", 43, 75]);
knapsackItems.add(["note-case", 22, 80]);
knapsackItems.add(["sunglasses", 7, 20]);
knapsackItems.add(["towel", 18, 12]);
knapsackItems.add(["socks", 4, 50]);
knapsackItems.add(["book", 30, 10]);
int maxWeight = 400;
Stopwatch sw = new Stopwatch.start();
List p = solveKnapsack(knapsackItems, maxWeight);
sw.stop();
int totalWeight = 0;
int totalValue = 0;
print(["item","profit","weight"]);
p.forEach((var i) { print("${i}"); totalWeight+=i[1]; totalValue+=i[2]; });
print("Total Value = ${totalValue}");
print("Total Weight = ${totalWeight}");
print("Elapsed Time = ${sw.elapsedInMs()}ms");
}</lang>
- Output:
[item, profit, weight] [socks, 4, 50] [sunglasses, 7, 20] [note-case, 22, 80] [waterproof overclothes, 43, 75] [waterproof trousers, 42, 70] [suntan cream, 11, 70] [banana, 27, 60] [glucose, 15, 60] [sandwich, 50, 160] [water, 153, 200] [compass, 13, 35] [map, 9, 150] Total Value = 1030 Total Weight = 396 Elapsed Time = 6ms
EasyLang
<lang>name$[] = [ "map" "compass" "water" "sandwich" "glucose" "tin" "banana" "apple" "cheese" "beer" "suntan cream" "camera" "t-shirt" "trousers" "umbrella" "waterproof trousers" "waterproof overclothes" "note-case" "sunglasses" "towel" "socks" "book" ] weight[] = [ 9 13 153 50 15 68 27 39 23 52 11 32 24 48 73 42 43 22 7 18 4 30 ] value[] = [ 150 35 200 160 60 45 60 40 30 10 70 30 15 10 40 70 75 80 20 12 50 10 ] max_w = 400
func solve i w . items[] wres vres .
if i < 0
wres = 0
vres = 0
items[] = [ ]
elif weight[i] > w
call solve i - 1 w items[] wres vres
else
call solve i - 1 w items[] wres vres
call solve i - 1 w - weight[i] items1[] w1 v1
v1 += value[i]
if v1 > vres
swap items[] items1[]
items[] &= i
wres = w1 + weight[i]
vres = v1
.
.
. call solve len weight[] - 1 max_w items[] w v print "weight: " & w print "value: " & v print "items:" for i range len items[]
print " " & name$[items[i]]
.</lang>
EchoLisp
<lang scheme> (require 'struct) (require 'hash) (require 'sql)
(define H (make-hash)) (define T (make-table (struct goodies (name poids valeur )))) (define-syntax-rule (name i) (table-xref T i 0)) (define-syntax-rule (poids i) (table-xref T i 1)) (define-syntax-rule (valeur i) (table-xref T i 2))
- make an unique hash-key from (i rest)
(define (t-idx i r) (string-append i "|" r))
- retrieve best score for item i, remaining r availbble weight
(define (t-get i r) (or (hash-ref H (t-idx i r)) 0))
- compute best score (i), assuming best (i-1 rest) is known
(define (score i restant) (if (< i 0) 0 (hash-ref! H (t-idx i restant) (if ( >= restant (poids i)) (max (score (1- i) restant) (+ (score (1- i) (- restant (poids i))) (valeur i))) (score (1- i) restant)))))
- compute best scores, starting from last item
(define (task W)
(define restant W)
(define N (1- (table-count T)))
(writeln 'total-value (score N W)) (for/list ((i (in-range N -1 -1))) #:continue (= (t-get i restant) (t-get (1- i) restant)) (set! restant (- restant (poids i))) (name i))) </lang>
- Output:
<lang scheme>
- init table
(define goodies
'((map 9 150) ; 9 is weight, 150 is value
(compass 13 35) (water 153 200) (sandwich 50 160)
(glucose 15 60) (tin 68 45)(banana 27 60) (apple 39 40)
(fromage 23 30) (beer 52 10) (🌞-suntan-cream 11 70) (camera 32 30)
(T-shirt 24 15) (pantalons 48 10) (umbrella 73 40)
(☔️-trousers 42 70) (☔️-overclothes 43 75) (note-case 22 80)
(🌞-sun-glasses 7 20) (towel 18 12) (socks 4 50) (book 30 10)))
(list->table goodies T)
(task 400)
total-value 1030
→ (socks 🌞-sun-glasses note-case ☔️-overclothes ☔️-trousers 🌞-suntan-cream banana glucose sandwich water compass map)
(length (hash-keys H))
→ 4939 ;; number of entries "i | weight" in hash table
</lang>
Eiffel
<lang Eiffel> class APPLICATION
create make
feature {NONE} -- Initialization
make local knapsack: KNAPSACKZEROONE do create knapsack.make (400) knapsack.add_item (create {ITEM}.make ("", 0, 0)) knapsack.add_item (create {ITEM}.make ("map", 9, 150)) knapsack.add_item (create {ITEM}.make ("compass", 13, 35)) knapsack.add_item (create {ITEM}.make ("water", 153, 200)) knapsack.add_item (create {ITEM}.make ("sandwich", 50, 160)) knapsack.add_item (create {ITEM}.make ("glucose", 15, 60)) knapsack.add_item (create {ITEM}.make ("tin", 68, 45)) knapsack.add_item (create {ITEM}.make ("banana", 27, 60)) knapsack.add_item (create {ITEM}.make ("apple", 39, 40)) knapsack.add_item (create {ITEM}.make ("cheese", 23, 30)) knapsack.add_item (create {ITEM}.make ("beer", 52, 10)) knapsack.add_item (create {ITEM}.make ("suntan cream", 11, 70)) knapsack.add_item (create {ITEM}.make ("camera", 32, 30)) knapsack.add_item (create {ITEM}.make ("T-shirt", 24, 15)) knapsack.add_item (create {ITEM}.make ("trousers", 48, 10)) knapsack.add_item (create {ITEM}.make ("umbrella, ella ella", 73, 40)) knapsack.add_item (create {ITEM}.make ("waterproof trousers", 42, 70)) knapsack.add_item (create {ITEM}.make ("waterproof overclothes", 43, 75)) knapsack.add_item (create {ITEM}.make ("note-case", 22, 80)) knapsack.add_item (create {ITEM}.make ("sunglasses", 7, 20)) knapsack.add_item (create {ITEM}.make ("towel", 18, 12)) knapsack.add_item (create {ITEM}.make ("socks", 4, 50)) knapsack.add_item (create {ITEM}.make ("book", 30, 10)) knapsack.compute_solution end
end </lang> <lang Eiffel> class ITEM
create make, make_from_other
feature
name: STRING
weight: INTEGER
value: INTEGER
make_from_other (other: ITEM) -- Item with name, weight and value set to 'other's name, weight and value. do name := other.name weight := other.weight value := other.value end
make (a_name: String; a_weight, a_value: INTEGER) -- Item with name, weight and value set to 'a_name', 'a_weight' and 'a_value'. require a_name /= Void a_weight >= 0 a_value >= 0 do name := a_name weight := a_weight value := a_value end
end </lang> <lang Eiffel> class KNAPSACKZEROONE
create make
feature
items: ARRAY [ITEM]
max_weight: INTEGER
feature
make (a_max_weight: INTEGER) -- Make an empty knapsack. require a_max_weight >= 0 do create items.make_empty max_weight := a_max_weight end
add_item (item: ITEM) -- Add 'item' to knapsack. local temp: ITEM do create temp.make_from_other (item) items.force (item, items.count + 1) end
compute_solution local M: ARRAY [INTEGER] n: INTEGER i, j: INTEGER w_i, v_i: INTEGER item_i: ITEM final_items: LINKED_LIST [ITEM] do n := items.count create M.make_filled (0, 1, n * max_weight) from i := 2 until (i > n) loop from j := 1 until j > max_weight loop item_i := items [i] w_i := item_i.weight if w_i <= j then v_i := item_i.value M [(i - 1) * max_weight + j] := max (M [(i - 2) * max_weight + j], M [(i - 2) * max_weight + j - w_i + 1] + v_i) else M [(i - 1) * max_weight + j] := M [(i - 2) * max_weight + j] end j := j + 1 end i := i + 1 end io.put_string ("The final value of the knapsack will be: ") io.put_integer (M [(n - 1) * max_weight + max_weight]); io.new_line --compute the items that fit into the knapsack create final_items.make io.put_string ("We'll take the following items: %N"); from i := n j := max_weight until i <= 1 or j <= 1 loop item_i := items [i] w_i := item_i.weight if w_i <= j then v_i := item_i.value if M [(i - 1) * max_weight + j] = M [(i - 2) * max_weight + j] then else final_items.extend (item_i) io.put_string (item_i.name) io.new_line j := j - w_i end else end i := i - 1 end end
feature {NONE}
max (a, b: INTEGER): INTEGER -- Max of 'a' and 'b'. do Result := a if a < b then Result := b end end
end </lang>
- Output:
The final value of the knapsack will be: 1030 We'll take the following items: socks sunglasses note-case waterproof overclothes waterproof trousers suntan cream banana glucose sandwich water compass map
Elixir
<lang elixir>defmodule Knapsack do
def solve([], _total_weight, item_acc, value_acc, weight_acc), do:
{item_acc, value_acc, weight_acc}
def solve([{_item, item_weight, _item_value} | t],
total_weight,
item_acc,
value_acc,
weight_acc) when item_weight > total_weight, do:
solve(t, total_weight, item_acc, value_acc, weight_acc)
def solve([{item_name, item_weight, item_value} | t],
total_weight,
item_acc,
value_acc,
weight_acc) do
{_tail_item_acc, tail_value_acc, _tail_weight_acc} = tail_res =
solve(t, total_weight, item_acc, value_acc, weight_acc)
{_head_item_acc, head_value_acc, _head_weight_acc} = head_res =
solve(t,
total_weight - item_weight,
[item_name | item_acc],
value_acc + item_value,
weight_acc + item_weight)
if tail_value_acc > head_value_acc, do: tail_res, else: head_res
end
end
stuff = [{"map", 9, 150},
{"compass", 13, 35},
{"water", 153, 200},
{"sandwich", 50, 160},
{"glucose", 15, 60},
{"tin", 68, 45},
{"banana", 27, 60},
{"apple", 39, 40},
{"cheese", 23, 30},
{"beer", 52, 10},
{"suntan cream", 11, 70},
{"camera", 32, 30},
{"T-shirt", 24, 15},
{"trousers", 48, 10},
{"umbrella", 73, 40},
{"waterproof trousers", 42, 70},
{"waterproof overclothes", 43, 75},
{"note-case", 22, 80},
{"sunglasses", 7, 20},
{"towel", 18, 12},
{"socks", 4, 50},
{"book", 30, 10}]
max_weight = 400
go = fn (stuff, max_weight) ->
{time, {item_list, total_value, total_weight}} = :timer.tc(fn ->
Knapsack.solve(stuff, max_weight, [], 0, 0)
end)
IO.puts "Items:"
Enum.each(item_list, fn item -> IO.inspect item end)
IO.puts "Total value: #{total_value}"
IO.puts "Total weight: #{total_weight}"
IO.puts "Time elapsed in milliseconds: #{time/1000}"
end go.(stuff, max_weight)</lang>
- Output:
Items: "socks" "sunglasses" "note-case" "waterproof overclothes" "waterproof trousers" "suntan cream" "banana" "glucose" "sandwich" "water" "compass" "map" Total value: 1030 Total weight: 396 Time elapsed in milliseconds: 733.0
Emacs Lisp
with changes (memoization without macro)
<lang lisp> (defun ks (max-w items)
(let ((cache (make-vector (1+ (length items)) nil)))
(dotimes (n (1+ (length items)))
(setf (aref cache n) (make-hash-table :test 'eql)))
(defun ks-emb (spc items)
(let ((slot (gethash spc (aref cache (length items)))))
(cond
((null items) (list 0 0 '()))
(slot slot)
(t (puthash spc
(let*
((i (car items))
(w (nth 1 i))
(v (nth 2 i))
(x (ks-emb spc (cdr items))))
(cond
((> w spc) x)
(t
(let* ((y (ks-emb (- spc w) (cdr items)))
(v (+ v (car y))))
(cond
((< v (car x)) x)
(t
(list v (+ w (nth 1 y)) (cons i (nth 2 y)))))))))
(aref cache (length items)))))))
(ks-emb max-w items)))
(ks 400
'((map 9 150) (compass 13 35) (water 153 200) (sandwich 50 160)
(glucose 15 60) (tin 68 45)(banana 27 60) (apple 39 40)
(cheese 23 30) (beer 52 10) (cream 11 70) (camera 32 30)
(T-shirt 24 15) (trousers 48 10) (umbrella 73 40)
(waterproof-trousers 42 70) (overclothes 43 75) (notecase 22 80)
(glasses 7 20) (towel 18 12) (socks 4 50) (book 30 10)))
</lang>
- Output:
(1030 396 ((map 9 150) (compass 13 35) (water 153 200) (sandwich 50 160) (glucose 15 60) (banana 27 60) (cream 11 70) (waterproof-trousers 42 70) (overclothes 43 75) (notecase 22 80) (glasses 7 20) (socks 4 50)))
Another way without cache : <lang lisp> (defun best-rate (l1 l2)
"predicate for sorting a list of elements regarding the value/weight rate"
(let*
((r1 (/ (* 1.0 (nth 2 l1)) (nth 1 l1)))
(r2 (/ (* 1.0 (nth 2 l2)) (nth 1 l2))))
(cond
((> r1 r2) t)
(t nil))))
(defun ks1 (l max)
"return a complete list - complete means 'less than max-weight
but add the next element is impossible'" (let ((l (sort l 'best-rate)))
(cond ((null l) l) ((<= (nth 1 (car l)) max) (cons (car l) (ks1 (cdr l) (- max (nth 1 (car l)))))) (t (ks1 (cdr l) max)))))
(defun totval (lol)
"totalize values of a list - lol is not for laughing
but for list of list"
(cond
((null lol) 0)
(t
(+
(nth 2 (car lol))
(totval (cdr lol))))))
(defun ks (l max)
"browse the list to find the best subset to put in the f***ing knapsack"
(cond
((null (cdr l)) (list (car l)))
(t
(let*
((x (ks1 l max))
(y (ks (cdr l) max)))
(cond
((> (totval x) (totval y)) x)
(t y))))))
(ks '((map 9 150) (compass 13 35) (water 153 200) (sandwich 50 160)
(glucose 15 60) (tin 68 45)(banana 27 60) (apple 39 40)
(cheese 23 30) (beer 52 10) (cream 11 70) (camera 32 30)
(T-shirt 24 15) (trousers 48 10) (umbrella 73 40)
(waterproof-trousers 42 70) (overclothes 43 75) (notecase 22 80)
(glasses 7 20) (towel 18 12) (socks 4 50) (book 30 10)) 400)
</lang>
- Output:
with org-babel in Emacs
| map | 9 | 150 | | socks | 4 | 50 | | cream | 11 | 70 | | glucose | 15 | 60 | | notecase | 22 | 80 | | sandwich | 50 | 160 | | glasses | 7 | 20 | | compass | 13 | 35 | | banana | 27 | 60 | | overclothes | 43 | 75 | | waterproof-trousers | 42 | 70 | | water | 153 | 200 | | | 396 | 1030 |
Erlang
<lang Erlang>
-module(knapsack_0_1).
-export([go/0,
solve/5]).
-define(STUFF,
[{"map", 9, 150},
{"compass", 13, 35},
{"water", 153, 200},
{"sandwich", 50, 160},
{"glucose", 15, 60},
{"tin", 68, 45},
{"banana", 27, 60},
{"apple", 39, 40},
{"cheese", 23, 30},
{"beer", 52, 10},
{"suntan cream", 11, 70},
{"camera", 32, 30},
{"T-shirt", 24, 15},
{"trousers", 48, 10},
{"umbrella", 73, 40},
{"waterproof trousers", 42, 70},
{"waterproof overclothes", 43, 75},
{"note-case", 22, 80},
{"sunglasses", 7, 20},
{"towel", 18, 12},
{"socks", 4, 50},
{"book", 30, 10}
]).
-define(MAX_WEIGHT, 400).
go() ->
StartTime = os:timestamp(),
{ItemList, TotalValue, TotalWeight} =
solve(?STUFF, ?MAX_WEIGHT, [], 0, 0),
TimeElapsed = timer:now_diff(os:timestamp(), StartTime),
io:format("Items: ~n"),
[io:format("~p~n", [Item]) || Item <- ItemList],
io:format(
"Total value: ~p~nTotal weight: ~p~nTime elapsed in milliseconds: ~p~n",
[TotalValue, TotalWeight, TimeElapsed/1000]).
solve([], _TotalWeight, ItemAcc, ValueAcc, WeightAcc) ->
{ItemAcc, ValueAcc, WeightAcc};
solve([{_Item, ItemWeight, _ItemValue} | T],
TotalWeight,
ItemAcc,
ValueAcc,
WeightAcc) when ItemWeight > TotalWeight ->
solve(T, TotalWeight, ItemAcc, ValueAcc, WeightAcc);
solve([{ItemName, ItemWeight, ItemValue} | T],
TotalWeight,
ItemAcc,
ValueAcc,
WeightAcc) ->
{_TailItemAcc, TailValueAcc, _TailWeightAcc} = TailRes =
solve(T, TotalWeight, ItemAcc, ValueAcc, WeightAcc),
{_HeadItemAcc, HeadValueAcc, _HeadWeightAcc} = HeadRes =
solve(T,
TotalWeight - ItemWeight,
[ItemName | ItemAcc],
ValueAcc + ItemValue,
WeightAcc + ItemWeight),
case TailValueAcc > HeadValueAcc of
true ->
TailRes;
false ->
HeadRes
end.
</lang>
- Output:
1> knapsack_0_1:go(). Items: "socks" "sunglasses" "note-case" "waterproof overclothes" "waterproof trousers" "suntan cream" "banana" "glucose" "sandwich" "water" "compass" "map" Total value: 1030 Total weight: 396 Time elapsed in milliseconds: 133.445 ok
Euler Math Toolbox
<lang Euler Math Toolbox> >items=["map","compass","water","sandwich","glucose", ... > "tin","banana","apple","cheese","beer","suntan creame", ... > "camera","t-shirt","trousers","umbrella","waterproof trousers", ... > "waterproof overclothes","note-case","sunglasses", ... > "towel","socks","book"]; >ws = [9,13,153,50,15,68,27,39,23,52,11, ... > 32,24,48,73,42,43,22,7,18,4,30]; >vs = [150,35,200,160,60,45,60,40,30,10,70, ... > 30,15,10,40,70,75,80,20,12,50,10]; >A=ws_id(cols(ws)); >c=vs; >b=[400]_ones(cols(vs),1); >sol = intsimplex(A,b,c,eq=-1,>max,>check); >items[nonzeros(sol)]
map compass water sandwich glucose banana suntan creame waterproof trousers waterproof overclothes note-case sunglasses socks
</lang>
F#
Using A* Algorithm
<lang fsharp> //Solve Knapsack 0-1 using A* algorithm //Nigel Galloway, August 3rd., 2018 let knapStar items maxW=
let l=List.length items
let p=System.Collections.Generic.SortedSet<float*int*float*float*list<int>>() //H*; level; value of items taken so far; weight so far
p.Add (0.0,0,0.0,0.0,[])|>ignore
let H items maxW=let rec H n g a=match g with |(_,w,v)::e->let t=n+w
if t<=maxW then H t e (a+v) else a+(v/w)*(maxW-n)
|_->a
H 0.0 items 0.0
let pAdd ((h,_,_,_,_) as n) bv=if h>bv then p.Add n |> ignore
let fH n (bv,t) w' v' t'=let _,w,v=List.item n items
let e=max bv (if w<=(maxW-w') then v'+v else bv)
let rt=n::t'
if n+1<l then pAdd ((v'+H (List.skip (n+1) items) maxW),n+1,v',w',t') bv
if w<=(maxW-w') then pAdd ((v'+v+H (List.skip (n+1) items) (maxW-w')),n+1,v'+v,w'+w,rt) bv
if e>bv then (e,rt) else (bv,t)
let rec fN (bv,t)=
let h,zl,zv,zw,zt as r=p.Max
p.Remove r |> ignore
if bv>=h then t else fN (fH zl (bv,t) zw zv zt)
fN (fH 0 (0.0,[]) 0.0 0.0 [])
</lang>
- Output:
<lang fsharp> let itemsf = [
"map", 9.0, 150.0; "compass", 13.0, 35.0; "water", 153.0, 200.0; "sandwich", 50.0, 160.0; "glucose", 15.0, 60.0; "tin", 68.0, 45.0; "banana", 27.0, 60.0; "apple", 39.0, 40.0; "cheese", 23.0, 30.0; "beer", 52.0, 10.0; "suntan cream", 11.0, 70.0; "camera", 32.0, 30.0; "t-shirt", 24.0, 15.0; "trousers", 48.0, 10.0; "umbrella", 73.0, 40.0; "waterproof trousers", 42.0, 70.0; "waterproof overclothes", 43.0, 75.0; "note-case", 22.0, 80.0; "sunglasses", 7.0, 20.0; "towel", 18.0, 12.0; "socks", 4.0, 50.0; "book", 30.0, 10.0;]|> List.sortBy(fun(_,n,g)->n/g)
</lang>
> let x=knapStar itemsf 400.0;;
> x|>Seq.map (fun n->Seq.item n itemsf)|>Seq.sumBy(fun (_,_,n)->(+n));;
val it : float = 1030.0
> x|>Seq.map (fun n->Seq.item n itemsf)|>Seq.sumBy(fun (_,n,_)->(+n));;
val it : float = 396.0
> x|>Seq.iter(fun n->printfn "%A" (List.item n itemsf));;
("map", 9.0, 150.0)
("socks", 4.0, 50.0)
("suntan cream", 11.0, 70.0)
("glucose", 15.0, 60.0)
("note-case", 22.0, 80.0)
("sandwich", 50.0, 160.0)
("sunglasses", 7.0, 20.0)
("compass", 13.0, 35.0)
("banana", 27.0, 60.0)
("waterproof overclothes", 43.0, 75.0)
("waterproof trousers", 42.0, 70.0)
("water", 153.0, 200.0)
Factor
Using dynamic programming: <lang factor>USING: accessors arrays fry io kernel locals make math math.order math.parser math.ranges sequences sorting ; IN: rosetta.knappsack.0-1
TUPLE: item
name weight value ;
CONSTANT: items {
T{ item f "map" 9 150 }
T{ item f "compass" 13 35 }
T{ item f "water" 153 200 }
T{ item f "sandwich" 50 160 }
T{ item f "glucose" 15 60 }
T{ item f "tin" 68 45 }
T{ item f "banana" 27 60 }
T{ item f "apple" 39 40 }
T{ item f "cheese" 23 30 }
T{ item f "beer" 52 10 }
T{ item f "suntan cream" 11 70 }
T{ item f "camera" 32 30 }
T{ item f "t-shirt" 24 15 }
T{ item f "trousers" 48 10 }
T{ item f "umbrella" 73 40 }
T{ item f "waterproof trousers" 42 70 }
T{ item f "waterproof overclothes" 43 75 }
T{ item f "note-case" 22 80 }
T{ item f "sunglasses" 7 20 }
T{ item f "towel" 18 12 }
T{ item f "socks" 4 50 }
T{ item f "book" 30 10 }
}
CONSTANT: limit 400
- make-table ( -- table )
items length 1 + [ limit 1 + 0 <array> ] replicate ;
- iterate ( item-no table -- )
item-no table nth :> prev
item-no 1 + table nth :> curr
item-no items nth :> item
limit [1,b] [| weight |
weight prev nth
weight item weight>> - dup 0 >=
[ prev nth item value>> + max ]
[ drop ] if
weight curr set-nth
] each ;
- fill-table ( table -- )
[ items length iota ] dip '[ _ iterate ] each ;
- extract-packed-items ( table -- items )
[
limit :> weight!
items length iota <reversed> [| item-no |
item-no table nth :> prev
item-no 1 + table nth :> curr
weight [ curr nth ] [ prev nth ] bi =
[
item-no items nth
[ name>> , ] [ weight>> weight swap - weight! ] bi
] unless
] each
] { } make ;
- solve-knappsack ( -- items value )
make-table [ fill-table ] [ extract-packed-items ] [ last last ] tri ;
- main ( -- )
solve-knappsack "Total value: " write number>string print "Items packed: " print natural-sort [ " " write print ] each ;</lang>
( scratchpad ) main
Total value: 1030
Items packed:
banana
compass
glucose
map
note-case
sandwich
socks
sunglasses
suntan cream
water
waterproof overclothes
waterproof trousers
Forth
<lang Forth> \ Rosetta Code Knapp-sack 0-1 problem. Tested under GForth 0.7.3. \ 22 items. On current processors a set fits nicely in one CELL (32 or 64 bits). \ Brute force approach: for every possible set of 22 items, \ check for admissible solution then for optimal set.
- offs HERE over - ;
400 VALUE WLIMIT
0 VALUE ITEM
0 VALUE VAL
0 VALUE /ITEM
0 VALUE ITEMS#
Create Sack HERE
9 , offs TO VAL
150 , offs TO ITEM
s" map " s, offs TO /ITEM
DROP
13 , 35 , s" compass " s,
153 , 200 , s" water " s,
50 , 160 , s" sandwich " s,
15 , 60 , s" glucose " s,
68 , 45 , s" tin " s,
27 , 60 , s" banana " s,
39 , 40 , s" apple " s,
23 , 30 , s" cheese " s,
52 , 10 , s" beer " s,
11 , 70 , s" suntan cream " s,
32 , 30 , s" camera " s,
24 , 15 , s" T-shirt " s,
48 , 10 , s" trousers " s,
73 , 40 , s" umbrella " s,
42 , 70 , s" wp trousers " s,
43 , 75 , s" wp overclothes " s,
22 , 80 , s" note-case " s,
7 , 20 , s" sunglasses " s,
18 , 12 , s" towel " s,
4 , 50 , s" socks " s,
30 , 10 , s" book " s,
HERE VALUE END-SACK
VARIABLE Sol \ Solution Set
VARIABLE Vmax \ Temporary Maximum Value
VARIABLE Sum \ Temporary Sum (for speed-up)
- ]sum ( Rtime: set -- sum ;Ctime: hilimit.a start.a -- )
\ Loop unwinding & precomputing addresses
]
]] Sum OFF [[
DO ]] dup 1 LITERAL AND IF I LITERAL @ Sum +! THEN 2/ [[
/ITEM +LOOP ]] drop Sum @ [[
- IMMEDIATE
- solve ( -- )
Vmax OFF
[ 1 END-SACK Sack - /ITEM / lshift 1- ]L 0
DO
I [ END-SACK Sack ]sum ( by weight ) WLIMIT <
IF
I [ END-SACK VAL + Sack VAL + ]sum ( by value )
dup Vmax @ >
IF Vmax ! I Sol ! ELSE drop THEN
THEN
LOOP
- .solution ( -- )
Sol @ END-SACK ITEM + Sack ITEM +
DO
dup 1 AND IF I count type cr THEN
2/
/ITEM +LOOP
drop
." Weight: " Sol @ [ END-SACK Sack ]sum . ." Value: " Sol @ [ END-SACK VAL + Sack VAL + ]sum .
</lang>
- Output:
map compass water sandwich glucose banana suntan cream wp trousers wp overclothes note-case sunglasses socks Weight: 396 Value: 1030
FreeBASIC
<lang freebasic>#define Tabu = Chr(9) Dim As Integer i, A, P, V, N Dim As Integer MejorArticulo, MejorValor = 0 Type Knapsack
articulo As String*22 peso As Integer valor As Integer
End Type Dim item(1 To 22) As Knapsack => { _ ("map ", 9, 150), ("compass ", 13, 35), _ ("water ", 153, 200), ("sandwich ", 50, 160), _ ("glucose ", 15, 60), ("tin ", 68, 45), _ ("banana ", 27, 60), ("apple ", 39, 40), _ ("cheese ", 23, 30), ("beer ", 52, 10), _ ("suntan cream ", 11, 70), ("camera ", 32, 30), _ ("T-shirt ", 24, 15), ("trousers ", 48, 10), _ ("umbrella ", 73, 40), ("waterproof trousers ", 42, 70), _ ("waterproof overclothes", 43, 75), ("note-case ", 22, 80), _ ("sunglasses ", 7, 20), ("towel ", 18, 12), _ ("socks ", 4, 50), ("book ", 30, 10)}
For i = 1 To (1 Shl 22)-1
A = i : P = 0 : V = 0 : N = 1
While A
If A And 1 Then
P += item(N).peso
V += item(N).valor
End If
A Shr= 1
N += 1
Wend
If V > MejorValor And P <= 400 Then
MejorValor = V
MejorArticulo = i
End If
Next
A = MejorArticulo : P = 0 : V = 0 : N = 1 While A
If A And 1 Then
Print " "; item(N).articulo; Tabu;
Print item(N).peso; Tabu; item(N).valor
P += item(N).peso
V += item(N).valor
End If
A Shr= 1 : N += 1
Wend Print "Totals:"; Spc(25); P; Tabu; V Sleep</lang>
- Output:
Same as XLP0 entry.
FutureBasic
<lang futurebasic> output file "Knapsack Problem Solution"
include "ConsoleWindow"
def tab 20
_numberOfObjects = 21 _weightOfKnapsack = 400
dim as short n : n = _numberOfObjects /* The number of objects available to pack */ dim as Str31 s(_numberOfObjects) /* The names of available objects */ dim as short c(_numberOfObjects) /* The *COST* of the ith object i.e. how much weight you must carry to pack the object */ dim as short v(_numberOfObjects) /* The *VALUE* of the ith object i.e. on a scale of 1 to 200, how important is it that the object included */ dim as short W : W = _weightOfKnapsack /* The maximum weight your knapsack will carry in ounces*/
s(0) = "map" s(1) = "compass" s(2) = "water" s(3) = "sandwich" s(4) = "glucose" s(5) = "tin" s(6) = "banana" s(7) = "apple" s(8) = "cheese" s(9) = "beer" s(10) = "suntan cream" s(11) = "camera" s(12) = "T-shirt" s(13) = "trousers" s(14) = "umbrella" s(15) = "waterproof pants" s(16) = "raincoat" s(17) = "note-case" s(18) = "sunglasses" s(19) = "towel" s(20) = "socks" s(21) = "socks"
c(0) = 9 c(1) = 13 c(2) = 153 c(3) = 50 c(4) = 15 c(5) = 68 c(6) = 27 c(7) = 39 c(8) = 23 c(9) = 52 c(10) = 11 c(11) = 32 c(12) = 24 c(13) = 48 c(14) = 73 c(15) = 42 c(16) = 43 c(17) = 22 c(18) = 7 c(19) = 18 c(20) = 4 c(21) = 30
v(0) = 150 v(1) = 35 v(2) = 200 v(3) = 160 v(4) = 60 v(5) = 45 v(6) = 60 v(7) = 40 v(8) = 30 v(9) = 10 v(10) = 70 v(11) = 30 v(12) = 15 v(13) = 10 v(14) = 40 v(15) = 70 v(16) = 75 v(17) = 80 v(18) = 20 v(19) = 12 v(20) = 50 v(21) = 10
local fn FillKnapsack
dim as short cur_w
dim as double tot_v : tot_v = 0
dim as short i, maxi, finalWeight : finalWeight = 0
dim as short finalValue : finalValue = 0
dim as short used(_numberOfObjects)
for i = 0 to n
used(i) = 0
next
cur_w = W while cur_w > -1
maxi = -1
BeginCCode
for ( i = 0; i < n; ++i)
if ((used[i] == 0) && ((maxi == -1) || ((float)v[i]/c[i] > (float)v[maxi]/c[maxi])))
maxi = i;
EndC
used(maxi) = 1 cur_w -= c(maxi) tot_v += v(maxi)
if (cur_w >= 0)
print s(maxi), c(maxi), v(maxi)
finalWeight = finalWeight + c(maxi)
finalValue = finalValue + v(maxi)
else
print
print "Add"; int( ( (double)cur_w/c(maxi) * 100 ) +100 ); "% more of "; s(maxi); " into the knapsack to fill remaining space."
tot_v -= v(maxi)
tot_v += (1 + (double )cur_w/c(maxi)) * v(maxi)
end if
wend
print print "Filled the bag with objects whose total value is"; finalValue; "." print "Total weight of packed objects is"; finalWeight; " ounces."
end fn
dim as short i, totalValue, totalWeight
print print "Available Items", "Weight in ounces", "Value (Scale of 1 to 200)" for i = 0 to _numberOfObjects
print s(i), c(i), v(i) totalValue += v(i) totalWeight += c(i)
next
print print "Total capacity of knapsack:"; W; " ounces"; "." print "Total value of all"; _numberOfObjects; " objects:"; totalValue; "." print "Total weight of all"; _numberOfObjects; " objects:"; totalWeight; " ounces." print print print "Most optimal packing considering weight and value:" print print "Item", "Weight", "Value"
fn FillKnapsack </lang>
Output:
Available Items Weight in ounces Value (Scale of 1 to 200) map 9 150 compass 13 35 water 153 200 sandwich 50 160 glucose 15 60 tin 68 45 banana 27 60 apple 39 40 cheese 23 30 beer 52 10 suntan cream 11 70 camera 32 30 T-shirt 24 15 trousers 48 10 umbrella 73 40 waterproof pants 42 70 raincoat 43 75 note-case 22 80 sunglasses 7 20 towel 18 12 socks 4 50 socks 30 10 Total capacity of knapsack: 400 ounces. Total value of all 21 objects: 1272. Total weight of all 21 objects: 803 ounces. Most optimal packing considering weight and value: Item Weight Value map 9 150 socks 4 50 suntan cream 11 70 glucose 15 60 note-case 22 80 sandwich 50 160 sunglasses 7 20 compass 13 35 banana 27 60 raincoat 43 75 waterproof pants 42 70 water 153 200 Add 17% more of cheese into the knapsack to fill remaining space. Filled the bag with objects whose total value is 1030. Total weight of packed objects is 396 ounces.
Go
From WP, "0-1 knapsack problem" under The Knapsack Problem, although the solution here simply follows the recursive defintion and doesn't even use the array optimization. <lang go>package main
import "fmt"
type item struct {
string w, v int
}
var wants = []item{
{"map", 9, 150},
{"compass", 13, 35},
{"water", 153, 200},
{"sandwich", 50, 160},
{"glucose", 15, 60},
{"tin", 68, 45},
{"banana", 27, 60},
{"apple", 39, 40},
{"cheese", 23, 30},
{"beer", 52, 10},
{"suntan cream", 11, 70},
{"camera", 32, 30},
{"T-shirt", 24, 15},
{"trousers", 48, 10},
{"umbrella", 73, 40},
{"waterproof trousers", 42, 70},
{"waterproof overclothes", 43, 75},
{"note-case", 22, 80},
{"sunglasses", 7, 20},
{"towel", 18, 12},
{"socks", 4, 50},
{"book", 30, 10},
}
const maxWt = 400
func main() {
items, w, v := m(len(wants)-1, maxWt)
fmt.Println(items)
fmt.Println("weight:", w)
fmt.Println("value:", v)
}
func m(i, w int) ([]string, int, int) {
if i < 0 || w == 0 {
return nil, 0, 0
} else if wants[i].w > w {
return m(i-1, w)
}
i0, w0, v0 := m(i-1, w)
i1, w1, v1 := m(i-1, w-wants[i].w)
v1 += wants[i].v
if v1 > v0 {
return append(i1, wants[i].string), w1 + wants[i].w, v1
}
return i0, w0, v0
}</lang>
- Output:
[map compass water sandwich glucose banana suntan cream waterproof trousers waterproof overclothes note-case sunglasses socks] weight: 396 value: 1030
Alternative test case
Data for which a greedy algorithm might give an incorrect result: <lang go> var wants = []item{
{"sunscreen", 15, 2},
{"GPS", 25, 2},
{"beer", 35, 3},
}
const maxWt = 40 </lang>
- Output:
[sunscreen GPS] weight: 40 value: 4
Groovy
Solution #1: brute force <lang groovy>def totalWeight = { list -> list*.weight.sum() } def totalValue = { list -> list*.value.sum() }
def knapsack01bf = { possibleItems ->
possibleItems.subsequences().findAll{ ss ->
def w = totalWeight(ss)
350 < w && w < 401
}.max(totalValue)
}</lang> Solution #2: dynamic programming <lang groovy>def knapsack01dp = { possibleItems ->
def n = possibleItems.size()
def m = (0..n).collect{ i -> (0..400).collect{ w -> []} }
(1..400).each { w ->
(1..n).each { i ->
def wi = possibleItems[i-1].weight
m[i][w] = wi > w ? m[i-1][w] : ([m[i-1][w], m[i-1][w-wi] + [possibleItems[i-1]]].max(totalValue))
}
}
m[n][400]
}</lang> Test: <lang groovy>def items = [
[name:"map", weight:9, value:150],
[name:"compass", weight:13, value:35],
[name:"water", weight:153, value:200],
[name:"sandwich", weight:50, value:160],
[name:"glucose", weight:15, value:60],
[name:"tin", weight:68, value:45],
[name:"banana", weight:27, value:60],
[name:"apple", weight:39, value:40],
[name:"cheese", weight:23, value:30],
[name:"beer", weight:52, value:10],
[name:"suntan cream", weight:11, value:70],
[name:"camera", weight:32, value:30],
[name:"t-shirt", weight:24, value:15],
[name:"trousers", weight:48, value:10],
[name:"umbrella", weight:73, value:40],
[name:"waterproof trousers", weight:42, value:70],
[name:"waterproof overclothes", weight:43, value:75],
[name:"note-case", weight:22, value:80],
[name:"sunglasses", weight:7, value:20],
[name:"towel", weight:18, value:12],
[name:"socks", weight:4, value:50],
[name:"book", weight:30, value:10],
]
[knapsack01bf, knapsack01dp].each { knapsack01 ->
def start = System.currentTimeMillis()
def packingList = knapsack01(items)
def elapsed = System.currentTimeMillis() - start
println "\n\n\nElapsed Time: ${elapsed/1000.0} s"
println "Total Weight: ${totalWeight(packingList)}"
println " Total Value: ${totalValue(packingList)}"
packingList.each {
printf (" item: %-25s weight:%4d value:%4d\n", it.name, it.weight, it.value)
}
}</lang>
- Output:
Elapsed Time: 132.267 s Total Weight: 396 Total Value: 1030 item: map weight: 9 value: 150 item: compass weight: 13 value: 35 item: water weight: 153 value: 200 item: sandwich weight: 50 value: 160 item: glucose weight: 15 value: 60 item: banana weight: 27 value: 60 item: suntan cream weight: 11 value: 70 item: waterproof trousers weight: 42 value: 70 item: waterproof overclothes weight: 43 value: 75 item: note-case weight: 22 value: 80 item: sunglasses weight: 7 value: 20 item: socks weight: 4 value: 50 Elapsed Time: 0.27 s Total Weight: 396 Total Value: 1030 item: map weight: 9 value: 150 item: compass weight: 13 value: 35 item: water weight: 153 value: 200 item: sandwich weight: 50 value: 160 item: glucose weight: 15 value: 60 item: banana weight: 27 value: 60 item: suntan cream weight: 11 value: 70 item: waterproof trousers weight: 42 value: 70 item: waterproof overclothes weight: 43 value: 75 item: note-case weight: 22 value: 80 item: sunglasses weight: 7 value: 20 item: socks weight: 4 value: 50
Haskell
Brute force: <lang haskell>inv = [("map",9,150), ("compass",13,35), ("water",153,200), ("sandwich",50,160), ("glucose",15,60), ("tin",68,45), ("banana",27,60), ("apple",39,40), ("cheese",23,30), ("beer",52,10), ("cream",11,70), ("camera",32,30), ("tshirt",24,15), ("trousers",48,10), ("umbrella",73,40), ("trousers",42,70), ("overclothes",43,75), ("notecase",22,80), ("sunglasses",7,20), ("towel",18,12), ("socks",4,50), ("book",30,10)]
-- get all combos of items under total weight sum; returns value sum and list combs [] _ = [ (0, []) ] combs ((name,w,v):rest) cap = combs rest cap ++ if w > cap then [] else map (prepend (name,w,v)) (combs rest (cap - w)) where prepend (name,w,v) (v2, lst) = (v2 + v, (name,w,v):lst)
main = do putStr "Total value: "; print value mapM_ print items where (value, items) = maximum $ combs inv 400</lang>
- Output:
Total value: 1030
("map",9,150)
("compass",13,35)
("water",153,200)
("sandwich",50,160)
("glucose",15,60)
("banana",27,60)
("cream",11,70)
("trousers",42,70)
("overclothes",43,75)
("notecase",22,80)
("sunglasses",7,20)
("socks",4,50)
Much faster brute force solution that computes the maximum before prepending, saving most of the prepends: <lang haskell>inv = [("map",9,150), ("compass",13,35), ("water",153,200), ("sandwich",50,160), ("glucose",15,60), ("tin",68,45), ("banana",27,60), ("apple",39,40), ("cheese",23,30), ("beer",52,10), ("cream",11,70), ("camera",32,30), ("tshirt",24,15), ("trousers",48,10), ("umbrella",73,40), ("trousers",42,70), ("overclothes",43,75), ("notecase",22,80), ("sunglasses",7,20), ("towel",18,12), ("socks",4,50), ("book",30,10)]
combs [] _ = (0, []) combs ((name,w,v):rest) cap | w <= cap = max skipthis $ prepend (name,w,v) (combs rest (cap - w)) | otherwise = skipthis where prepend (name,w,v) (v2, lst) = (v2 + v, (name,w,v):lst) skipthis = combs rest cap
main = do print $ combs inv 400</lang>
- Output:
(1030,[("map",9,150),("compass",13,35),("water",153,200),("sandwich",50,160),("glucose",15,60),("banana",27,60),("cream",11,70),("trousers",42,70),("overclothes",43,75),("notecase",22,80),("sunglasses",7,20),("socks",4,50)])
Dynamic programming with a list for caching (this can be adapted to bounded problem easily): <lang haskell>inv = [("map",9,150), ("compass",13,35), ("water",153,200), ("sandwich",50,160),
("glucose",15,60), ("tin",68,45), ("banana",27,60), ("apple",39,40),
("cheese",23,30), ("beer",52,10), ("cream",11,70), ("camera",32,30),
("tshirt",24,15), ("trousers",48,10), ("umbrella",73,40),
("waterproof trousers",42,70), ("overclothes",43,75), ("notecase",22,80),
("sunglasses",7,20), ("towel",18,12), ("socks",4,50), ("book",30,10)]
knapsack = foldr addItem (repeat (0,[])) where addItem (name,w,v) list = left ++ zipWith max right newlist where newlist = map (\(val, names)->(val + v, name:names)) list (left,right) = splitAt w list
main = print $ (knapsack inv) !! 400</lang>
- Output:
(1030,["map","compass","water","sandwich","glucose","banana","cream","waterproof trousers","overclothes","notecase","sunglasses","socks"])
Icon and Unicon
Translation from Wikipedia pseudo-code. Memoization can be enabled with a command line argument that causes the procedure definitions to be swapped which effectively hooks the procedure. <lang Icon>link printf
global wants # items wanted for knapsack
procedure main(A) # kanpsack 0-1
if !A == ("--trace"|"-t") then &trace := -1 # trace everything (debug)
if !A == ("--memoize"|"-m") then m :=: Memo_m # hook (swap) procedure
printf("Knapsack-0-1: with maximum weight allowed=%d.\n",maxw := 400)
showwanted(wants := get_wants())
showcontents(bag := m(*wants,maxw))
printf("Performance: time=%d ms collections=%d\n",&time,&collections)
end
record packing(items,weight,value)
procedure Memo_m(i,w) #: Hook procedure to memoize the knapsack static memoT initial memoT := table()
return \memoT[k := i||","||w] | ( memoT[k] := Memo_m(i,w) )
end
procedure m(i,w) #: Solve the Knapsack 0-1 as per Wikipedia static nil initial nil := packing([],0,0)
if 0 = (i | w) then
return nil
else if wants[i].weight > w then
return m(i-1, w)
else {
x0 := m(i-1,w)
x1 := m(i-1,w-wants[i].weight)
if ( x1.value + wants[i].value) > x0.value then
return packing(x1.items ||| wants[i].items,
x1.weight + wants[i].weight,
x1.value + wants[i].value)
else
return x0
}
end
procedure showwanted(wants) #: show the list of wanted items
every (tw := 0) +:= (!wants).weight
printf("Packing list has total weight=%d and includes %d items [",tw,*wants)
every printf(" %s",!(!wants).items|"]\n")
end
procedure showcontents(bag) #: show the list of the packed bag
printf("The bag weighs=%d holding %d items [",bag.weight,*bag.items)
every printf(" %s",!bag.items|"]\n")
end
procedure get_wants() #: setup list of wanted items
return [ packing(["map"], 9, 150),
packing(["compass"], 13, 35),
packing(["water"], 153, 200),
packing(["sandwich"], 50, 160),
packing(["glucose"], 15, 60),
packing(["tin"], 68, 45),
packing(["banana"], 27, 60),
packing(["apple"], 39, 40),
packing(["cheese"], 23, 30),
packing(["beer"], 52, 10),
packing(["suntan cream"], 11, 70),
packing(["camera"], 32, 30),
packing(["T-shirt"], 24, 15),
packing(["trousers"], 48, 10),
packing(["umbrella"], 73, 40),
packing(["waterproof trousers"], 42, 70),
packing(["waterproof overclothes"], 43, 75),
packing(["note-case"], 22, 80),
packing(["sunglasses"], 7, 20),
packing(["towel"], 18, 12),
packing(["socks"], 4, 50),
packing(["book"], 30, 10) ]
end</lang>
- Output:
Knapsack-0-1: with maximum weight allowed=400. Packing list has total weight=803 and includes 22 items [ map compass water sandwich glucose tin banana apple cheese beer suntan cream camera T-shirt trousers umbrella waterproof trousers waterproof overclothes note-case sunglasses towel socks book ] The bag weighs=396 holding 12 items [ map compass water sandwich glucose banana suntan cream waterproof trousers waterproof overclothes note-case sunglasses socks ] Performance: time=37 ms collections=0
The above shows memoized performance. Un-memoized results on the same PC took time=9728 ms collections=4.
J
Static solution: <lang J>'names values'=:|:".;._2]0 :0
'map'; 9 150 'compass'; 13 35 'water'; 153 200 'sandwich'; 50 160 'glucose'; 15 60 'tin'; 68 45 'banana'; 27 60 'apple'; 39 40 'cheese'; 23 30 'beer'; 52 10 'suntan cream'; 11 70 'camera'; 32 30 'tshirt'; 24 15 'trousers'; 48 10 'umbrella'; 73 40 'waterproof trousers'; 42 70 'waterproof overclothes'; 43 75 'notecase'; 22 80 'sunglasses'; 7 20 'towel'; 18 12 'socks'; 4 50 'book'; 30 10
)
X=: +/ .*"1 plausible=: (] (] #~ 400 >: X) #:@i.@(2&^)@#)@:({."1) best=: (plausible ([ {~ [ (i. >./)@:X {:"1@]) ]) values</lang> Illustration of answer: <lang J> +/best#values NB. total weight and value 396 1030
best#names
map compass water sandwich glucose banana suntan cream waterproof trousers waterproof overclothes notecase sunglasses socks </lang>
Alternative test case
<lang J>'names values'=:|:".;._2]0 :0
'sunscreen'; 15 2 'GPS'; 25 2 'beer'; 35 3
)
X=: +/ .*"1 plausible=: (] (] #~ 40 >: X) #:@i.@(2&^)@#)@:({."1) best=: (plausible ([ {~ [ (i. >./)@:X {:"1@]) ]) values</lang>
Illustration:
<lang J> +/best#values 40 4
best#names
sunscreen GPS </lang>
Java
General dynamic solution after wikipedia. <lang java>package hu.pj.alg.test;
import hu.pj.alg.ZeroOneKnapsack; import hu.pj.obj.Item; import java.util.*; import java.text.*;
public class ZeroOneKnapsackForTourists {
public ZeroOneKnapsackForTourists() {
ZeroOneKnapsack zok = new ZeroOneKnapsack(400); // 400 dkg = 400 dag = 4 kg
// making the list of items that you want to bring
zok.add("map", 9, 150);
zok.add("compass", 13, 35);
zok.add("water", 153, 200);
zok.add("sandwich", 50, 160);
zok.add("glucose", 15, 60);
zok.add("tin", 68, 45);
zok.add("banana", 27, 60);
zok.add("apple", 39, 40);
zok.add("cheese", 23, 30);
zok.add("beer", 52, 10);
zok.add("suntan cream", 11, 70);
zok.add("camera", 32, 30);
zok.add("t-shirt", 24, 15);
zok.add("trousers", 48, 10);
zok.add("umbrella", 73, 40);
zok.add("waterproof trousers", 42, 70);
zok.add("waterproof overclothes", 43, 75);
zok.add("note-case", 22, 80);
zok.add("sunglasses", 7, 20);
zok.add("towel", 18, 12);
zok.add("socks", 4, 50);
zok.add("book", 30, 10);
// calculate the solution:
List<Item> itemList = zok.calcSolution();
// write out the solution in the standard output
if (zok.isCalculated()) {
NumberFormat nf = NumberFormat.getInstance();
System.out.println(
"Maximal weight = " +
nf.format(zok.getMaxWeight() / 100.0) + " kg"
);
System.out.println(
"Total weight of solution = " +
nf.format(zok.getSolutionWeight() / 100.0) + " kg"
);
System.out.println(
"Total value = " +
zok.getProfit()
);
System.out.println();
System.out.println(
"You can carry the following materials " +
"in the knapsack:"
);
for (Item item : itemList) {
if (item.getInKnapsack() == 1) {
System.out.format(
"%1$-23s %2$-3s %3$-5s %4$-15s \n",
item.getName(),
item.getWeight(), "dag ",
"(value = " + item.getValue() + ")"
);
}
}
} else {
System.out.println(
"The problem is not solved. " +
"Maybe you gave wrong data."
);
}
}
public static void main(String[] args) {
new ZeroOneKnapsackForTourists();
}
} // class</lang> <lang java>package hu.pj.alg;
import hu.pj.obj.Item; import java.util.*;
public class ZeroOneKnapsack {
protected List<Item> itemList = new ArrayList<Item>(); protected int maxWeight = 0; protected int solutionWeight = 0; protected int profit = 0; protected boolean calculated = false;
public ZeroOneKnapsack() {}
public ZeroOneKnapsack(int _maxWeight) {
setMaxWeight(_maxWeight);
}
public ZeroOneKnapsack(List<Item> _itemList) {
setItemList(_itemList);
}
public ZeroOneKnapsack(List<Item> _itemList, int _maxWeight) {
setItemList(_itemList);
setMaxWeight(_maxWeight);
}
// calculte the solution of 0-1 knapsack problem with dynamic method:
public List<Item> calcSolution() {
int n = itemList.size();
setInitialStateForCalculation();
if (n > 0 && maxWeight > 0) {
List< List<Integer> > c = new ArrayList< List<Integer> >();
List<Integer> curr = new ArrayList<Integer>();
c.add(curr);
for (int j = 0; j <= maxWeight; j++)
curr.add(0);
for (int i = 1; i <= n; i++) {
List<Integer> prev = curr;
c.add(curr = new ArrayList<Integer>());
for (int j = 0; j <= maxWeight; j++) {
if (j > 0) {
int wH = itemList.get(i-1).getWeight();
curr.add(
(wH > j)
?
prev.get(j)
:
Math.max(
prev.get(j),
itemList.get(i-1).getValue() + prev.get(j-wH)
)
);
} else {
curr.add(0);
}
} // for (j...)
} // for (i...)
profit = curr.get(maxWeight);
for (int i = n, j = maxWeight; i > 0 && j >= 0; i--) {
int tempI = c.get(i).get(j);
int tempI_1 = c.get(i-1).get(j);
if (
(i == 0 && tempI > 0)
||
(i > 0 && tempI != tempI_1)
)
{
Item iH = itemList.get(i-1);
int wH = iH.getWeight();
iH.setInKnapsack(1);
j -= wH;
solutionWeight += wH;
}
} // for()
calculated = true;
} // if()
return itemList;
}
// add an item to the item list
public void add(String name, int weight, int value) {
if (name.equals(""))
name = "" + (itemList.size() + 1);
itemList.add(new Item(name, weight, value));
setInitialStateForCalculation();
}
// add an item to the item list
public void add(int weight, int value) {
add("", weight, value); // the name will be "itemList.size() + 1"!
}
// remove an item from the item list
public void remove(String name) {
for (Iterator<Item> it = itemList.iterator(); it.hasNext(); ) {
if (name.equals(it.next().getName())) {
it.remove();
}
}
setInitialStateForCalculation();
}
// remove all items from the item list
public void removeAllItems() {
itemList.clear();
setInitialStateForCalculation();
}
public int getProfit() {
if (!calculated)
calcSolution();
return profit;
}
public int getSolutionWeight() {return solutionWeight;}
public boolean isCalculated() {return calculated;}
public int getMaxWeight() {return maxWeight;}
public void setMaxWeight(int _maxWeight) {
maxWeight = Math.max(_maxWeight, 0);
}
public void setItemList(List<Item> _itemList) {
if (_itemList != null) {
itemList = _itemList;
for (Item item : _itemList) {
item.checkMembers();
}
}
}
// set the member with name "inKnapsack" by all items:
private void setInKnapsackByAll(int inKnapsack) {
for (Item item : itemList)
if (inKnapsack > 0)
item.setInKnapsack(1);
else
item.setInKnapsack(0);
}
// set the data members of class in the state of starting the calculation:
protected void setInitialStateForCalculation() {
setInKnapsackByAll(0);
calculated = false;
profit = 0;
solutionWeight = 0;
}
} // class</lang> <lang java>package hu.pj.obj;
public class Item {
protected String name = ""; protected int weight = 0; protected int value = 0; protected int bounding = 1; // the maximal limit of item's pieces protected int inKnapsack = 0; // the pieces of item in solution
public Item() {}
public Item(Item item) {
setName(item.name);
setWeight(item.weight);
setValue(item.value);
setBounding(item.bounding);
}
public Item(int _weight, int _value) {
setWeight(_weight);
setValue(_value);
}
public Item(int _weight, int _value, int _bounding) {
setWeight(_weight);
setValue(_value);
setBounding(_bounding);
}
public Item(String _name, int _weight, int _value) {
setName(_name);
setWeight(_weight);
setValue(_value);
}
public Item(String _name, int _weight, int _value, int _bounding) {
setName(_name);
setWeight(_weight);
setValue(_value);
setBounding(_bounding);
}
public void setName(String _name) {name = _name;}
public void setWeight(int _weight) {weight = Math.max(_weight, 0);}
public void setValue(int _value) {value = Math.max(_value, 0);}
public void setInKnapsack(int _inKnapsack) {
inKnapsack = Math.min(getBounding(), Math.max(_inKnapsack, 0));
}
public void setBounding(int _bounding) {
bounding = Math.max(_bounding, 0);
if (bounding == 0)
inKnapsack = 0;
}
public void checkMembers() {
setWeight(weight);
setValue(value);
setBounding(bounding);
setInKnapsack(inKnapsack);
}
public String getName() {return name;}
public int getWeight() {return weight;}
public int getValue() {return value;}
public int getInKnapsack() {return inKnapsack;}
public int getBounding() {return bounding;}
} // class</lang>
- Output:
Maximal weight = 4 kg Total weight of solution = 3,96 kg Total value = 1030 You can carry te following materials in the knapsack: map 9 dag (value = 150) compass 13 dag (value = 35) water 153 dag (value = 200) sandwich 50 dag (value = 160) glucose 15 dag (value = 60) banana 27 dag (value = 60) suntan cream 11 dag (value = 70) waterproof trousers 42 dag (value = 70) waterproof overclothes 43 dag (value = 75) note-case 22 dag (value = 80) sunglasses 7 dag (value = 20) socks 4 dag (value = 50)
JavaScript
Also available at gist. <lang javascript>/*global portviz:false, _:false */ /*
* 0-1 knapsack solution, recursive, memoized, approximate. * * credits: * * the Go implementation here: * http://rosettacode.org/mw/index.php?title=Knapsack_problem/0-1 * * approximation details here: * http://math.mit.edu/~goemans/18434S06/knapsack-katherine.pdf */
portviz.knapsack = {}; (function() {
this.combiner = function(items, weightfn, valuefn) {
// approximation guarantees result >= (1-e) * optimal
var _epsilon = 0.01;
var _p = _.max(_.map(items,valuefn));
var _k = _epsilon * _p / items.length;
var _memo = (function(){
var _mem = {};
var _key = function(i, w) {
return i + '::' + w;
};
return {
get: function(i, w) {
return _mem[_key(i,w)];
},
put: function(i, w, r) {
_mem[_key(i,w)]=r;
return r;
}
};
})();
var _m = function(i, w) {
i = Math.round(i);
w = Math.round(w);
if (i < 0 || w === 0) {
// empty base case
return {items: [], totalWeight: 0, totalValue: 0};
}
var mm = _memo.get(i,w);
if (!_.isUndefined(mm)) {
return mm;
}
var item = items[i];
if (weightfn(item) > w) {
//item does not fit, try the next item
return _memo.put(i, w, _m(i-1, w));
}
// this item could fit.
// are we better off excluding it?
var excluded = _m(i-1, w);
// or including it?
var included = _m(i-1, w - weightfn(item));
if (included.totalValue + Math.floor(valuefn(item)/_k) > excluded.totalValue) {
// better off including it
// make a copy of the list
var i1 = included.items.slice();
i1.push(item);
return _memo.put(i, w,
{items: i1,
totalWeight: included.totalWeight + weightfn(item),
totalValue: included.totalValue + Math.floor(valuefn(item)/_k)});
}
//better off excluding it
return _memo.put(i,w, excluded);
};
return {
/* one point */
one: function(maxweight) {
var scaled = _m(items.length - 1, maxweight);
return {
items: scaled.items,
totalWeight: scaled.totalWeight,
totalValue: scaled.totalValue * _k
};
},
/* the entire EF */
ef: function(maxweight, step) {
return _.map(_.range(0, maxweight+1, step), function(weight) {
var scaled = _m(items.length - 1, weight);
return {
items: scaled.items,
totalWeight: scaled.totalWeight,
totalValue: scaled.totalValue * _k
};
});
}
};
};
}).apply(portviz.knapsack);
/*global portviz:false, _:false */ /*
* after rosettacode.org/mw/index.php?title=Knapsack_problem/0-1 */
var allwants = [
{name:"map", weight:9, value: 150},
{name:"compass", weight:13, value: 35},
{name:"water", weight:153, value: 200},
{name:"sandwich", weight: 50, value: 160},
{name:"glucose", weight:15, value: 60},
{name:"tin", weight:68, value: 45},
{name:"banana", weight:27, value: 60},
{name:"apple", weight:39, value: 40},
{name:"cheese", weight:23, value: 30},
{name:"beer", weight:52, value: 10},
{name:"suntan cream", weight:11, value: 70},
{name:"camera", weight:32, value: 30},
{name:"T-shirt", weight:24, value: 15},
{name:"trousers", weight:48, value: 10},
{name:"umbrella", weight:73, value: 40},
{name:"waterproof trousers", weight:42, value: 70},
{name:"waterproof overclothes", weight:43, value: 75},
{name:"note-case", weight:22, value: 80},
{name:"sunglasses", weight:7, value: 20},
{name:"towel", weight:18, value: 12},
{name:"socks", weight:4, value: 50},
{name:"book", weight:30, value: 10}
];
var near = function(actual, expected, tolerance) {
if (expected === 0 && actual === 0) return true;
if (expected === 0) {
return Math.abs(expected - actual) / actual < tolerance;
}
return Math.abs(expected - actual) / expected < tolerance;
};
test("one knapsack", function() {
var combiner =
portviz.knapsack.combiner(allwants,
function(x){return x.weight;},
function(x){return x.value;});
var oneport = combiner.one(400);
ok(near(oneport.totalValue, 1030, 0.01), "correct total value");
ok(near(oneport.totalValue, 1030, 0.01), "correct total value");
equal(oneport.totalWeight, 396, "correct total weight");
});
test("frontier", function() {
var combiner =
portviz.knapsack.combiner(allwants,
function(x){return x.weight;},
function(x){return x.value;});
var ef = combiner.ef(400, 1);
equal(ef.length, 401, "401 because it includes the endpoints");
ef = combiner.ef(400, 40);
equal(ef.length, 11, "11 because it includes the endpoints");
var expectedTotalValue = [
0,
330,
445,
590,
685,
755,
810,
860,
902,
960,
1030
] ;
_.each(ef, function(element, index) {
// 15% error! bleah!
ok(near(element.totalValue, expectedTotalValue[index], 0.15),
'actual ' + element.totalValue + ' expected ' + expectedTotalValue[index]);
});
deepEqual(_.pluck(ef, 'totalWeight'), [
0,
39,
74,
118,
158,
200,
236,
266,
316,
354,
396
]);
deepEqual(_.map(ef, function(x){return x.items.length;}), [
0,
4,
6,
7,
9,
10,
10,
12,
14,
11,
12
]);
});</lang>
jq
"dynamic_knapsack(W)" implements a dynamic programming algorithm based on computing m[i,W] as the maximum value that can be attained with weight no greater than W using the first i items (with i = 0 corresponding to no items). Here, m[i,W] is set to [V, ary] where ary is an array of the names of the accepted items. <lang jq># Input should be the array of objects giving name, weight and value.
- Because of the way addition is defined on null and because of the
- way setpath works, there is no need to initialize the matrix m in
- detail.
def dynamic_knapsack(W):
. as $objects
| length as $n
| reduce range(1; $n+1) as $i # i is the number of items
# state: m[i][j] is an array of [value, array_of_object_names]
(null; # see above remark about initialization of m
$objects[$i-1] as $o
| reduce range(0; W+1) as $j
( .;
if $o.weight <= $j then
.[$i-1][$j][0] as $v1 # option 1: do not add this object
| (.[$i-1][$j - $o.weight][0] + $o.value) as $v2 # option 2: add it
| (if $v1 > $v2 then
[$v1, .[$i-1][$j][1]] # do not add this object
else [$v2, .[$i-1][$j - $o.weight][1]+[$o.name]] # add it
end) as $mx
| .[$i][$j] = $mx
else
.[$i][$j] = .[$i-1][$j]
end))
| .[$n][W];</lang>
Example: <lang jq>def objects: [
{name: "map", "weight": 9, "value": 150},
{name: "compass", "weight": 13, "value": 35},
{name: "water", "weight": 153, "value": 200},
{name: "sandwich", "weight": 50, "value": 160},
{name: "glucose", "weight": 15, "value": 60},
{name: "tin", "weight": 68, "value": 45},
{name: "banana", "weight": 27, "value": 60},
{name: "apple", "weight": 39, "value": 40},
{name: "cheese", "weight": 23, "value": 30},
{name: "beer", "weight": 52, "value": 10},
{name: "suntancream", "weight": 11, "value": 70},
{name: "camera", "weight": 32, "value": 30},
{name: "T-shirt", "weight": 24, "value": 15},
{name: "trousers", "weight": 48, "value": 10},
{name: "umbrella", "weight": 73, "value": 40},
{name: "waterproof trousers", "weight": 42, "value": 70},
{name: "waterproof overclothes", "weight": 43, "value": 75},
{name: "note-case", "weight": 22, "value": 80},
{name: "sunglasses", "weight": 7, "value": 20},
{name: "towel", "weight": 18, "value": 12},
{name: "socks", "weight": 4, "value": 50},
{name: "book", "weight": 30, "value": 10}
];
objects | dynamic_knapsack(400)[]</lang>
- Output:
<lang sh>$jq -M -c -n -f knapsack.jq 1030 ["map","compass","water","sandwich","glucose","banana","suntancream","waterproof trousers","waterproof overclothes","note-case","sunglasses","socks"]</lang>
Julia
This solution uses the MathProgBase package (with the Cbc solver package installed). It is the mixintprog function from this package that does the heavy lifting of this solution.
KPDSupply has one more field than is needed, quant. This field is may be useful in a solution to the bounded version of this task.
Type and Functions: <lang julia>struct KPDSupply{T<:Integer}
item::String weight::T value::T quant::T
end
KPDSupply{T<:Integer}(itm::AbstractString, w::T, v::T, q::T=one(T)) = KPDSupply(itm, w, v, q) Base.show(io::IO, kdps::KPDSupply) = print(io, kdps.quant, " ", kdps.item, " ($(kdps.weight) kg, $(kdps.value) €)")
using MathProgBase, Cbc function solve(gear::Vector{<:KPDSupply}, capacity::Integer)
w = getfield.(gear, :weight) v = getfield.(gear, :value) sol = mixintprog(-v, w', '<', capacity, :Bin, 0, 1, CbcSolver()) gear[sol.sol .≈ 1]
end</lang>
Main: <lang julia>gear = [KPDSupply("map", 9, 150),
KPDSupply("compass", 13, 35),
KPDSupply("water", 153, 200),
KPDSupply("sandwich", 50, 160),
KPDSupply("glucose", 15, 60),
KPDSupply("tin", 68, 45),
KPDSupply("banana", 27, 60),
KPDSupply("apple", 39, 40),
KPDSupply("cheese", 23, 30),
KPDSupply("beer", 52, 10),
KPDSupply("suntan cream", 11, 70),
KPDSupply("camera", 32, 30),
KPDSupply("T-shirt", 24, 15),
KPDSupply("trousers", 48, 10),
KPDSupply("umbrella", 73, 40),
KPDSupply("waterproof trousers", 42, 70),
KPDSupply("waterproof overclothes", 43, 75),
KPDSupply("note-case", 22, 80),
KPDSupply("sunglasses", 7, 20),
KPDSupply("towel", 18, 12),
KPDSupply("socks", 4, 50),
KPDSupply("book", 30, 10)]
pack = solve(gear, 400) println("The hicker should pack: \n - ", join(pack, "\n - ")) println("\nPacked weight: ", mapreduce(x -> x.weight, +, pack), " kg") println("Packed value: ", mapreduce(x -> x.value, +, pack), " €")</lang>
- Output:
The hicker should pack: - 1 map (9 kg, 150 €) - 1 compass (13 kg, 35 €) - 1 water (153 kg, 200 €) - 1 sandwich (50 kg, 160 €) - 1 glucose (15 kg, 60 €) - 1 banana (27 kg, 60 €) - 1 suntan cream (11 kg, 70 €) - 1 waterproof trousers (42 kg, 70 €) - 1 waterproof overclothes (43 kg, 75 €) - 1 note-case (22 kg, 80 €) - 1 sunglasses (7 kg, 20 €) - 1 socks (4 kg, 50 €) Packed weight: 396 kg Packed value: 1030 €
Kotlin
<lang scala>// version 1.1.2
data class Item(val name: String, val weight: Int, val value: Int)
val wants = listOf(
Item("map", 9, 150),
Item("compass", 13, 35),
Item("water", 153, 200),
Item("sandwich", 50, 160),
Item("glucose", 15, 60),
Item("tin", 68, 45),
Item("banana", 27, 60),
Item("apple", 39, 40),
Item("cheese", 23, 30),
Item("beer", 52, 10),
Item("suntan cream", 11, 70),
Item("camera", 32, 30),
Item("T-shirt", 24, 15),
Item("trousers", 48, 10),
Item("umbrella", 73, 40),
Item("waterproof trousers", 42, 70),
Item("waterproof overclothes", 43, 75),
Item("note-case", 22, 80),
Item("sunglasses", 7, 20),
Item("towel", 18, 12),
Item("socks", 4, 50),
Item("book", 30, 10)
)
const val MAX_WEIGHT = 400
fun m(i: Int, w: Int): Triple<MutableList<Item>, Int, Int> {
val chosen = mutableListOf<Item>()
if (i < 0 || w == 0) return Triple(chosen, 0, 0)
else if (wants[i].weight > w) return m(i - 1, w)
val (l0, w0, v0) = m(i - 1, w)
var (l1, w1, v1) = m(i - 1, w - wants[i].weight)
v1 += wants[i].value
if (v1 > v0) {
l1.add(wants[i])
return Triple(l1, w1 + wants[i].weight, v1)
}
return Triple(l0, w0, v0)
}
fun main(args: Array<String>) {
val (chosenItems, totalWeight, totalValue) = m(wants.size - 1, MAX_WEIGHT)
println("Knapsack Item Chosen Weight Value")
println("---------------------- ------ -----")
for (item in chosenItems.sortedByDescending { it.value} )
println("${item.name.padEnd(24)} ${"%3d".format(item.weight)} ${"%3d".format(item.value)}")
println("---------------------- ------ -----")
println("Total ${chosenItems.size} Items Chosen $totalWeight $totalValue")
}</lang>
- Output:
Knapsack Item Chosen Weight Value ---------------------- ------ ----- water 153 200 sandwich 50 160 map 9 150 note-case 22 80 waterproof overclothes 43 75 suntan cream 11 70 waterproof trousers 42 70 glucose 15 60 banana 27 60 socks 4 50 compass 13 35 sunglasses 7 20 ---------------------- ------ ----- Total 12 Items Chosen 396 1030
LSL
To test it yourself, rez a box on the ground, add the following as a New Script, create a notecard named "Knapsack_Problem_0_1_Data.txt" with the data shown below. <lang LSL>string sNOTECARD = "Knapsack_Problem_0_1_Data.txt"; integer iMAX_WEIGHT = 400; integer iSTRIDE = 4; list lList = []; default { integer iNotecardLine = 0; state_entry() { llOwnerSay("Reading '"+sNOTECARD+"'"); llGetNotecardLine(sNOTECARD, iNotecardLine); } dataserver(key kRequestId, string sData) { if(sData==EOF) { //llOwnerSay("EOF"); lList = llListSort(lList, iSTRIDE, FALSE); integer iTotalWeight = 0; integer iTotalValue = 0; list lKnapsack = []; integer x = 0; while(x*iSTRIDE<llGetListLength(lList)) { float fValueWeight = (float)llList2String(lList, x*iSTRIDE); string sItem = (string)llList2String(lList, x*iSTRIDE+1); integer iWeight = (integer)llList2String(lList, x*iSTRIDE+2); integer iValue = (integer)llList2String(lList, x*iSTRIDE+3); if(iTotalWeight+iWeight<iMAX_WEIGHT) { iTotalWeight += iWeight; iTotalValue += iValue; lKnapsack += [sItem, iWeight, iValue, fValueWeight]; } x++; } for(x=0 ; x*iSTRIDE<llGetListLength(lKnapsack) ; x++) { llOwnerSay((string)x+": "+llList2String(lList, x*iSTRIDE+1)+", "+llList2String(lList, x*iSTRIDE+2)+", "+llList2String(lList, x*iSTRIDE+3));
} llOwnerSay("iTotalWeight="+(string)iTotalWeight); llOwnerSay("iTotalValue="+(string)iTotalValue); } else { //llOwnerSay((string)iNotecardLine+": "+sData); if(llStringTrim(sData, STRING_TRIM)!="") { list lParsed = llParseString2List(sData, [","], []); string sItem = llStringTrim(llList2String(lParsed, 0), STRING_TRIM); integer iWeight = (integer)llStringTrim(llList2String(lParsed, 1), STRING_TRIM); integer iValue = (integer)llStringTrim(llList2String(lParsed, 2), STRING_TRIM); float fValueWeight = (1.0*iValue)/iWeight; lList += [fValueWeight, sItem, iWeight, iValue]; } llGetNotecardLine(sNOTECARD, ++iNotecardLine); } } }</lang> Notecard:
map, 9, 150 compass, 13, 35 water, 153, 200 sandwich, 50, 160 glucose, 15, 60 tin, 68, 45 banana, 27, 60 apple, 39, 40 cheese, 23, 30 beer, 52, 10 suntan cream, 11, 70 camera, 32, 30 T-shirt, 24, 15 trousers, 48, 10 umbrella, 73, 40 waterproof trousers, 42, 70 waterproof overclothes, 43, 75 note-case, 22, 80 sunglasses, 7, 20 towel, 18, 12 socks, 4, 50 book, 30, 10
- Output:
Reading 'Knapsack_Problem_0_1_Data.txt' 0: map, 9, 150 1: socks, 4, 50 2: suntan cream, 11, 70 3: glucose, 15, 60 4: note-case, 22, 80 5: sandwich, 50, 160 6: sunglasses, 7, 20 7: compass, 13, 35 8: banana, 27, 60 9: waterproof overclothes, 43, 75 10: waterproof trousers, 42, 70 11: water, 153, 200 iTotalWeight=396 iTotalValue=1030
Lua
This version is adapted from the Clojure version. <lang lua>items = {
{"map", 9, 150},
{"compass", 13, 35},
{"water", 153, 200},
{"sandwich", 50, 160},
{"glucose", 15, 60},
{"tin", 68, 45},
{"banana", 27, 60},
{"apple", 39, 40},
{"cheese", 23, 30},
{"beer", 52, 10},
{"suntan cream", 11, 70},
{"camera", 32, 30},
{"t-shirt", 24, 15},
{"trousers", 48, 10},
{"umbrella", 73, 40},
{"waterproof trousers", 42, 70},
{"waterproof overclothes", 43, 75},
{"note-case", 22, 80},
{"sunglasses", 7, 20},
{"towel", 18, 12},
{"socks", 4, 50},
{"book", 30, 10},
}
local unpack = table.unpack
function m(i, w)
if i<1 or w==0 then
return 0, {}
else
local _, wi, vi = unpack(items[i])
if wi > w then
return mm(i - 1, w)
else
local vn, ln = mm(i - 1, w)
local vy, ly = mm(i - 1, w - wi)
if vy + vi > vn then
return vy + vi, { i, ly }
else
return vn, ln
end
end
end
end
local memo, mm_calls = {}, 0 function mm(i, w) -- memoization function for m
mm_calls = mm_calls + 1
local key = 10000*i + w
local result = memo[key]
if not result then
result = { m(i, w) }
memo[key] = result
end
return unpack(result)
end
local total_value, index_list = m(#items, 400)
function list_items(head) -- makes linked list iterator function
return function()
local item, rest = unpack(head)
head = rest
return item
end
end
local names = {} local total_weight = 0 for i in list_items(index_list) do
local name, weight = unpack(items[i]) table.insert(names, 1, name) total_weight = total_weight + weight
end
local function printf(fmt, ...) print(string.format(fmt, ...)) end printf("items to pack: %s", table.concat(names, ", ")) printf("total value: %d", total_value) printf("total weight: %d", total_weight)
-- out of curiosity local count = 0 for k,v in pairs(memo) do count = count + 1 end printf("\n(memo count: %d; mm call count: %d)", count, mm_calls) </lang>
- Output:
items to pack: map, compass, water, sandwich, glucose, banana, suntan cream, waterproof trousers, waterproof overclothes, note-case, sunglasses, socks total value: 1030 total weight: 396 (memo count: 5329; mm call count: 9485)
Maple
<lang Maple>weights := [9,13,153,50,15,68,27,39,23,52,11,32,24,48,73,42,43,22,7,18,4,30]: vals := [150,35,200,160,60,45,60,40,30,10,70,30,15,10,40,70,75,80,20,12,50,10]: items := ["map","compass","water","sandwich","glucose","tin","banana","apple","cheese","beer","suntan cream","camera","T-shirt","trousers","umbrella","waterproof trousers","waterproof overclothes","note-case","sunglasses","towel","socks","book"]: acc := Array(1..numelems(vals)+1,1..400+1,1,fill=0): len := numelems(weights): for i from 2 to len+1 do #number of items picked + 1 for j from 2 to 401 do #weight capacity left + 1 if weights[i-1] > j-1 then acc[i,j] := acc[i-1, j]: else acc[i,j] := max(acc[i-1,j], acc[i-1, j-weights[i-1]]+vals[i-1]): end if: end do: end do: printf("Total Value is %d\n", acc[len+1, 401]): count := 0: i := len+1: j := 401: while (i>1 and j>1) do if acc[i,j] <> acc[i-1,j] then printf("Item: %s\n", items[i-1]): count := count+weights[i-1]: j := j-weights[i-1]: i := i-1: else i := i-1: end if: end do: printf("Total Weight is %d\n", count):</lang>
- Output:
Total Value is 1030 Item: socks Item: sunglasses Item: note-case Item: waterproof overclothes Item: waterproof trousers Item: suntan cream Item: banana Item: glucose Item: sandwich Item: water Item: compass Item: map Total Weight is 396
Mathematica/Wolfram Language
Used the <lang mathematica>#[[Flatten@
Position[LinearProgramming[-#;; , 3, -{#;; , 2}, -{400}, {0, 1} & /@ #, Integers], 1], 1]] &@ {{"map", 9, 150}, {"compass", 13, 35}, {"water", 153, 200}, {"sandwich", 50, 160}, {"glucose", 15, 60}, {"tin", 68, 45}, {"banana", 27, 60}, {"apple", 39, 40}, {"cheese", 23, 30}, {"beer", 52, 10}, {"suntan cream", 11, 70}, {"camera", 32, 30}, {"T-shirt", 24, 15}, {"trousers", 48, 10}, {"umbrella", 73, 40}, {"waterproof trousers", 42, 70}, {"waterproof overclothes", 43, 75}, {"note-case", 22, 80}, {"sunglasses", 7, 20}, {"towel", 18, 12}, {"socks", 4, 50}, {"book", 30, 10}}</lang>
- Output:
{"map", "compass", "water", "sandwich", "glucose", "banana", "suntan cream", "waterproof trousers", "waterproof overclothes", "note-case", "sunglasses", "socks"}
Mathprog
<lang mathprog>/*Knapsack
This model finds the integer optimal packing of a knapsack Nigel_Galloway January 9th., 2012
- /
set Items; param weight{t in Items}; param value{t in Items};
var take{t in Items}, binary;
knap_weight : sum{t in Items} take[t] * weight[t] <= 400;
maximize knap_value: sum{t in Items} take[t] * value[t];
data;
param : Items : weight value :=
map 9 150
compass 13 35
water 153 200
sandwich 50 160
glucose 15 60
tin 68 45
banana 27 60
apple 39 40
cheese 23 30
beer 52 10
suntancream 11 70
camera 32 30
T-shirt 24 15
trousers 48 10
umbrella 73 40
w-trousers 42 70
w-overclothes 43 75
note-case 22 80
sunglasses 7 20
towel 18 12
socks 4 50
book 30 10
end;</lang> The solution may be found at Knapsack problem/0-1/Mathprog. Activity=1 means take, Activity=0 means don't take.
MAXScript
<lang MAXScript> global globalItems = #() global usedMass = 0 global neededItems = #() global totalValue = 0 struct kn_item ( item, weight, value )
itemStrings = #("map#9#150","compass#13#35","water#153#200", \ "sandwich#50#160","glucose#15#60","tin#68#45", \ "banana#27#60","apple#39#40","cheese#23#30", \ "beer#52#10","suntan cream#11#70","camera#32#30", \ "T-shirt#24#15","trousers#48#10","umbrella#73#40", \ "waterproof trousers#42#70","waterproof overclothes#43#75", \ "note-case#22#80","sunglasses#7#20", "towel#18#12", \ "socks#4#50","book#30#10")
fn sortByValue a b = ( if a[1].value > b[1].value then return -1 else ( if a[1].value == b[1].value then return 0 else return 1 ) ) fn chooseBestItem maximumWeight: items: = ( local itemsCopy = deepcopy items local possibleItems = #() for i = 1 to itemsCopy.count do ( if itemsCopy[i].weight <= maximumWeight do append possibleItems (#(itemsCopy[i],i)) ) qsort possibleItems sortByValue if possibleItems.count > 0 then return possibleItems[1] else return 0 )
for i = 1 to itemStrings.count do ( local split = filterstring itemStrings[i] "#" local itemStruct = kn_item item:split[1] weight:(split[2] as integer) \ value:(split[3] as integer) appendifunique globalItems itemstruct )
while usedMass < 400 do ( local item = chooseBestItem maximumweight:(400-usedMass) items:(globalItems) if item != 0 then ( deleteitem globalItems (item[2]) appendifunique neededItems item[1] usedMass += item[1].weight ) else exit ) for i in neededitems do ( format "Item name: %, weight: %, value:%\n" i.item i.weight i.value totalValue += i.value ) format "Total mass: %, Total Value: %\n" usedMass totalValue </lang>
- Output:
<lang MAXScript> Item name: water, weight: 153, value:200 Item name: sandwich, weight: 50, value:160 Item name: map, weight: 9, value:150 Item name: note-case, weight: 22, value:80 Item name: waterproof overclothes, weight: 43, value:75 Item name: suntan cream, weight: 11, value:70 Item name: waterproof trousers, weight: 42, value:70 Item name: glucose, weight: 15, value:60 Item name: banana, weight: 27, value:60 Item name: socks, weight: 4, value:50 Item name: compass, weight: 13, value:35 Item name: sunglasses, weight: 7, value:20 OK Total mass: 396, Total Value: 1030 OK </lang>
MiniZinc
<lang MiniZinc> %Knapsack 0/1. Nigel Galloway: October 5th., 2020. enum Items={map,compass,water,sandwich,glucose,tin,banana,apple,cheese,beer,suntan_cream,camera,t_shirt,trousers,umbrella,waterproof_trousers,waterproof_overclothes,note_case,sunglasses,towel,socks,book}; array[Items] of int: weight=[9,13,153,50,15,68,27,39,23,52,11,32,24,48,73,42,43,22,7,18,4,30]; array[Items] of int: value =[150,35,200,160,60,45,60,40,30,10,70,30,15,10,40,70,75,80,20,12,50,10]; int: maxWeight=400; var int: wTaken=sum(n in take)(weight[n]); var int: wValue=sum(n in take)(value[n]); var set of Items: take; constraint wTaken <= maxWeight; solve maximize wValue; output["Take "++show(take)++"\nTotal Weight=\(wTaken) Total Value=\(wValue)"] </lang>
- Output:
Take {map, compass, water, sandwich, glucose, banana, suntan_cream, waterproof_trousers, waterproof_overclothes, note_case, sunglasses, socks}
Total Weight=396 Total Value=1030
Nim
This solution uses the same algorithm as Go and Kotlin, with some modifications to improve performance:
- – use item indexes rather than items themselves;
- – rather than using sequences of item indexes, use sets of item indexes which are represented as bit sets;
- – use a cache for memoization.
<lang Nim>
- Knapsack. Recursive algorithm.
import algorithm import sequtils import tables
- Description of an item.
type Item = tuple[name: string; weight, value: int]
- List of available items.
const Items: seq[Item] = @[("map", 9, 150),
("compass", 13, 35),
("water", 153, 200),
("sandwich", 50, 160),
("glucose", 15, 60),
("tin", 68, 45),
("banana", 27, 60),
("apple", 39, 40),
("cheese", 23, 30),
("beer", 52, 10),
("suntan cream", 11, 70),
("camera", 32, 30),
("T-shirt", 24, 15),
("trousers", 48, 10),
("umbrella", 73, 40),
("waterproof trousers", 42, 70),
("waterproof overclothes", 43, 75),
("note-case", 22, 80),
("sunglasses", 7, 20),
("towel", 18, 12),
("socks", 4, 50),
("book", 30, 10)
]
type
# Item numbers (used rather than items themselves). Number = range[0..Items.high]
# Chosen items and their total value. Choice = tuple[nums: set[Number]; weight, value: int]
- Cache used to speed up the search.
var cache: Table[tuple[num, weight: int], Choice]
- ---------------------------------------------------------------------------------------------------
proc select(num, weightLimit: int): Choice =
## Find the best choice starting from item at index "num".
if num < 0 or weightLimit == 0: return
if (num, weightLimit) in cache: return cache[(num, weightLimit)]
let weight = Items[num].weight if weight > weightLimit: return select(num - 1, weightLimit)
# Try by leaving this item and selecting among remaining items. result = select(num - 1, weightLimit)
# Try by taking this item and completing with some remaining items. var result1 = select(num - 1, weightLimit - weight) inc result1.value, Items[num].value
# Select the best choice (giving the greater value).
if result1.value > result.value:
result = (result1.nums + {num.Number}, result1.weight + weight, result1.value)
cache[(num, weightLimit)] = result
- ---------------------------------------------------------------------------------------------------
let (nums, weight, value) = select(Items.high, 400) echo "List of items:" for num in sorted(toSeq(nums)):
echo "– ", Items[num].name
echo "" echo "Total weight: ", weight echo "Total value: ", value </lang>
- Output:
List of items: – map – compass – water – sandwich – glucose – banana – suntan cream – waterproof trousers – waterproof overclothes – note-case – sunglasses – socks Total weight: 396 Total value: 1030
OCaml
A brute force solution: <lang ocaml>let items = [
"map", 9, 150; "compass", 13, 35; "water", 153, 200; "sandwich", 50, 160; "glucose", 15, 60; "tin", 68, 45; "banana", 27, 60; "apple", 39, 40; "cheese", 23, 30; "beer", 52, 10; "suntan cream", 11, 70; "camera", 32, 30; "t-shirt", 24, 15; "trousers", 48, 10; "umbrella", 73, 40; "waterproof trousers", 42, 70; "waterproof overclothes", 43, 75; "note-case", 22, 80; "sunglasses", 7, 20; "towel", 18, 12; "socks", 4, 50; "book", 30, 10;
]
let comb =
List.fold_left (fun acc x -> let acc2 = List.rev_map (fun li -> x::li) acc in
List.rev_append acc acc2) [[]]
let score =
List.fold_left (fun (w_tot,v_tot) (_,w,v) -> (w + w_tot, v + v_tot)) (0,0)
let () =
let combs = comb items in
let vals = List.rev_map (fun this -> (score this, this)) combs in
let poss = List.filter (fun ((w,_), _) -> w <= 400) vals in
let _, res = List.fold_left (fun (((_,s1),_) as v1) (((_,s2),_) as v2) ->
if s2 > s1 then v2 else v1)
(List.hd poss) (List.tl poss) in
List.iter (fun (name,_,_) -> print_endline name) res;
- </lang>
Oz
Using constraint programming. <lang oz>declare
%% maps items to pairs of Weight(hectogram) and Value
Problem = knapsack('map':9#150
'compass':13#35
'water':153#200
'sandwich':50#160
'glucose':15#60
'tin':68#45
'banana':27#60
'apple':39#40
'cheese':23#30
'beer':52#10
'suntan cream':11#70
'camera':32#30
't-shirt':24#15
'trousers':48#10
'umbrella':73#40
'waterproof trousers':42#70
'waterproof overclothes':43#75
'note-case':22#80
'sunglasses':7#20
'towel':18#12
'socks':4#50
'book':30#10
)
%% item -> Weight
Weights = {Record.map Problem fun {$ X} X.1 end}
%% item -> Value
Values = {Record.map Problem fun {$ X} X.2 end}
proc {Knapsack Solution}
%% a solution maps items to finite domain variables
%% with the domain {0,1}
Solution = {Record.map Problem fun {$ _} {FD.int 0#1} end}
%% no more than 400 hectograms
{FD.sumC Weights Solution '=<:' 400}
%% search through valid solutions
{FD.distribute naive Solution}
end
proc {PropagateLargerValue Old New}
%% propagate that new solutions must yield a higher value
%% than previously found solutions (essential for performance)
{FD.sumC Values New '>:' {Value Old}}
end
fun {Value Candidate}
{Record.foldL {Record.zip Candidate Values Number.'*'} Number.'+' 0}
end
fun {Weight Candidate}
{Record.foldL {Record.zip Candidate Weights Number.'*'} Number.'+' 0}
end
[Best] = {SearchBest Knapsack PropagateLargerValue}
in
{System.showInfo "Items: "}
{ForAll
{Record.arity {Record.filter Best fun {$ T} T == 1 end}}
System.showInfo}
{System.printInfo "\n"}
{System.showInfo "total value: "#{Value Best}}
{System.showInfo "total weight: "#{Weight Best}}</lang>
- Output:
Items: banana compass glucose map note-case sandwich socks sunglasses suntan cream water waterproof overclothes waterproof trousers total value: 1030 total weight: 396
Typically runs in less than 150 milliseconds.
Pascal
Uses a stringlist to store the items. I used the algorithm given on Wikipedia (Knapsack problem) to find the maximum value. It is written in pseudocode that translates very easily to Pascal. <lang pascal> program project1; uses
sysutils, classes, math;
const
MaxWeight = 400; N = 21;
type
TMaxArray = array[0..N, 0..MaxWeight] of integer;
TEquipment = record Description : string; Weight : integer; Value : integer; end;
TEquipmentList = array[1..N] of TEquipment;
var
M:TMaxArray; MaxValue, WeightLeft, i, j, Sum : integer; S,KnapSack:TStringList; L:string; List:TEquipmentList;
begin
//Put all the items into an array called List L:='map ,9 ,150,compass ,13 ,35 ,water ,153 ,200 ,sandwich,50 ,160 ,glucose ,15 ,60 ,tin,68 ,45 ,banana,27,60 ,apple ,39 ,40 ,cheese ,23 ,30 ,beer ,52 ,10 ,suntancreme ,11 ,70 ,camera ,32 ,30 ,T-shirt ,24 ,15 ,trousers ,48 ,40 ,waterprooftrousers ,42 ,70 ,waterproofoverclothes ,43 ,75 ,notecase ,22 ,80 ,sunglasses ,7 ,20 ,towel ,18 ,12 ,socks ,4 ,50 ,book ,30 ,10'; S:=TStringList.create; S.Commatext:=L;
For i:= 1 to N do
begin
List[i].Description:=S[3*i - 3];
List[i].Weight:=strtoint(S[3*i - 2]);
List[i].Value:=strtoint(S[3*i - 1]);
end;
//create M, a table linking the possible items for each weight
//and recording the value at that point
for j := 0 to MaxWeight do
M[0, j] := 0
for i := 1 to N do
for j := 0 to MaxWeight do
if List[i].weight > j then
M[i, j] := M[i-1, j]
else
M[i, j] := max(M[i-1, j], M[i-1, j-List[i].weight] + List[i].value);
//get the highest total value by testing every value in table M
for i:=1 to N do
for j:= 0 to MaxWeight do
If M[i,j] > MaxValue then
MaxValue := m[i,j];
writeln('Highest total value : ',MaxValue);
//Work backwards through the items to find those items that go in the Knapsack (a stringlist)
KnapSack := TStringList.create;
WeightLeft := MaxWeight;
For i:= N downto 1 do
if M[i,WeightLeft] = MaxValue then
if M[i-1, WeightLeft - List[i].Weight] = MaxValue - List[i].Value then
begin
Knapsack.add(List[i].Description + ' ' + IntToStr(List[i].Weight)+ ' ' + inttostr(List[i].Value));
MaxValue := MaxValue - List[i].Value;
WeightLeft := WeightLeft - List[i].Weight;
end
//Show the items in the knapsack
writeln('Number of items : ',KnapSack.count);
writeln('-------------------------');
For i:= KnapSack.count-1 downto 0 do
writeln(KnapSack[i]);
KnapSack.free;
S.free;
writeln('-------------------------');
writeln('done');
readln;
end. </lang>
Output
Highest total value : 1030 Number of items : 12 ------------------------- map 9 150 compass 13 35 water 153 200 sandwich 50 160 glucose 15 60 banana 27 60 suntancreme 11 70 waterprooftrousers 42 70 waterproofoverclothes 43 75 notecase 22 80 sunglasses 7 20 socks 4 50 ------------------------- done
Perl
The dynamic programming solution from Wikipedia. <lang perl>my $raw = <<'TABLE'; map 9 150 compass 13 35 water 153 200 sandwich 50 160 glucose 15 60 tin 68 45 banana 27 60 apple 39 40 cheese 23 30 beer 52 10 suntancream 11 70 camera 32 30 T-shirt 24 15 trousers 48 10 umbrella 73 40 waterproof trousers 42 70 waterproof overclothes 43 75 note-case 22 80 sunglasses 7 20 towel 18 12 socks 4 50 book 30 10 TABLE
my (@name, @weight, @value); for (split "\n", $raw) {
for ([ split /\t+/ ]) {
push @name, $_->[0];
push @weight, $_->[1];
push @value, $_->[2];
}
}
my $max_weight = 400; my @p = map [map undef, 0 .. 1+$max_weight], 0 .. $#name;
sub optimal {
my ($i, $w) = @_; return [0, []] if $i < 0; return $p[$i][$w] if $p[$i][$w];
if ($weight[$i] > $w) {
$p[$i][$w] = optimal($i - 1, $w)
} else {
my $x = optimal($i - 1, $w);
my $y = optimal($i - 1, $w - $weight[$i]);
if ($x->[0] > $y->[0] + $value[$i]) {
$p[$i][$w] = $x
} else {
$p[$i][$w] = [ $y->[0] + $value[$i], [ @{$y->[1]}, $name[$i] ]]
}
}
return $p[$i][$w]
}
my $solution = optimal($#name, $max_weight); print "$solution->[0]: @{$solution->[1]}\n";</lang>
- Output:
1030: map compass water sandwich glucose banana suntancream waterproof trousers waterproof overclothes note-case sunglasses socks
Phix
Trivial simplification of the Knapsack/Bounded solution. By careful ordering we can ensure we find the optimal solution first (see terminate flag).
-- demo\rosetta\knapsack0.exw with javascript_semantics bool terminate = false integer attempts = 0 function knapsack(sequence res, goodies, atom points, weight, at=1, sequence chosen={}) atom {?,witem,pitem} = goodies[at] integer n = iff(witem<=weight?1:0) chosen &= n points += n*pitem -- increase value weight -= n*witem -- decrease weight left if at=length(goodies) then attempts += 1 if length(res)=0 or res<{points,weight} then res = {points,weight,chosen} end if terminate = (n=1) else -- while n>=0 do -- full exhaustive search while n>=0 and not terminate do -- optimised res = knapsack(res,goodies,points,weight,at+1,deep_copy(chosen)) n -= 1 chosen[$] = n points -= pitem weight += witem end while end if return res end function function byweightedvalue(object a, b) -- sort by weight/value return compare(a[2]/a[3],b[2]/b[3]) -- nb other sort orders break the optimisation end function constant goodies = custom_sort(byweightedvalue,{ -- item weight value {"map", 9, 150}, {"compass", 13, 35 }, {"water", 153, 200}, {"sandwich", 50, 160}, {"glucose", 15, 60 }, {"tin", 68, 45 }, {"banana", 27, 60 }, {"apple", 39, 40 }, {"cheese", 23, 30 }, {"beer", 52, 10 }, {"suntan cream", 11, 70 }, {"camera", 32, 30 }, {"T-shirt", 24, 15 }, {"trousers", 48, 10 }, {"umbrella", 73, 40 }, {"waterproof trousers", 42, 70 }, {"waterproof overclothes", 43, 75 }, {"note-case", 22, 80 }, {"sunglasses", 7, 20 }, {"towel", 18, 12 }, {"socks", 4, 50 }, {"book", 30, 10 }}) atom t0 = time() object {points,weight,counts} = knapsack({},goodies,0,400) printf(1,"Value %d, weight %g [%d attempts, %3.2fs]:\n",{points,400-weight,attempts,time()-t0}) for i=1 to length(counts) do integer c = counts[i] if c then printf(1,"%s\n",{goodies[i][1]}) end if end for
- Output:
Value 1030, weight 396 [9 attempts, 0.00s]: map socks suntan cream glucose note-case sandwich sunglasses compass banana waterproof overclothes waterproof trousers water
without the optimisation (ie "and not terminate" removed, full exhaustive search, further lines as above):
Value 1030, weight 396 [1216430 attempts, 0.84s]:
PHP
<lang php>#########################################################
- 0-1 Knapsack Problem Solve with memoization optimize and index returns
- $w = weight of item
- $v = value of item
- $i = index
- $aW = Available Weight
- $m = Memo items array
- PHP Translation from Python, Memoization,
- and index return functionality added by Brian Berneker
-
function knapSolveFast2($w, $v, $i, $aW, &$m) {
global $numcalls;
$numcalls ++;
// echo "Called with i=$i, aW=$aW
";
// Return memo if we have one if (isset($m[$i][$aW])) { return array( $m[$i][$aW], $m['picked'][$i][$aW] ); } else {
// At end of decision branch if ($i == 0) { if ($w[$i] <= $aW) { // Will this item fit? $m[$i][$aW] = $v[$i]; // Memo this item $m['picked'][$i][$aW] = array($i); // and the picked item return array($v[$i],array($i)); // Return the value of this item and add it to the picked list
} else { // Won't fit $m[$i][$aW] = 0; // Memo zero $m['picked'][$i][$aW] = array(); // and a blank array entry... return array(0,array()); // Return nothing } }
// Not at end of decision branch.. // Get the result of the next branch (without this one) list ($without_i, $without_PI) = knapSolveFast2($w, $v, $i-1, $aW, $m);
if ($w[$i] > $aW) { // Does it return too many?
$m[$i][$aW] = $without_i; // Memo without including this one $m['picked'][$i][$aW] = $without_PI; // and a blank array entry... return array($without_i, $without_PI); // and return it
} else {
// Get the result of the next branch (WITH this one picked, so available weight is reduced) list ($with_i,$with_PI) = knapSolveFast2($w, $v, ($i-1), ($aW - $w[$i]), $m); $with_i += $v[$i]; // ..and add the value of this one..
// Get the greater of WITH or WITHOUT if ($with_i > $without_i) { $res = $with_i; $picked = $with_PI; array_push($picked,$i); } else { $res = $without_i; $picked = $without_PI; }
$m[$i][$aW] = $res; // Store it in the memo $m['picked'][$i][$aW] = $picked; // and store the picked item return array ($res,$picked); // and then return it } } }
$items4 = array("map","compass","water","sandwich","glucose","tin","banana","apple","cheese","beer","suntan cream","camera","t-shirt","trousers","umbrella","waterproof trousers","waterproof overclothes","note-case","sunglasses","towel","socks","book"); $w4 = array(9,13,153,50,15,68,27,39,23,52,11,32,24,48,73,42,43,22,7,18,4,30); $v4 = array(150,35,200,160,60,45,60,40,30,10,70,30,15,10,40,70,75,80,20,12,50,10);
- Initialize
$numcalls = 0; $m = array(); $pickedItems = array();
- Solve
list ($m4,$pickedItems) = knapSolveFast2($w4, $v4, sizeof($v4) -1, 400, $m);
- Display Result
echo "Items:
".join(", ",$items4)."
";
echo "Max Value Found:
$m4 (in $numcalls calls)
";
echo "Array Indices:
".join(",",$pickedItems)."
";
echo "Chosen Items:
";
echo "
"; echo "";$totalVal = $totalWt = 0; foreach($pickedItems as $key) { $totalVal += $v4[$key]; $totalWt += $w4[$key];
echo "";}
echo ""; echo "| Item | Value | Weight |
| ".$items4[$key]." | ".$v4[$key]." | ".$w4[$key]." |
| Totals | $totalVal | $totalWt |
";</lang>
- Output:
map, compass, water, sandwich, glucose, tin, banana, apple, cheese, beer, suntan cream, camera, t-shirt, trousers, umbrella, waterproof trousers, waterproof overclothes, note-case, sunglasses, towel, socks, book
Max Value Found:
1030 (in 8725 calls)
Array Indices:
0,1,2,3,4,6,10,15,16,17,18,20
Chosen Items:
| Item | Value | Weight |
| map | 150 | 9 |
| compass | 35 | 13 |
| water | 200 | 153 |
| sandwich | 160 | 50 |
| glucose | 60 | 15 |
| banana | 60 | 27 |
| suntan cream | 70 | 11 |
| waterproof trousers | 70 | 42 |
| waterproof overclothes | 75 | 43 |
| note-case | 80 | 22 |
| sunglasses | 20 | 7 |
| socks | 50 | 4 |
| Totals | 1030 | 396 |
Minimal PHP Algorithm for totals only translated from Python version as discussed in the YouTube posted video at: http://www.youtube.com/watch?v=ZKBUu_ahSR4 <lang php>#########################################################
- 0-1 Knapsack Problem Solve
- $w = weight of item
- $v = value of item
- $i = index
- $aW = Available Weight
- PHP Translation by Brian Berneker
function knapSolve($w,$v,$i,$aW) {
global $numcalls;
$numcalls ++;
// echo "Called with i=$i, aW=$aW
";
if ($i == 0) { if ($w[$i] <= $aW) { return $v[$i]; } else { return 0; } }
$without_i = knapSolve($w, $v, $i-1, $aW); if ($w[$i] > $aW) { return $without_i; } else { $with_i = $v[$i] + knapSolve($w, $v, ($i-1), ($aW - $w[$i])); return max($with_i, $without_i); }
}
- 0-1 Knapsack Problem Solve (with "memo"-ization optimization)
- $w = weight of item
- $v = value of item
- $i = index
- $aW = Available Weight
- $m = 'memo' array
- PHP Translation by Brian Berneker
function knapSolveFast($w,$v,$i,$aW,&$m) { // Note: We use &$m because the function writes to the $m array
global $numcalls;
$numcalls ++;
// echo "Called with i=$i, aW=$aW
";
// Return memo if we have one if (isset($m[$i][$aW])) { return $m[$i][$aW]; } else {
if ($i == 0) { if ($w[$i] <= $aW) { $m[$i][$aW] = $v[$i]; // save memo return $v[$i]; } else { $m[$i][$aW] = 0; // save memo return 0; } }
$without_i = knapSolveFast($w, $v, $i-1, $aW,$m); if ($w[$i] > $aW) { $m[$i][$aW] = $without_i; // save memo return $without_i; } else { $with_i = $v[$i] + knapSolveFast($w, $v, ($i-1), ($aW - $w[$i]),$m); $res = max($with_i, $without_i); $m[$i][$aW] = $res; // save memo return $res; } } }
$w3 = array(1, 1, 1, 2, 2, 2, 4, 4, 4, 44, 96, 96, 96);
$v3 = array(1, 1, 1, 2, 2, 2, 4, 4, 4, 44, 96, 96, 96);
$numcalls = 0;
$m = array();
$m3 = knapSolveFast($w3, $v3, sizeof($v3) -1, 54,$m);
print_r($w3); echo "
FAST: ";
echo "Max: $m3 ($numcalls calls)
";
$numcalls = 0;
$m = array();
$m3 = knapSolve($w3, $v3, sizeof($v3) -1, 54 );
print_r($w3); echo "
";
echo "Max: $m3 ($numcalls calls)
";</lang>
- Output:
Array ( [0] => 1 [1] => 1 [2] => 1 [3] => 2 [4] => 2 [5] => 2 [6] => 4 [7] => 4 [8] => 4 [9] => 44 [10] => 96 [11] => 96 [12] => 96 ) FAST: Max: 54 (191 calls) Array ( [0] => 1 [1] => 1 [2] => 1 [3] => 2 [4] => 2 [5] => 2 [6] => 4 [7] => 4 [8] => 4 [9] => 44 [10] => 96 [11] => 96 [12] => 96 ) Max: 54 (828 calls)
Picat
<lang Picat>import mip,util.
go =>
items(AllItems,MaxTotalWeight), [Items,Weights,Values] = transpose(AllItems), knapsack_01(Weights,Values,MaxTotalWeight, X,TotalWeight,TotalValues), print_solution(Items,Weights,Values, X,TotalWeight,TotalValues), nl.
% Print the solution print_solution(Items,Weights,Values, X,TotalWeight,TotalValues) =>
println("\nThese are the items to pick:"),
println(" Item Weight Value"),
foreach(I in 1..Items.len)
if X[I] == 1 then
printf("* %-25w %3d %3d\n", Items[I],Weights[I], Values[I])
end
end,
nl,
printf("Total weight: %d\n", TotalWeight),
printf("Total value: %d\n", TotalValues),
nl.
% Solve the knapsack problem knapsack_01(Weights,Values,MaxTotalWeight, X,TotalWeight,TotalValues) =>
NumItems = length(Weights),
% Variables X = new_list(NumItems), X :: 0..1,
% Constraints scalar_product(Weights,X,TotalWeight), scalar_product(Values,X,TotalValues), TotalWeight #=< MaxTotalWeight,
% Search Vars = X ++ [TotalWeight, TotalValues], solve($[max(TotalValues)], Vars).
% data items(Items,MaxTotalWeight) =>
% Item Weight Value
Items = [["map", 9, 150],
["compass", 13, 35],
["water", 153, 200],
["sandwich", 50, 160],
["glucose", 15, 60],
["tin", 68, 45],
["banana", 27, 60],
["apple", 39, 40],
["cheese", 23, 30],
["beer", 52, 10],
["suntancream", 11, 70],
["camera", 32, 30],
["T-shirt", 24, 15],
["trousers", 48, 10],
["umbrella", 73, 40],
["waterproof trousers", 42, 70],
["waterproof overclothes", 43, 75],
["note-case", 22, 80],
["sunglasses", 7, 20],
["towel", 18, 12],
["socks", 4, 50],
["book", 30, 10]],
MaxTotalWeight = 400.</lang>
- Output:
These are the items to pick: Item Weight Value * map 9 150 * compass 13 35 * water 153 200 * sandwich 50 160 * glucose 15 60 * banana 27 60 * suntancream 11 70 * waterproof trousers 42 70 * waterproof overclothes 43 75 * note-case 22 80 * sunglasses 7 20 * socks 4 50 Total weight: 396 Total value: 1030
PicoLisp
<lang PicoLisp>(de *Items
("map" 9 150) ("compass" 13 35)
("water" 153 200) ("sandwich" 50 160)
("glucose" 15 60) ("tin" 68 45)
("banana" 27 60) ("apple" 39 40)
("cheese" 23 30) ("beer" 52 10)
("suntan cream" 11 70) ("camera" 32 30)
("t-shirt" 24 15) ("trousers" 48 10)
("umbrella" 73 40) ("waterproof trousers" 42 70)
("waterproof overclothes" 43 75) ("note-case" 22 80)
("sunglasses" 7 20) ("towel" 18 12)
("socks" 4 50) ("book" 30 10) )
- Dynamic programming solution
(de knapsack (Lst W)
(when Lst
(cache '*KnapCache (cons W Lst)
(let X (knapsack (cdr Lst) W)
(if (ge0 (- W (cadar Lst)))
(let Y (cons (car Lst) (knapsack (cdr Lst) @))
(if (> (sum caddr X) (sum caddr Y)) X Y) )
X ) ) ) ) )
(let K (knapsack *Items 400)
(for I K
(apply tab I (3 -24 6 6) NIL) )
(tab (27 6 6) NIL (sum cadr K) (sum caddr K)) )</lang>
- Output:
map 9 150
compass 13 35
water 153 200
sandwich 50 160
glucose 15 60
banana 27 60
suntan cream 11 70
waterproof trousers 42 70
waterproof overclothes 43 75
note-case 22 80
sunglasses 7 20
socks 4 50
396 1030
Prolog
Using the clpfd library
<lang Prolog>:- use_module(library(clpfd)).
knapsack :-
L = [
item(map, 9, 150),
item(compass, 13, 35),
item(water, 153, 200),
item(sandwich, 50, 160),
item(glucose, 15, 60),
item(tin, 68, 45),
item(banana, 27, 60),
item(apple, 39, 40),
item(cheese, 23, 30),
item(beer, 52, 10),
item('suntan cream', 11, 70),
item(camera, 32, 30),
item('t-shirt', 24, 15),
item(trousers, 48, 10),
item(umbrella, 73, 40),
item('waterproof trousers', 42, 70),
item('waterproof overclothes', 43, 75),
item('note-case',22, 80),
item(sunglasses, 7, 20),
item(towel, 18, 12),
item(socks, 4, 50),
item(book, 30, 10 )],
length(L, N),
length(R, N),
R ins 0..1,
maplist(arg(2), L, LW),
maplist(arg(3), L, LV),
scalar_product(LW, R, #=<, 400),
scalar_product(LV, R, #=, VM),
labeling([max(VM)], R),
scalar_product(LW, R, #=, WM),
%% affichage des résultats
compute_lenword(L, 0, Len),
sformat(A1, '~~w~~t~~~w|', [Len]),
sformat(A2, '~~t~~w~~~w|', [4]),
sformat(A3, '~~t~~w~~~w|', [5]),
print_results(A1,A2,A3, L, R, WM, VM).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % to show the results in a good way %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % compute_lenword([], N, N). compute_lenword([item(Name, _, _)|T], N, NF):-
atom_length(Name, L),
( L > N -> N1 = L; N1 = N),
compute_lenword(T, N1, NF).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % print_results(A1,A2,A3, [], [], WM, WR) :-
sformat(W1, A1, [' ']),
sformat(W2, A2, [WM]),
sformat(W3, A3, [WR]),
format('~w~w~w~n', [W1,W2,W3]).
print_results(A1,A2,A3, [_H|T], [0|TR], WM, VM) :-
print_results(A1,A2,A3, T, TR, WM, VM).
print_results(A1, A2, A3, [item(Name, W, V)|T], [1|TR], WM, VM) :-
sformat(W1, A1, [Name]),
sformat(W2, A2, [W]),
sformat(W3, A3, [V]),
format('~w~w~w~n', [W1,W2,W3]),
print_results(A1, A2, A3, T, TR, WM, VM).</lang>
- Output:
?- knapsack
map 9 150
compass 13 35
water 153 200
sandwich 50 160
glucose 15 60
banana 27 60
suntan cream 11 70
waterproof trousers 42 70
waterproof overclothes 43 75
note-case 22 80
sunglasses 7 20
socks 4 50
396 1030
Using the simplex library
Library written by Markus Triska. The problem is solved in about 3 seconds. <lang Prolog>:- use_module(library(simplex)).
knapsack :- L = [ (map, 9, 150), (compass, 13, 35), (water, 153, 200), (sandwich, 50, 160), (glucose, 15, 60), (tin, 68, 45), (banana, 27, 60), (apple, 39, 40), (cheese, 23, 30), (beer, 52, 10), ('suntan cream', 11, 70), (camera, 32, 30), ('t-shirt', 24, 15), (trousers, 48, 10), (umbrella, 73, 40), ('waterproof trousers', 42, 70), ('waterproof overclothes', 43, 75), ('note-case',22, 80), (sunglasses, 7, 20), (towel, 18, 12), (socks, 4, 50), (book, 30, 10 )], gen_state(S0), length(L, N), numlist(1, N, LN), time(( create_constraint_N(LN, S0, S1), maplist(create_constraint_WV, LN, L, LW, LV), constraint(LW =< 400, S1, S2), maximize(LV, S2, S3) )), compute_lenword(L, 0, Len), sformat(A1, '~~w~~t~~~w|', [Len]), sformat(A2, '~~t~~w~~~w|', [4]), sformat(A3, '~~t~~w~~~w|', [5]), print_results(S3, A1,A2,A3, L, LN, 0, 0).
create_constraint_N([], S, S).
create_constraint_N([HN|TN], S1, SF) :- constraint(integral(x(HN)), S1, S2), constraint([x(HN)] =< 1, S2, S3), constraint([x(HN)] >= 0, S3, S4), create_constraint_N(TN, S4, SF).
create_constraint_WV(N, (_, W, V), W * x(N), V * x(N)).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % compute_lenword([], N, N). compute_lenword([(Name, _, _)|T], N, NF):- atom_length(Name, L), ( L > N -> N1 = L; N1 = N), compute_lenword(T, N1, NF).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % print_results(_S, A1, A2, A3, [], [], WM, VM) :- sformat(W1, A1, [' ']), sformat(W2, A2, [WM]), sformat(W3, A3, [VM]), format('~w~w~w~n', [W1,W2,W3]).
print_results(S, A1, A2, A3, [(Name, W, V)|T], [N|TN], W1, V1) :-
variable_value(S, x(N), X),
( X = 0 -> W1 = W2, V1 = V2
; sformat(S1, A1, [Name]),
sformat(S2, A2, [W]),
sformat(S3, A3, [V]),
format('~w~w~w~n', [S1,S2,S3]),
W2 is W1 + W,
V2 is V1 + V),
print_results(S, A1, A2, A3, T, TN, W2, V2).</lang>
PureBasic
Solution uses memoization. <lang PureBasic>Structure item
name.s weight.i ;units are dekagrams (dag) Value.i
EndStructure
Structure memo
Value.i List picked.i()
EndStructure
Global itemCount = 0 ;this will be increased as needed to match count Global Dim items.item(itemCount)
Procedure addItem(name.s, weight, Value)
If itemCount >= ArraySize(items()) Redim items.item(itemCount + 10) EndIf With items(itemCount) \name = name \weight = weight \Value = Value EndWith itemCount + 1
EndProcedure
- build item list
addItem("map", 9, 150) addItem("compass", 13, 35) addItem("water", 153, 200) addItem("sandwich", 50, 160) addItem("glucose", 15, 60) addItem("tin", 68, 45) addItem("banana", 27, 60) addItem("apple", 39, 40) addItem("cheese", 23, 30) addItem("beer", 52, 10) addItem("suntan cream", 11, 70) addItem("camera", 32, 30) addItem("t-shirt", 24, 15) addItem("trousers", 48, 10) addItem("umbrella", 73, 40) addItem("waterproof trousers", 42, 70) addItem("waterproof overclothes", 43, 75) addItem("note-case", 22, 80) addItem("sunglasses", 7, 20) addItem("towel", 18, 12) addItem("socks", 4, 50) addItem("book", 30, 10)
Procedure knapsackSolveFast(Array item.item(1), i, aw, Map m.memo())
Protected.memo without_i, with_i, result, *tmp, memoIndex.s = Hex((i << 16) + aw, #PB_Long)
If FindMapElement(m(), memoIndex)
ProcedureReturn @m()
Else
If i = 0
If item(0)\weight <= aw
;item fits
m(memoIndex)\Value = item(0)\Value ;memo this item's value
AddElement(m()\picked())
m()\picked() = 0 ;memo item's index also
Else
;item doesn't fit, memo a zero Value
m(memoIndex)\Value = 0
EndIf
ProcedureReturn @m()
EndIf
;test if a greater value results with or without item included
*tmp = knapsackSolveFast(item(), i - 1, aw, m()) ;find value without this item
CopyStructure(*tmp, @without_i, memo)
If item(i)\weight > aw
;item weighs too much, memo without including this item
m(memoIndex) = without_i
ProcedureReturn @m()
Else
*tmp = knapsackSolveFast(item(), i - 1, aw - item(i)\weight, m()) ;find value when item is included
CopyStructure(*tmp, @with_i, memo)
with_i\Value + item(i)\Value
AddElement(with_i\picked())
with_i\picked() = i ;add item to with's picked list
EndIf
;set the result to the larger value
If with_i\Value > without_i\Value
result = with_i
Else
result = without_i
EndIf
m(memoIndex) = result ;memo the result
ProcedureReturn @m()
EndIf
EndProcedure
Procedure.s knapsackSolve(Array item.item(1), i, aw)
Protected *result.memo, output.s, totalWeight
NewMap m.memo()
*result = knapsackSolveFast(item(), i, aw, m())
output = "Knapsack:" + #CRLF$
ForEach *result\picked()
output + LSet(item(*result\picked())\name, 24) + RSet(Str(item(*result\picked())\weight), 5) + RSet(Str(item(*result\picked())\Value), 5) + #CRLF$
totalWeight + item(*result\picked())\weight
Next
output + LSet("TOTALS:", 24) + RSet(Str(totalWeight), 5) + RSet(Str(*result\Value), 5)
ProcedureReturn output
EndProcedure
If OpenConsole()
#maxWeight = 400 Define *result.memo PrintN(knapsackSolve(items(), itemCount - 1, #maxWeight)) Print(#CRLF$ + #CRLF$ + "Press ENTER to exit"): Input() CloseConsole()
EndIf </lang>
- Output:
Knapsack: map 9 150 compass 13 35 water 153 200 sandwich 50 160 glucose 15 60 banana 27 60 suntan cream 11 70 waterproof trousers 42 70 waterproof overclothes 43 75 note-case 22 80 sunglasses 7 20 socks 4 50 TOTALS: 396 1030
QB64
Russian Knapsack 0-1 synthesizes all ciphers from 0 & 1 adding left +1 register and 0 remain on left in cipher
Number of comparisons decreases from N! to 2^N for example N=8 N!=40320 >> 2^N=256
Random values origin are automatically assigned create integral of quantity and quality
Program write results to qb64 directory <lang QB64>Open "knapsack.txt" For Output As #1 N=7: L=5: a = 2^(N+1): Randomize Timer 'knapsack.bas DANILIN Dim L(N), C(N), j(N), q(a), q$(a), d(a)
For m=a-1 To (a-1)/2 Step -1: g=m: Do ' cipher 0-1
q$(m)=LTrim$(Str$(g Mod 2))+q$(m) g=g\2: Loop Until g=0 q$(m)=Mid$(q$(m), 2, Len(q$(m)))
Next
For i=1 To N: L(i)=Int(Rnd*3+1) ' mass & cost C(i)=10+Int(Rnd*9): Print #1, i, L(i), C(i): Next ' origin
For h=a-1 To (a-1)/2 Step -1
For k=1 To N: j(k)=Val(Mid$(q$(h), k, 1)) ' j() = cipher
q(h)=q(h)+L(k)*j(k)*C(k) ' 0 or 1
d(h)=d(h)+L(k)*j(k)
Next
If d(h) <= L Then Print #1, d(h), q(h), q$(h)
Next max=0: m=1: For i=1 To a
If d(i)<=L Then If q(i) > max Then max=q(i): m=i
Next Print #1,: Print #1, d(m), q(m), q$(m): End</lang>
<lang QB64>Main thing is very brief and clear to even all
Results is reduced manually: </lang>
- Output:
# Mass Cost 1 2 17 2 2 14 3 2 17 4 1 11 5 2 18 6 3 14 7 3 10 Mass Cost Chifer 5 73 1101000 2 28 0100000 5 81 0011100 !!! 3 45 0011000 5 76 0010010 2 36 0000100 Mass MAX Chifer 5 81 0011100
Python
Brute force algorithm
<lang python>from itertools import combinations
def anycomb(items):
' return combinations of any length from the items '
return ( comb
for r in range(1, len(items)+1)
for comb in combinations(items, r)
)
def totalvalue(comb):
' Totalise a particular combination of items'
totwt = totval = 0
for item, wt, val in comb:
totwt += wt
totval += val
return (totval, -totwt) if totwt <= 400 else (0, 0)
items = (
("map", 9, 150), ("compass", 13, 35), ("water", 153, 200), ("sandwich", 50, 160),
("glucose", 15, 60), ("tin", 68, 45), ("banana", 27, 60), ("apple", 39, 40),
("cheese", 23, 30), ("beer", 52, 10), ("suntan cream", 11, 70), ("camera", 32, 30),
("t-shirt", 24, 15), ("trousers", 48, 10), ("umbrella", 73, 40),
("waterproof trousers", 42, 70), ("waterproof overclothes", 43, 75),
("note-case", 22, 80), ("sunglasses", 7, 20), ("towel", 18, 12),
("socks", 4, 50), ("book", 30, 10),
)
bagged = max( anycomb(items), key=totalvalue) # max val or min wt if values equal print("Bagged the following items\n " +
'\n '.join(sorted(item for item,_,_ in bagged)))
val, wt = totalvalue(bagged) print("for a total value of %i and a total weight of %i" % (val, -wt))</lang>
- Output:
Bagged the following items banana compass glucose map note-case sandwich socks sunglasses suntan cream water waterproof overclothes waterproof trousers for a total value of 1030 and a total weight of 396
Dynamic programming solution
<lang python>try:
xrange
except:
xrange = range
def totalvalue(comb):
' Totalise a particular combination of items'
totwt = totval = 0
for item, wt, val in comb:
totwt += wt
totval += val
return (totval, -totwt) if totwt <= 400 else (0, 0)
items = (
("map", 9, 150), ("compass", 13, 35), ("water", 153, 200), ("sandwich", 50, 160),
("glucose", 15, 60), ("tin", 68, 45), ("banana", 27, 60), ("apple", 39, 40),
("cheese", 23, 30), ("beer", 52, 10), ("suntan cream", 11, 70), ("camera", 32, 30),
("t-shirt", 24, 15), ("trousers", 48, 10), ("umbrella", 73, 40),
("waterproof trousers", 42, 70), ("waterproof overclothes", 43, 75),
("note-case", 22, 80), ("sunglasses", 7, 20), ("towel", 18, 12),
("socks", 4, 50), ("book", 30, 10),
)
def knapsack01_dp(items, limit):
table = [[0 for w in range(limit + 1)] for j in xrange(len(items) + 1)]
for j in xrange(1, len(items) + 1):
item, wt, val = items[j-1]
for w in xrange(1, limit + 1):
if wt > w:
table[j][w] = table[j-1][w]
else:
table[j][w] = max(table[j-1][w],
table[j-1][w-wt] + val)
result = []
w = limit
for j in range(len(items), 0, -1):
was_added = table[j][w] != table[j-1][w]
if was_added:
item, wt, val = items[j-1]
result.append(items[j-1])
w -= wt
return result
bagged = knapsack01_dp(items, 400)
print("Bagged the following items\n " +
'\n '.join(sorted(item for item,_,_ in bagged)))
val, wt = totalvalue(bagged) print("for a total value of %i and a total weight of %i" % (val, -wt))</lang>
Recursive dynamic programming algorithm
<lang python>def total_value(items, max_weight):
return sum([x[2] for x in items]) if sum([x[1] for x in items]) <= max_weight else 0
cache = {} def solve(items, max_weight):
if not items:
return ()
if (items,max_weight) not in cache:
head = items[0]
tail = items[1:]
include = (head,) + solve(tail, max_weight - head[1])
dont_include = solve(tail, max_weight)
if total_value(include, max_weight) > total_value(dont_include, max_weight):
answer = include
else:
answer = dont_include
cache[(items,max_weight)] = answer
return cache[(items,max_weight)]
items = (
("map", 9, 150), ("compass", 13, 35), ("water", 153, 200), ("sandwich", 50, 160),
("glucose", 15, 60), ("tin", 68, 45), ("banana", 27, 60), ("apple", 39, 40),
("cheese", 23, 30), ("beer", 52, 10), ("suntan cream", 11, 70), ("camera", 32, 30),
("t-shirt", 24, 15), ("trousers", 48, 10), ("umbrella", 73, 40),
("waterproof trousers", 42, 70), ("waterproof overclothes", 43, 75),
("note-case", 22, 80), ("sunglasses", 7, 20), ("towel", 18, 12),
("socks", 4, 50), ("book", 30, 10),
)
max_weight = 400
solution = solve(items, max_weight) print "items:" for x in solution:
print x[0]
print "value:", total_value(solution, max_weight) print "weight:", sum([x[1] for x in solution])</lang>
Python Russian Binary ciphers
Russian Knapsack 0-1 synthesizes all ciphers from 0 & 1 adding left +1 register and 0 remain on left in cipher
Number of comparisons decreases from N! to 2^N for example N=8 N!=40320 >> 2^N=256
Random values origin are automatically assigned create integral of quantity and quality
<lang python>n=5; N=n+1; G=5; a=2**N # KNAPSACK 0-1 DANILIN L=[];C=[];e=[];j=[];q=[];s=[] # rextester.com/BCKP19591 d=[];L=[1]*n;C=[1]*n;e=[1]*a j=[1]*n;q=[0]*a;s=[0]*a;d=[0]*a
from random import randint for i in range(0,n):
L[i]=randint(1,3) C[i]=10+randint(1,9) print(i+1,L[i],C[i])
print()
for h in range(a-1,(a-1)//2,-1):
b=str(bin(h))
e[h]=b[3:len(b)]
for k in range (n):
j[k]=int(e[h][k])
q[h]=q[h]+L[k]*j[k]*C[k]
d[h]=d[h]+L[k]*j[k]
if d[h]<= G:
print(e[h], G, d[h], q[h])
print()
max=0; m=1 for i in range(a):
if d[i]<=G and q[i]>max:
max=q[i]; m=i
print (d[m], q[m], e[m])</lang>
- Output:
# Mass Cost 1 2 12 2 3 17 3 1 14 4 3 17 5 1 13 Chifer Mass Cost 11000 5 5 75 10101 5 4 51 01001 5 4 64 00111 5 5 78 !!! 00110 5 4 65 00101 5 2 27 00000 5 0 0 Mass MAX Chifer 5 78 00111
R
<lang r> Full_Data<-structure(list(item = c("map", "compass", "water", "sandwich", "glucose", "tin", "banana", "apple", "cheese", "beer", "suntan_cream", "camera", "T-shirt", "trousers", "umbrella", "waterproof_trousers", "waterproof_overclothes", "note-case", "sunglasses", "towel", "socks", "book"), weigth = c(9, 13, 153, 50, 15, 68, 27, 39, 23, 52, 11, 32, 24, 48, 73, 42, 43, 22, 7, 18, 4, 30), value = c(150, 35, 200, 160, 60, 45, 60, 40, 30, 10, 70, 30, 15, 10, 40, 70, 75, 80, 20, 12, 50, 10)), .Names = c("item", "weigth", "value" ), row.names = c(NA, 22L), class = "data.frame")
Bounded_knapsack<-function(Data,W)
{
K<-matrix(NA,nrow=W+1,ncol=dim(Data)[1]+1)
0->K[1,]->K[,1]
matrix_item<-matrix(,nrow=W+1,ncol=dim(Data)[1]+1)
for(j in 1:dim(Data)[1])
{
for(w in 1:W)
{
wj<-Data$weigth[j]
item<-Data$item[j]
value<-Data$value[j]
if( wj > w )
{
K[w+1,j+1]<-K[w+1,j]
matrix_item[w+1,j+1]<-matrix_item[w+1,j]
}
else
{
if( K[w+1,j] >= K[w+1-wj,j]+value )
{
K[w+1,j+1]<-K[w+1,j]
matrix_item[w+1,j+1]<-matrix_item[w+1,j]
}
else
{
K[w+1,j+1]<-K[w+1-wj,j]+value
matrix_item[w+1,j+1]<-item
}
}
}
}
return(list(K=K,Item=matrix_item))
}
backtracking<-function(knapsack, Data) { W<-dim(knapsack$K)[1] itens<-c() col<-dim(knapsack$K)[2] selected_item<-knapsack$Item[W,col] while(selected_item!=) { selected_item<-knapsack$Item[W,col] if(selected_item!=) { selected_item_value<-Data[Data$item == selected_item,] if(-knapsack$K[W - selected_item_value$weigth,col-1]+knapsack$K[W,col]==selected_item_value$value) { W <- W - selected_item_value$weigth itens<-c(itens,selected_item) } col <- col - 1 } } return(itens) }
print_output<-function(Data,W) { Bounded_knapsack(Data,W)->Knap backtracking(Knap, Data)->Items output<-paste('You must carry:', paste(Items, sep = ', '), sep=' ' ) return(output) }
print_output(Full_Data, 400)
</lang>
- Output:
<lang>
[1] "You must carry: socks" [2] "You must carry: sunglasses" [3] "You must carry: note-case" [4] "You must carry: waterproof_overclothes" [5] "You must carry: waterproof_trousers" [6] "You must carry: suntan_cream" [7] "You must carry: banana" [8] "You must carry: glucose" [9] "You must carry: sandwich"
[10] "You must carry: water" [11] "You must carry: compass" [12] "You must carry: map" </lang>
Racket
<lang racket>
- lang racket
- An ITEM a list of three elements
- a name, a weight, and, a value
- A SOLUTION to a knapsack01 problems is a list of three elements
- the total value, the total weight, and, names of the items to bag
(define (add i s) ; add an item i to the solution s
(match-define (list in iw iv) i) (match-define (list v w is) s) (list (+ v iv) (+ w iw) (cons in is)))
(define (knapsack max-weight items)
; return a solution to the knapsack01 problem
(define ht (make-hash)) ; (weight number-of-items) -> items
(define (get w no-items) (hash-ref ht (list w no-items) #f))
(define (update w is x) (hash-set! ht (list w (length is)) is) x)
(define (knapsack1 left items)
; return a solution to the (left, items) problem
(cond
; if there are no items, then bag no items:
[(empty? items) (list 0 0 '())]
; look up the best solution:
[(or (get left (length items))
; the solution haven't been cached, so we
; must compute it and update the cache:
(update
left items
(match items
; let us name the first item
[(cons (and (list i w v) x) more)
; if the first item weighs more than the space left,
; we simply find a solution, where it is omitted:
(cond [(> w left) (knapsack left more)]
; there is room for the first item, but
; we need to choose the best solutions
; between those with it and that without:
[else
(define without (knapsack left more))
(define value-without (first without))
(define with (knapsack (- left w) more))
(define value-with (+ (first with) v))
; choose the solutions with greatest value
(if (> value-without value-with)
without
(add x with))])])))]))
(knapsack1 max-weight items))
(knapsack 400
'((map 9 150) ; 9 is weight, 150 is value
(compass 13 35) (water 153 200) (sandwich 50 160)
(glucose 15 60) (tin 68 45)(banana 27 60) (apple 39 40)
(cheese 23 30) (beer 52 10) (cream 11 70) (camera 32 30)
(T-shirt 24 15) (trousers 48 10) (umbrella 73 40)
(trousers 42 70) (overclothes 43 75) (notecase 22 80)
(glasses 7 20) (towel 18 12) (socks 4 50) (book 30 10)))
</lang>
- Output:
<lang racket> '(1030 396 (map compass water sandwich glucose banana cream trousers overclothes notecase glasses socks)) </lang> Brute Force and Memoized Recursive Alternate <lang racket>
- lang racket
(define items '((map 9 150) (compass 13 35) (water 153 200) (sandwich 50 160)
(glucose 15 60) (tin 68 45)(banana 27 60) (apple 39 40)
(cheese 23 30) (beer 52 10) (cream 11 70) (camera 32 30)
(T-shirt 24 15) (trousers 48 10) (umbrella 73 40)
(trousers 42 70) (overclothes 43 75) (notecase 22 80)
(glasses 7 20) (towel 18 12) (socks 4 50) (book 30 10)))
(define max-weight 400)
(define (item-value item)
(caddr item))
(define (item-weight item)
(cadr item))
(define (pack-weight pack)
(apply + (map item-weight pack)))
(define (pack-value pack)
(apply + (map item-value pack)))
(define (max-pack-value pack-with pack-without max-weight)
(if (and
(not (> (pack-weight pack-with) max-weight))
(> (pack-value pack-with) (pack-value pack-without)))
pack-with pack-without))
(define (display-solution pack)
(displayln (list 'weight: (pack-weight pack)
'value: (pack-value pack)
'items: (map car pack))))
</lang> Brute Force <lang racket> (define (show-brute)
(define empty-accumulator '())
(define (knapsack-brute included items)
(cond
((null? items) included)
(else
(max-pack-value
(knapsack-brute (cons (car items) included) (cdr items))
(knapsack-brute included (cdr items))
max-weight
))))
(display-solution (reverse (knapsack-brute empty-accumulator items))))
(show-brute); takes around five seconds on my machine </lang> Recursive Alternate <lang racket> (define (show-memoized)
(define (memoize func)
(let ([result-ht (make-hash)])
(lambda args ; this is the rest-id pattern
(when (not (hash-has-key? result-ht args))
(hash-set! result-ht args (apply func args)))
(hash-ref result-ht args))))
(define knapsack
(memoize
(lambda (max-weight items)
(cond
((null? items) '())
(else
(let ([item (car items)] [items (cdr items)])
(max-pack-value
(cons item (knapsack (- max-weight (item-weight item)) items))
(knapsack max-weight items)
max-weight)))))))
(display-solution (knapsack max-weight items)))
(show-memoized) </lang>
- Output:
<lang racket> (weight: 396 value: 1030 items: (map compass water sandwich glucose banana cream trousers overclothes notecase glasses socks)) </lang>
Raku
(formerly Perl 6)
Brute force
<lang perl6>my class KnapsackItem { has $.name; has $.weight; has $.unit; }
multi sub pokem ([], $, $v = 0) { $v } multi sub pokem ([$, *@], 0, $v = 0) { $v } multi sub pokem ([$i, *@rest], $w, $v = 0) {
my $key = "{+@rest} $w $v";
(state %cache){$key} or do {
my @skip = pokem @rest, $w, $v;
if $w >= $i.weight { # next one fits
my @put = pokem @rest, $w - $i.weight, $v + $i.unit;
return (%cache{$key} = |@put, $i.name).list if @put[0] > @skip[0];
}
return (%cache{$key} = |@skip).list;
}
}
my $MAX_WEIGHT = 400; my @table = flat map -> $name, $weight, $unit {
KnapsackItem.new: :$name, :$weight, :$unit;
},
'map', 9, 150, 'compass', 13, 35, 'water', 153, 200, 'sandwich', 50, 160, 'glucose', 15, 60, 'tin', 68, 45, 'banana', 27, 60, 'apple', 39, 40, 'cheese', 23, 30, 'beer', 52, 10, 'suntan cream', 11, 70, 'camera', 32, 30, 'T-shirt', 24, 15, 'trousers', 48, 10, 'umbrella', 73, 40, 'waterproof trousers', 42, 70, 'waterproof overclothes', 43, 75, 'note-case', 22, 80, 'sunglasses', 7, 20, 'towel', 18, 12, 'socks', 4, 50, 'book', 30, 10;
my ($value, @result) = pokem @table, $MAX_WEIGHT; say "Value = $value\nTourist put in the bag:\n " ~ @result;</lang>
- Output:
Value = 1030 Tourist put in the bag: socks sunglasses note-case waterproof overclothes waterproof trousers suntan cream banana glucose sandwich water compass map
Dynamic programming
Much faster than the previous example.
<lang perl6>my $raw = q:to/TABLE/;
map 9 150 compass 13 35 water 153 200 sandwich 50 160 glucose 15 60 tin 68 45 banana 27 60 apple 39 40 cheese 23 30 beer 52 10 suntancream 11 70 camera 32 30 T-shirt 24 15 trousers 48 10 umbrella 73 40 waterproof trousers 42 70 waterproof overclothes 43 75 note-case 22 80 sunglasses 7 20 towel 18 12 socks 4 50 book 30 10
TABLE
my (@name, @weight, @value);
for $raw.lines.sort({-($_ ~~ m/\d+/)}).comb(/\S+[\s\S+]*/) -> $n, $w, $v {
@name.push: $n; @weight.push: $w; @value.push: $v;
}
my $sum = 400;
my @subset;
sub optimal ($item, $sub-sum) {
return 0, [] if $item < 0; return |@subset[$item][$sub-sum] if @subset[$item][$sub-sum];
my @next = optimal($item-1, $sub-sum);
if @weight[$item] > $sub-sum {
@subset[$item][$sub-sum] = @next
} else {
my @skip = optimal($item-1, $sub-sum - @weight[$item]);
if (@next[0] > @skip[0] + @value[$item] ) {
@subset[$item][$sub-sum] = @next;
} else {
@subset[$item][$sub-sum] = @skip[0] + @value[$item], [|@skip[1], @name[$item]];
}
}
|@subset[$item][$sub-sum]
}
my @solution = optimal(@name.end, $sum); put "@solution[0]: ", @solution[1].sort.join(', ');</lang>
- Output:
1030: banana, compass, glucose, map, note-case, sandwich, socks, sunglasses, suntancream, water, waterproof overclothes, waterproof trousers
REXX
Originally, the combination generator/checker subroutine was recursive and made the program solution generic (and more concise).
However, a recursive solution also made the solution much more slower, so the combination generator/checker was "unrolled" and converted into discrete combination checks (based on the number of allowable items). The unused combinatorial checks were discarded and only the pertinent code was retained. It made no sense to include all the unused subroutines here as space appears to be a premium for some entries in Rosetta Code.
The term allowable items refers to all items that are of allowable weight (those that weigh within the weight criteria). An half metric─ton anvil was added to the list to show how overweight items are pruned from the list of items. <lang rexx>/*REXX program solves a knapsack problem (22 {+1} items with a weight restriction). */ maxWeight=400 /*the maximum weight for the knapsack. */
say 'maximum weight allowed for a knapsack:' commas(maxWeight); say
call gen@ /*generate the @ array of choices. */ call sortD /* sort " " " " " */ call build /*build some associative arrays from @.*/ call findBest /*go ye forth and find the best choises*/ call results /*display the best choices for knapsack*/ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ build: do j=1 for obj; parse var @.j x w v . /*construct a list of knapsack choices.*/
if w>maxWeight then iterate /*Is weight greater than max? Ignore.*/
totW=totW + w; totV=totV + v /*add the totals (for output alignment)*/
maxL=max(maxL, length(x) ) /*determine maximum width for an item. */
#=#+1; i.#=x; w.#=w; v.#=v /*bump the number of items (choices). */
end /*j*/ /* [↑] build indexable arrays of items*/
maxL= maxL + maxL%4 + 4 /*extend width of name for shown table.*/
maxW= max(maxW, length( commas(totW) ) ) /*find the maximum width for weight. */
maxV= max(maxV, length( commas(totV) ) ) /* " " " " " value. */
call hdr 'potential knapsack items' /*display a header for list of choices.*/
do j=1 for obj; parse var @.j i w v . /*show all the choices in a nice format*/
if w<=maxWeight then call show i,w,v /*Is weight within limits? Then show. */
end /*j*/ /* [↑] display the list of choices. */
$=0
say; say 'number of allowable items: ' #
return
/*──────────────────────────────────────────────────────────────────────────────────────*/ commas: procedure; parse arg _; n=_'.9'; #=123456789; b=verify(n, #, "M")
e=verify(n, #'0', , verify(n, #"0.", 'M')) - 4; comma=','
do j=e to b by -3; _=insert(comma, _, j); end /*j*/; return _
/*──────────────────────────────────────────────────────────────────────────────────────*/ findBest: m=maxWeight /*items are in decreasing weight.*/
do j1 =0 for #+1; w1 = w.j1 ; z1 = v.j1
do j2 =j1 +(j1 >0) to #; if w.j2 +w1 >m then iterate j1 ; w2 =w1 +w.j2 ; z2 =z1 +v.j2
do j3 =j2 +(j2 >0) to #; if w.j3 +w2 >m then iterate j2 ; w3 =w2 +w.j3 ; z3 =z2 +v.j3
do j4 =j3 +(j3 >0) to #; if w.j4 +w3 >m then iterate j3 ; w4 =w3 +w.j4 ; z4 =z3 +v.j4
do j5 =j4 +(j4 >0) to #; if w.j5 +w4 >m then iterate j4 ; w5 =w4 +w.j5 ; z5 =z4 +v.j5
do j6 =j5 +(j5 >0) to #; if w.j6 +w5 >m then iterate j5 ; w6 =w5 +w.j6 ; z6 =z5 +v.j6
do j7 =j6 +(j6 >0) to #; if w.j7 +w6 >m then iterate j6 ; w7 =w6 +w.j7 ; z7 =z6 +v.j7
do j8 =j7 +(j7 >0) to #; if w.j8 +w7 >m then iterate j7 ; w8 =w7 +w.j8 ; z8 =z7 +v.j8
do j9 =j8 +(j8 >0) to #; if w.j9 +w8 >m then iterate j8 ; w9 =w8 +w.j9 ; z9 =z8 +v.j9
do j10=j9 +(j9 >0) to #; if w.j10+w9 >m then iterate j9 ; w10=w9 +w.j10; z10=z9 +v.j10
do j11=j10+(j10>0) to #; if w.j11+w10>m then iterate j10; w11=w10+w.j11; z11=z10+v.j11
do j12=j11+(j11>0) to #; if w.j12+w11>m then iterate j11; w12=w11+w.j12; z12=z11+v.j12
do j13=j12+(j12>0) to #; if w.j13+w12>m then iterate j12; w13=w12+w.j13; z13=z12+v.j13
do j14=j13+(j13>0) to #; if w.j14+w13>m then iterate j13; w14=w13+w.j14; z14=z13+v.j14
do j15=j14+(j14>0) to #; if w.j15+w14>m then iterate j14; w15=w14+w.j15; z15=z14+v.j15
do j16=j15+(j15>0) to #; if w.j16+w15>m then iterate j15; w16=w15+w.j16; z16=z15+v.j16
do j17=j16+(j16>0) to #; if w.j17+w16>m then iterate j16; w17=w16+w.j17; z17=z16+v.j17
do j18=j17+(j17>0) to #; if w.j18+w17>m then iterate j17; w18=w17+w.j18; z18=z17+v.j18
do j19=j18+(j18>0) to #; if w.j19+w18>m then iterate j18; w19=w18+w.j19; z19=z18+v.j19
do j20=j19+(j19>0) to #; if w.j20+w19>m then iterate j19; w20=w19+w.j20; z20=z19+v.j20
do j21=j20+(j20>0) to #; if w.j21+w20>m then iterate j20; w21=w20+w.j21; z21=z20+v.j21
do j22=j21+(j21>0) to #; if w.j22+w21>m then iterate j21; w22=w21+w.j22; z22=z21+v.j22
if z22>$ then do; ?=; $=z22; do j=1 for 22; ?=? value("J"j); end /*j*/; end
end;end;end;end;end;end;end;end;end;end;end;end;end;end;end;end;end;end;end;end;end;end
return /*──────────────────────────────────────────────────────────────────────────────────────*/ gen@: @. = ; @.12= 'camera 32 30'
@.1 = 'map 9 150' ; @.13= 'T-shirt 24 15'
@.2 = 'compass 13 35' ; @.14= 'trousers 48 10'
@.3 = 'water 153 200' ; @.15= 'umbrella 73 40'
@.4 = 'sandwich 50 160' ; @.16= 'waterproof_trousers 42 70'
@.5 = 'glucose 15 60' ; @.17= 'waterproof_overclothes 43 75'
@.6 = 'tin 68 45' ; @.18= 'note-case 22 80'
@.7 = 'banana 27 60' ; @.19= 'sunglasses 7 20'
@.8 = 'apple 39 40' ; @.20= 'towel 18 12'
@.9 = 'cheese 23 30' ; @.21= 'socks 4 50'
@.10= 'beer 52 10' ; @.22= 'book 30 10'
@.11= 'suntan_cream 11 70' ; @.23= 'anvil 100000 1'
maxL = length('potential knapsack items') /*maximum width for the table items. */
maxW = length('weight') /* " " " " " weights. */
maxV = length('value') /* " " " " " values. */
#=0; i.=; w.=0; v.=0; totW=0; totV=0 /*initialize some REX variables stuff. */
return
/*──────────────────────────────────────────────────────────────────────────────────────*/ hdr: say; call show center(arg(1),maxL),center('weight',maxW),center("value",maxV) hdr2: call show copies('═',maxL),copies('═',maxW),copies('═',maxV); return /*──────────────────────────────────────────────────────────────────────────────────────*/ results: do #; ?=strip( space(?), "L", 0); end /*h*/ /*elide leading zeroes*/
bestC=?; bestW=0; totP=words(bestC); say; call hdr 'best choice'
do j=1 for totP; _=word(bestC, j); _w=w._; _v=v._
do k=j+1 to totP; __=word(bestC, k); if i._\==i.__ then leave
j=j+1; w._=w._ + _w; v._=v._ + _v
end /*k*/
call show i._, w._, v._; bestW=bestW + w._
end /*j*/
call hdr2 ; say; @bestTK= 'best total knapsack'
call show @bestTK 'weight' , bestW /*display a nicely formatted winner wt.*/
call show @bestTK 'value' ,, $ /* " " " " winner val*/
call show @bestTK 'items' ,,, totP /* " " " " pieces. */
return
/*──────────────────────────────────────────────────────────────────────────────────────*/ show: parse arg _i,_w,_v,_p; say translate( left(_i,maxL,'─'), , "_") ,
right(commas(_w),maxW) right(commas(_v),maxV) _p; return
/*──────────────────────────────────────────────────────────────────────────────────────*/ sortD: do j=1 while @.j\==; y=word(@.j,2) /*process each of the knapsack choices.*/
do k=j+1 while @.k\== /*find a possible heavier knapsack item*/
?=word(@.k,2); if ?>y then do; _=@.k; @.k=@.j; @.j=_; y=?; end /*swap*/
end /*k*/
end /*j*/ /* [↑] sort choices by decreasing wt. */
obj=j-1; return /*decrement J for the DO loop index*/</lang>
- output when using the default input:
maximum weight allowed for a knapsack: 400
potential knapsack items weight value
══════════════════════════════════ ══════ ═════
water───────────────────────────── 153 200
umbrella────────────────────────── 73 40
tin─────────────────────────────── 68 45
beer────────────────────────────── 52 10
sandwich────────────────────────── 50 160
trousers────────────────────────── 48 10
waterproof overclothes──────────── 43 75
waterproof trousers─────────────── 42 70
apple───────────────────────────── 39 40
camera──────────────────────────── 32 30
book────────────────────────────── 30 10
banana──────────────────────────── 27 60
T-shirt─────────────────────────── 24 15
cheese──────────────────────────── 23 30
note-case───────────────────────── 22 80
towel───────────────────────────── 18 12
glucose─────────────────────────── 15 60
compass─────────────────────────── 13 35
suntan cream────────────────────── 11 70
map─────────────────────────────── 9 150
sunglasses──────────────────────── 7 20
socks───────────────────────────── 4 50
number of allowable items: 22
best choice weight value
══════════════════════════════════ ══════ ═════
water───────────────────────────── 153 200
sandwich────────────────────────── 50 160
waterproof overclothes──────────── 43 75
waterproof trousers─────────────── 42 70
banana──────────────────────────── 27 60
note-case───────────────────────── 22 80
glucose─────────────────────────── 15 60
compass─────────────────────────── 13 35
suntan cream────────────────────── 11 70
map─────────────────────────────── 9 150
sunglasses──────────────────────── 7 20
socks───────────────────────────── 4 50
══════════════════════════════════ ══════ ═════
best total knapsack weight──────── 396
best total knapsack value───────── 1,030
best total knapsack items───────── 12
Ruby
Brute force
<lang ruby>KnapsackItem = Struct.new(:name, :weight, :value) potential_items = [
KnapsackItem['map', 9, 150], KnapsackItem['compass', 13, 35], KnapsackItem['water', 153, 200], KnapsackItem['sandwich', 50, 160], KnapsackItem['glucose', 15, 60], KnapsackItem['tin', 68, 45], KnapsackItem['banana', 27, 60], KnapsackItem['apple', 39, 40], KnapsackItem['cheese', 23, 30], KnapsackItem['beer', 52, 10], KnapsackItem['suntan cream', 11, 70], KnapsackItem['camera', 32, 30], KnapsackItem['t-shirt', 24, 15], KnapsackItem['trousers', 48, 10], KnapsackItem['umbrella', 73, 40], KnapsackItem['waterproof trousers', 42, 70], KnapsackItem['waterproof overclothes', 43, 75], KnapsackItem['note-case', 22, 80], KnapsackItem['sunglasses', 7, 20], KnapsackItem['towel', 18, 12], KnapsackItem['socks', 4, 50], KnapsackItem['book', 30, 10],
] knapsack_capacity = 400
class Array
# do something for each element of the array's power set
def power_set
yield [] if block_given?
self.inject([[]]) do |ps, elem|
ps.each_with_object([]) do |i,r|
r << i
new_subset = i + [elem]
yield new_subset if block_given?
r << new_subset
end
end
end
end
maxval, solutions = potential_items.power_set.group_by {|subset|
weight = subset.inject(0) {|w, elem| w + elem.weight}
weight>knapsack_capacity ? 0 : subset.inject(0){|v, elem| v + elem.value}
}.max
puts "value: #{maxval}" solutions.each do |set|
wt, items = 0, []
set.each {|elem| wt += elem.weight; items << elem.name}
puts "weight: #{wt}"
puts "items: #{items.join(',')}"
end</lang>
- Output:
value: 1030 weight: 396 items: map,compass,water,sandwich,glucose,banana,suntan cream,waterproof trousers,waterproof overclothes,note-case,sunglasses,socks
Dynamic Programming
Translated from http://sites.google.com/site/mikescoderama/Home/0-1-knapsack-problem-in-p <lang ruby>KnapsackItem = Struct.new(:name, :weight, :value)
def dynamic_programming_knapsack(items, max_weight)
num_items = items.size
cost_matrix = Array.new(num_items){Array.new(max_weight+1, 0)}
num_items.times do |i|
(max_weight + 1).times do |j|
if(items[i].weight > j)
cost_matrix[i][j] = cost_matrix[i-1][j]
else
cost_matrix[i][j] = [cost_matrix[i-1][j], items[i].value + cost_matrix[i-1][j-items[i].weight]].max
end
end
end
used_items = get_used_items(items, cost_matrix)
[get_list_of_used_items_names(items, used_items), # used items names
items.zip(used_items).map{|item,used| item.weight*used}.inject(:+), # total weight
cost_matrix.last.last] # total value
end
def get_used_items(items, cost_matrix)
i = cost_matrix.size - 1
currentCost = cost_matrix[0].size - 1
marked = cost_matrix.map{0}
while(i >= 0 && currentCost >= 0)
if(i == 0 && cost_matrix[i][currentCost] > 0 ) || (cost_matrix[i][currentCost] != cost_matrix[i-1][currentCost])
marked[i] = 1
currentCost -= items[i].weight
end
i -= 1
end
marked
end
def get_list_of_used_items_names(items, used_items)
items.zip(used_items).map{|item,used| item.name if used>0}.compact.join(', ')
end
if $0 == __FILE__
items = [
KnapsackItem['map' , 9, 150], KnapsackItem['compass' , 13, 35],
KnapsackItem['water' , 153, 200], KnapsackItem['sandwich' , 50, 160],
KnapsackItem['glucose' , 15, 60], KnapsackItem['tin' , 68, 45],
KnapsackItem['banana' , 27, 60], KnapsackItem['apple' , 39, 40],
KnapsackItem['cheese' , 23, 30], KnapsackItem['beer' , 52, 10],
KnapsackItem['suntan cream' , 11, 70], KnapsackItem['camera' , 32, 30],
KnapsackItem['t-shirt' , 24, 15], KnapsackItem['trousers' , 48, 10],
KnapsackItem['umbrella' , 73, 40], KnapsackItem['waterproof trousers', 42, 70],
KnapsackItem['waterproof overclothes', 43, 75], KnapsackItem['note-case' , 22, 80],
KnapsackItem['sunglasses' , 7, 20], KnapsackItem['towel' , 18, 12],
KnapsackItem['socks' , 4, 50], KnapsackItem['book' , 30, 10]
]
names, weight, value = dynamic_programming_knapsack(items, 400)
puts
puts 'Dynamic Programming:'
puts
puts "Found solution: #{names}"
puts "total weight: #{weight}"
puts "total value: #{value}"
end</lang>
- Output:
Dynamic Programming: Found solution: map, compass, water, sandwich, glucose, banana, suntan cream, waterproof trousers, waterproof overclothes, note-case, sunglasses, socks total weight: 396 total value: 1030
Rust
Dynamic Programming solution. <lang rust>use std::cmp;
struct Item {
name: &'static str, weight: usize, value: usize
}
fn knapsack01_dyn(items: &[Item], max_weight: usize) -> Vec<&Item> {
let mut best_value = vec![vec![0; max_weight + 1]; items.len() + 1];
for (i, it) in items.iter().enumerate() {
for w in 1 .. max_weight + 1 {
best_value[i + 1][w] =
if it.weight > w {
best_value[i][w]
} else {
cmp::max(best_value[i][w], best_value[i][w - it.weight] + it.value)
}
}
}
let mut result = Vec::with_capacity(items.len()); let mut left_weight = max_weight;
for (i, it) in items.iter().enumerate().rev() {
if best_value[i + 1][left_weight] != best_value[i][left_weight] {
result.push(it);
left_weight -= it.weight;
}
}
result
}
fn main () {
const MAX_WEIGHT: usize = 400;
const ITEMS: &[Item] = &[
Item { name: "map", weight: 9, value: 150 },
Item { name: "compass", weight: 13, value: 35 },
Item { name: "water", weight: 153, value: 200 },
Item { name: "sandwich", weight: 50, value: 160 },
Item { name: "glucose", weight: 15, value: 60 },
Item { name: "tin", weight: 68, value: 45 },
Item { name: "banana", weight: 27, value: 60 },
Item { name: "apple", weight: 39, value: 40 },
Item { name: "cheese", weight: 23, value: 30 },
Item { name: "beer", weight: 52, value: 10 },
Item { name: "suntancream", weight: 11, value: 70 },
Item { name: "camera", weight: 32, value: 30 },
Item { name: "T-shirt", weight: 24, value: 15 },
Item { name: "trousers", weight: 48, value: 10 },
Item { name: "umbrella", weight: 73, value: 40 },
Item { name: "waterproof trousers", weight: 42, value: 70 },
Item { name: "waterproof overclothes", weight: 43, value: 75 },
Item { name: "note-case", weight: 22, value: 80 },
Item { name: "sunglasses", weight: 7, value: 20 },
Item { name: "towel", weight: 18, value: 12 },
Item { name: "socks", weight: 4, value: 50 },
Item { name: "book", weight: 30, value: 10 }
];
let items = knapsack01_dyn(ITEMS, MAX_WEIGHT);
// We reverse the order because we solved the problem backward.
for it in items.iter().rev() {
println!("{}", it.name);
}
println!("Total weight: {}", items.iter().map(|w| w.weight).sum::<usize>());
println!("Total value: {}", items.iter().map(|w| w.value).sum::<usize>());
}</lang>
- Output:
map compass water sandwich glucose banana suntancream waterproof trousers waterproof overclothes note-case sunglasses socks Total weight: 396 Total value: 1030
SAS
Use MILP solver in SAS/OR: <lang sas>/* create SAS data set */ data mydata;
input item $1-23 weight value; datalines;
map 9 150 compass 13 35 water 153 200 sandwich 50 160 glucose 15 60 tin 68 45 banana 27 60 apple 39 40 cheese 23 30 beer 52 10 suntan cream 11 70 camera 32 30 T-shirt 24 15 trousers 48 10 umbrella 73 40 waterproof trousers 42 70 waterproof overclothes 43 75 note-case 22 80 sunglasses 7 20 towel 18 12 socks 4 50 book 30 10
/* call OPTMODEL procedure in SAS/OR */ proc optmodel;
/* declare sets and parameters, and read input data */
set <str> ITEMS;
num weight {ITEMS};
num value {ITEMS};
read data mydata into ITEMS=[item] weight value;
/* declare variables, objective, and constraints */
var NumSelected {ITEMS} binary;
max TotalValue = sum {i in ITEMS} value[i] * NumSelected[i];
con WeightCon:
sum {i in ITEMS} weight[i] * NumSelected[i] <= 400;
/* call mixed integer linear programming (MILP) solver */ solve;
/* print optimal solution */
print TotalValue;
print {i in ITEMS: NumSelected[i].sol > 0.5} NumSelected;
quit;</lang>
Output:
TotalValue 1030 [1] NumSelected banana 1 compass 1 glucose 1 map 1 note-case 1 sandwich 1 socks 1 sunglasses 1 suntan cream 1 water 1 waterproof overclothes 1 waterproof trousers 1
Scala
<lang Scala>object Knapsack extends App {
case class Item(name: String, weight: Int, value: Int)
val elapsed: (=> Unit) => Long = f => {val s = System.currentTimeMillis; f; (System.currentTimeMillis - s)/1000}
//===== brute force (caution: increase the heap!) ====================================
val ks01b: List[Item] => Unit = loi => {
val tw:Set[Item]=>Int=ps=>(ps:\0)((a,b)=>a.weight+b) //total weight
val tv:Set[Item]=>Int=ps=>(ps:\0)((a,b)=>a.value+b) //total value
val pis = (loi.toSet.subsets).toList.filterNot(_==Set())
#[test]
fn test_dp_results() {
let dp_results = knap_01_dp(items, 400); let dp_weights= dp_results.iter().fold(0, |a, &b| a + b.weight); let dp_values = dp_results.iter().fold(0, |a, &b| a + b.value); assert_eq!(dp_weights, 396); assert_eq!(dp_values, 1030);
} val res = pis.map(ss=>Pair(ss,tw(ss)))
.filter(p=>p._2>350 && p._2<401).map(p=>Pair(p,tv(p._1)))
.sortWith((s,t)=>s._2.compareTo(t._2) < 0)
.last
println{val h = "packing list of items (brute force):"; h+"\n"+"="*h.size}
res._1._1.foreach{p=>print(" "+p.name+": weight="+p.weight+" value="+p.value+"\n")}
println("\n"+" resulting items: "+res._1._1.size+" of "+loi.size)
println(" total weight: "+res._1._2+", total value: "+res._2)
}
//===== dynamic programming ==========================================================
val ks01d: List[Item] => Unit = loi => {
val W = 400
val N = loi.size
val m = Array.ofDim[Int](N+1,W+1)
val plm = (List((for {w <- 0 to W} yield Set[Item]()).toArray)++(
for {
n <- 0 to N-1
colN = (for {w <- 0 to W} yield Set[Item](loi(n))).toArray
} yield colN)).toArray
1 to N foreach {n =>
0 to W foreach {w =>
def in = loi(n-1)
def wn = loi(n-1).weight
def vn = loi(n-1).value
if (w<wn) {
m(n)(w) = m(n-1)(w)
plm(n)(w) = plm(n-1)(w)
}
else {
if (m(n-1)(w)>=m(n-1)(w-wn)+vn) {
m(n)(w) = m(n-1)(w)
plm(n)(w) = plm(n-1)(w)
}
else {
m(n)(w) = m(n-1)(w-wn)+vn
plm(n)(w) = plm(n-1)(w-wn)+in } }
} }
println{val h = "packing list of items (dynamic programming):"; h+"\n"+"="*h.size}
plm(N)(W).foreach{p=>print(" "+p.name+": weight="+p.weight+" value="+p.value+"\n")}
println("\n"+" resulting items: "+plm(N)(W).size+" of "+loi.size)
println(" total weight: "+(0/:plm(N)(W).toVector.map{item=>item.weight})(_+_)+", total value: "+m(N)(W))
}
val items = List(
Item("map", 9, 150)
,Item("compass", 13, 35)
,Item("water", 153, 200)
,Item("sandwich", 50, 160)
,Item("glucose", 15, 60)
,Item("tin", 68, 45)
,Item("banana", 27, 60)
,Item("apple", 39, 40)
,Item("cheese", 23, 30)
,Item("beer", 52, 10)
,Item("suntan cream", 11, 70)
,Item("camera", 32, 30)
,Item("t-shirt", 24, 15)
,Item("trousers", 48, 10)
,Item("umbrella", 73, 40)
,Item("waterproof trousers", 42, 70)
,Item("waterproof overclothes", 43, 75)
,Item("note-case", 22, 80)
,Item("sunglasses", 7, 20)
,Item("towel", 18, 12)
,Item("socks", 4, 50)
,Item("book", 30, 10)
)
List(ks01b, ks01d).foreach{f=>
val t = elapsed{f(items)}
println(" elapsed time: "+t+" sec"+"\n")
}
}</lang>
- Output:
packing list of items (brute force): ==================================== waterproof overclothes: weight=43 value=75 note-case: weight=22 value=80 socks: weight=4 value=50 sandwich: weight=50 value=160 banana: weight=27 value=60 glucose: weight=15 value=60 map: weight=9 value=150 water: weight=153 value=200 suntan cream: weight=11 value=70 sunglasses: weight=7 value=20 waterproof trousers: weight=42 value=70 compass: weight=13 value=35 resulting items: 12 of 22 total weight: 396, total value: 1030 elapsed time: 19 sec packing list of items (dynamic programming): ============================================ waterproof overclothes: weight=43 value=75 note-case: weight=22 value=80 socks: weight=4 value=50 sandwich: weight=50 value=160 banana: weight=27 value=60 glucose: weight=15 value=60 map: weight=9 value=150 water: weight=153 value=200 suntan cream: weight=11 value=70 sunglasses: weight=7 value=20 waterproof trousers: weight=42 value=70 compass: weight=13 value=35 resulting items: 12 of 22 total weight: 396, total value: 1030 elapsed time: 0 sec
Sidef
<lang ruby>var raw = <<'TABLE' map, 9, 150 compass, 13, 35 water, 153, 200 sandwich, 50, 160 glucose, 15, 60 tin, 68, 45 banana, 27, 60 apple, 39, 40 cheese, 23, 30 beer, 52, 10 suntancream, 11, 70 camera, 32, 30 T-shirt, 24, 15 trousers, 48, 10 umbrella, 73, 40 waterproof trousers, 42, 70 waterproof overclothes, 43, 75 note-case, 22, 80 sunglasses, 7, 20 towel, 18, 12 socks, 4, 50 book, 30, 10 TABLE
struct KnapsackItem {
String name, Number weight, Number value,
}
var items = [] raw.each_line{ |row|
var fields = row.split(/\s*,\s*/)
items << KnapsackItem(
name: fields[0],
weight: fields[1].to_n,
value: fields[2].to_n,
)
}
var max_weight = 400 var p = [
items.len.of { [[0, []], max_weight.of(nil)...] }...,
max_weight.inc.of {[0, []]}
]
func optimal(i, w) {
if (!defined p[i][w]) {
var item = items[i];
if (item.weight > w) {
p[i][w] = optimal(i.dec, w)
}
else {
var x = optimal(i.dec, w)
var y = optimal(i.dec, w - item.weight)
if (x[0] > (y[0] + item.value)) {
p[i][w] = x;
}
else {
p[i][w] = [y[0] + item.value, [y[1]..., item.name]]
}
}
}
return p[i][w]
}
var sol = optimal(items.end, max_weight) say "#{sol[0]}: #{sol[1]}"</lang>
- Output:
1030: map compass water sandwich glucose banana suntancream waterproof trousers waterproof overclothes note-case sunglasses socks
SQL
A brute force solution that runs in SQL Server 2005 or later using a recursive CTE. Displays the top 5 solutions and runs in about 39 seconds.
<lang SQL> WITH KnapsackItems (item, [weight], value) AS (
SELECT 'map',9, 150 UNION ALL SELECT 'compass',13, 35 UNION ALL SELECT 'water',153, 200 UNION ALL SELECT 'sandwich',50, 160 UNION ALL SELECT 'glucose',15, 60 UNION ALL SELECT 'tin',68, 45 UNION ALL SELECT 'banana',27, 60 UNION ALL SELECT 'apple',39, 40 UNION ALL SELECT 'cheese',23, 30 UNION ALL SELECT 'beer',52, 10 UNION ALL SELECT 'suntan cream',11, 70 UNION ALL SELECT 'camera',32, 30 UNION ALL SELECT 'T-shirt',24, 15 UNION ALL SELECT 'trousers',48, 10 UNION ALL SELECT 'umbrella',73, 40 UNION ALL SELECT 'waterproof trousers',42, 70 UNION ALL SELECT 'waterproof overclothes',43, 75 UNION ALL SELECT 'note-case',22, 80 UNION ALL SELECT 'sunglasses',7, 20 UNION ALL SELECT 'towel',18, 12 UNION ALL SELECT 'socks',4, 50 UNION ALL SELECT 'book',30, 10
) SELECT * INTO #KnapsackItems FROM KnapsackItems;
WITH UNIQUEnTuples (n, Tuples, ID, [weight], value) AS (
SELECT 1, CAST(item AS VARCHAR(8000)), item, [weight], value
FROM #KnapsackItems
UNION ALL
SELECT 1 + n.n, t.item + ',' + n.Tuples, item, n.[weight] + t.[weight], n.value + t.value
FROM UNIQUEnTuples n
CROSS APPLY (
SELECT item, [weight], value
FROM #KnapsackItems t
WHERE t.item < n.ID AND n.[weight] + t.[weight] < 400) t
)
SELECT TOP 5 * FROM UNIQUEnTuples ORDER BY value DESC, n, Tuples;
GO DROP TABLE #KnapsackItems; </lang>
- Output:
weight value Solution 396 1030 banana,compass,glucose,map,note-case,sandwich,socks,sunglasses,suntan cream,water,waterproof overclothes,waterproof trousers 389 1010 banana,compass,glucose,map,note-case,sandwich,socks,suntan cream,water,waterproof overclothes,waterproof trousers 399 1005 banana,cheese,glucose,map,note-case,sandwich,socks,suntan cream,water,waterproof overclothes,waterproof trousers 395 1002 banana,cheese,compass,glucose,map,note-case,sandwich,socks,sunglasses,suntan cream,towel,water,waterproof overclothes 393 1000 apple,banana,compass,glucose,map,note-case,sandwich,socks,sunglasses,suntan cream,water,waterproof overclothes
Swift
Dynamic Programming
<lang swift>struct KnapsackItem {
var name: String var weight: Int var value: Int
}
func knapsack(items: [KnapsackItem], limit: Int) -> [KnapsackItem] {
var table = Array(repeating: Array(repeating: 0, count: limit + 1), count: items.count + 1)
for j in 1..<items.count+1 {
let item = items[j-1]
for w in 1..<limit+1 {
if item.weight > w {
table[j][w] = table[j-1][w]
} else {
table[j][w] = max(table[j-1][w], table[j-1][w-item.weight] + item.value)
}
}
}
var result = [KnapsackItem]()
var w = limit
for j in stride(from: items.count, to: 0, by: -1) where table[j][w] != table[j-1][w] {
let item = items[j-1]
result.append(item)
w -= item.weight
}
return result
}
let items = [
KnapsackItem(name: "map", weight: 9, value: 150), KnapsackItem(name: "compass", weight: 13, value: 35), KnapsackItem(name: "water", weight: 153, value: 200), KnapsackItem(name: "sandwich", weight: 50, value: 160), KnapsackItem(name: "glucose", weight: 15, value: 60), KnapsackItem(name: "tin", weight: 68, value: 45), KnapsackItem(name: "banana", weight: 27, value: 60), KnapsackItem(name: "apple", weight: 39, value: 40), KnapsackItem(name: "cheese", weight: 23, value: 30), KnapsackItem(name: "beer", weight: 52, value: 10), KnapsackItem(name: "suntan cream", weight: 11, value: 70), KnapsackItem(name: "camera", weight: 32, value: 30), KnapsackItem(name: "t-shirt", weight: 24, value: 15), KnapsackItem(name: "trousers", weight: 48, value: 10), KnapsackItem(name: "umbrella", weight: 73, value: 40), KnapsackItem(name: "waterproof trousers", weight: 42, value: 70), KnapsackItem(name: "waterproof overclothes", weight: 43, value: 75), KnapsackItem(name: "note-case", weight: 22, value: 80), KnapsackItem(name: "sunglasses", weight: 7, value: 20), KnapsackItem(name: "towel", weight: 18, value: 12), KnapsackItem(name: "socks", weight: 4, value: 50), KnapsackItem(name: "book", weight: 30, value: 10)
]
let kept = knapsack(items: items, limit: 400)
print("Kept: ")
for item in kept {
print(" \(item.name)")
}
let (tValue, tWeight) = kept.reduce((0, 0), { ($0.0 + $1.value, $0.1 + $1.weight) })
print("For a total value of \(tValue) and a total weight of \(tWeight)")</lang>
- Output:
Kept: socks sunglasses note-case waterproof overclothes waterproof trousers suntan cream banana glucose sandwich water compass map For a total value of 1030 and a total weight of 396
Tcl
As the saying goes, “when in doubt, try brute force”. Since there's only 22 items we can simply iterate over all possible choices. <lang tcl># The list of items to consider, as list of lists set items {
{map 9 150}
{compass 13 35}
{water 153 200}
{sandwich 50 160}
{glucose 15 60}
{tin 68 45}
{banana 27 60}
{apple 39 40}
{cheese 23 30}
{beer 52 10}
{{suntan cream} 11 70}
{camera 32 30}
{t-shirt 24 15}
{trousers 48 10}
{umbrella 73 40}
{{waterproof trousers} 42 70}
{{waterproof overclothes} 43 75}
{note-case 22 80}
{sunglasses 7 20}
{towel 18 12}
{socks 4 50}
{book 30 10}
}
- Simple extraction functions
proc names {chosen} {
set names {}
foreach item $chosen {lappend names [lindex $item 0]}
return $names
} proc weight {chosen} {
set weight 0
foreach item $chosen {incr weight [lindex $item 1]}
return $weight
} proc value {chosen} {
set value 0
foreach item $chosen {incr value [lindex $item 2]}
return $value
}
- Recursive function for searching over all possible choices of items
proc knapsackSearch {items {chosen {}}} {
# If we've gone over the weight limit, stop now
if {[weight $chosen] > 400} {
return
}
# If we've considered all of the items (i.e., leaf in search tree)
# then see if we've got a new best choice.
if {[llength $items] == 0} {
global best max set v [value $chosen] if {$v > $max} { set max $v set best $chosen } return
} # Branch, so recurse for chosing the current item or not set this [lindex $items 0] set rest [lrange $items 1 end] knapsackSearch $rest $chosen knapsackSearch $rest [lappend chosen $this]
}
- Initialize a few global variables
set best {} set max 0
- Do the brute-force search
knapsackSearch $items
- Pretty-print the results
puts "Best filling has weight of [expr {[weight $best]/100.0}]kg and score [value $best]" puts "Best items:\n\t[join [lsort [names $best]] \n\t]"</lang>
- Output:
Best filling has weight of 3.96kg and score 1030 Best items: banana compass glucose map note-case sandwich socks sunglasses suntan cream water waterproof overclothes waterproof trousers
Ursala
This solution follows a very similar approach to the one used in Knapsack problem/Bounded#Ursala, which is to treat it as a mixed integer programming problem and solve it using an off-the-shelf library (lpsolve). <lang Ursala>#import std
- import nat
- import flo
- import lin
- import nat
items = # name: (weight,value)
<
'map': (9,150), 'compass': (13,35), 'water': (153,200), 'sandwich': (50,160), 'glucose': (15,60), 'tin': (68,45), 'banana': (27,60), 'apple': (39,40), 'cheese': (23,30), 'beer': (52,10), 'suntan cream': (11,70), 'camera': (32,30), 't-shirt': (24,15), 'trousers': (48,10), 'umbrella': (73,40), 'waterproof trousers': (42,70), 'waterproof overclothes': (43,75), 'note-case': (22,80), 'sunglasses': (7,20), 'towel': (18,12), 'socks': (4,50), 'book': (30,10)>
system =
linear_system$[
binaries: ~&nS,
lower_bounds: {'(slack)': 0.}!,
costs: * ^|/~& negative+ float@r,
equations: ~&iNC\400.+ :/(1.,'(slack)')+ * ^|rlX/~& float@l]
- show+
main = ~&tnS solution system items</lang>
Binary valued variables are a more specific constraint than the general mixed integer programming problem, but can be accommodated as shown using the binaries field in the linear_system specification. The additional slack variable is specified as continuous and non-negative with no cost or benefit so as to make the constraint equation solvable without affecting the solution.
- Output:
banana compass glucose map note-case sandwich socks sunglasses suntan cream water waterproof overclothes waterproof trousers
VBA
<lang vb>'Knapsack problem/0-1 - 12/02/2017 Option Explicit Const maxWeight = 400 Dim DataList As Variant Dim xList(64, 3) As Variant Dim nItems As Integer Dim s As String, xss As String Dim xwei As Integer, xval As Integer, nn As Integer
Sub Main()
Dim i As Integer, j As Integer
DataList = Array("map", 9, 150, "compass", 13, 35, "water", 153, 200, "sandwich", 50, 160, _
"glucose", 15, 60, "tin", 68, 45, "banana", 27, 60, "apple", 39, 40, _
"cheese", 23, 30, "beer", 52, 10, "suntan cream", 11, 70, "camera", 32, 30, _
"T-shirt", 24, 15, "trousers", 48, 10, "umbrella", 73, 40, "book", 30, 10, _
"waterproof trousers", 42, 70, "waterproof overclothes", 43, 75, _
"note-case", 22, 80, "sunglasses", 7, 20, "towel", 18, 12, "socks", 4, 50)
nItems = (UBound(DataList) + 1) / 3
j = 0
For i = 1 To nItems
xList(i, 1) = DataList(j)
xList(i, 2) = DataList(j + 1)
xList(i, 3) = DataList(j + 2)
j = j + 3
Next i
s = ""
For i = 1 To nItems
s = s & Chr(i)
Next
nn = 0
Call ChoiceBin(1, "")
For i = 1 To Len(xss)
j = Asc(Mid(xss, i, 1))
Debug.Print xList(j, 1)
Next i
Debug.Print "count=" & Len(xss), "weight=" & xwei, "value=" & xval
End Sub 'Main
Private Sub ChoiceBin(n As String, ss As String)
Dim r As String
Dim i As Integer, j As Integer, iwei As Integer, ival As Integer
Dim ipct As Integer
If n = Len(s) + 1 Then
iwei = 0: ival = 0
For i = 1 To Len(ss)
j = Asc(Mid(ss, i, 1))
iwei = iwei + xList(j, 2)
ival = ival + xList(j, 3)
Next
If iwei <= maxWeight And ival > xval Then
xss = ss: xwei = iwei: xval = ival
End If
Else
r = Mid(s, n, 1)
Call ChoiceBin(n + 1, ss & r)
Call ChoiceBin(n + 1, ss)
End If
End Sub 'ChoiceBin</lang>
- Output:
map compass water sandwich glucose banana suntan cream waterproof trousers waterproof overclothes note-case sunglasses socks count=12 weight=396 value=1030
VBScript
Non recurvive unfolded force version. Created by an other script. It runs 13 times faster than the recursive one. <lang vb>' Knapsack problem/0-1 - 13/02/2017 dim w(22),v(22),m(22) data=array( "map", 9, 150, "compass", 13, 35, "water", 153, 200, _
"sandwich", 50, 160 , "glucose", 15, 60, "tin", 68, 45, _ "banana", 27, 60, "apple", 39, 40 , "cheese", 23, 30, "beer", 52, 10, _ "suntan cream", 11, 70, "camera", 32, 30 , "T-shirt", 24, 15, _ "trousers", 48, 10, "umbrella", 73, 40, "book", 30, 10 , _ "waterproof trousers", 42, 70, "waterproof overclothes", 43, 75 , _ "note-case", 22, 80, "sunglasses", 7, 20, "towel", 18, 12, "socks", 4, 50)
ww=400 xw=0:iw=0:iv=0 w(1)=iw:v(1)=iv for i1=0 to 1:m(1)=i1:j=0
if i1=1 then
iw=w(1)+data(j*3+1):iv=v(1)+data(j*3+2)
if iv>xv and iw<=ww then xw=iw:xv=iv:l=m
end if 'i1
if iw<=ww then
w(2)=iw: v(2)=iv
for i2=0 to 1:m(2)=i2:j=1
if i2=1 then
iw=w(2)+data(j*3+1):iv=v(2)+data(j*3+2)
if iv>xv and iw<=ww then xw=iw:xv=iv:l=m
end if 'i2
if iw<=ww then
w(3)=iw: v(3)=iv
for i3=0 to 1:m(3)=i3:j=2
if i3=1 then
iw=w(3)+data(j*3+1):iv=v(3)+data(j*3+2)
if iv>xv and iw<=ww then xw=iw:xv=iv:l=m
end if 'i3
if iw<=ww then
w(4)=iw: v(4)=iv
for i4=0 to 1:m(4)=i4:j=3
if i4=1 then
iw=w(4)+data(j*3+1):iv=v(4)+data(j*3+2)
if iv>xv and iw<=ww then xw=iw:xv=iv:l=m
end if 'i4
if iw<=ww then
w(5)=iw: v(5)=iv
for i5=0 to 1:m(5)=i5:j=4
if i5=1 then
iw=w(5)+data(j*3+1):iv=v(5)+data(j*3+2)
if iv>xv and iw<=ww then xw=iw:xv=iv:l=m
end if 'i5
if iw<=ww then
w(6)=iw: v(6)=iv
for i6=0 to 1:m(6)=i6:j=5
if i6=1 then
iw=w(6)+data(j*3+1):iv=v(6)+data(j*3+2)
if iv>xv and iw<=ww then xw=iw:xv=iv:l=m
end if 'i6
if iw<=ww then
w(7)=iw: v(7)=iv
for i7=0 to 1:m(7)=i7:j=6
if i7=1 then
iw=w(7)+data(j*3+1):iv=v(7)+data(j*3+2)
if iv>xv and iw<=ww then xw=iw:xv=iv:l=m
end if 'i7
if iw<=ww then
w(8)=iw: v(8)=iv
for i8=0 to 1:m(8)=i8:j=7
if i8=1 then
iw=w(8)+data(j*3+1):iv=v(8)+data(j*3+2)
if iv>xv and iw<=ww then xw=iw:xv=iv:l=m
end if 'i8
if iw<=ww then
w(9)=iw: v(9)=iv
for i9=0 to 1:m(9)=i9:j=8
if i9=1 then
iw=w(9)+data(j*3+1):iv=v(9)+data(j*3+2)
if iv>xv and iw<=ww then xw=iw:xv=iv:l=m
end if 'i9
if iw<=ww then
w(10)=iw: v(10)=iv
for i10=0 to 1:m(10)=i10:j=9
if i10=1 then
iw=w(10)+data(j*3+1):iv=v(10)+data(j*3+2)
if iv>xv and iw<=ww then xw=iw:xv=iv:l=m
end if 'i10
if iw<=ww then
w(11)=iw: v(11)=iv
for i11=0 to 1:m(11)=i11:j=10
if i11=1 then
iw=w(11)+data(j*3+1):iv=v(11)+data(j*3+2)
if iv>xv and iw<=ww then xw=iw:xv=iv:l=m
end if 'i11
if iw<=ww then
w(12)=iw: v(12)=iv
for i12=0 to 1:m(12)=i12:j=11
if i12=1 then
iw=w(12)+data(j*3+1):iv=v(12)+data(j*3+2)
if iv>xv and iw<=ww then xw=iw:xv=iv:l=m
end if 'i12
if iw<=ww then
w(13)=iw: v(13)=iv
for i13=0 to 1:m(13)=i13:j=12
if i13=1 then
iw=w(13)+data(j*3+1):iv=v(13)+data(j*3+2)
if iv>xv and iw<=ww then xw=iw:xv=iv:l=m
end if 'i13
if iw<=ww then
w(14)=iw: v(14)=iv
for i14=0 to 1:m(14)=i14:j=13
if i14=1 then
iw=w(14)+data(j*3+1):iv=v(14)+data(j*3+2)
if iv>xv and iw<=ww then xw=iw:xv=iv:l=m
end if 'i14
if iw<=ww then
w(15)=iw: v(15)=iv
for i15=0 to 1:m(15)=i15:j=14
if i15=1 then
iw=w(15)+data(j*3+1):iv=v(15)+data(j*3+2)
if iv>xv and iw<=ww then xw=iw:xv=iv:l=m
end if 'i15
if iw<=ww then
w(16)=iw: v(16)=iv
for i16=0 to 1:m(16)=i16:j=15
if i16=1 then
iw=w(16)+data(j*3+1):iv=v(16)+data(j*3+2)
if iv>xv and iw<=ww then xw=iw:xv=iv:l=m
end if 'i16
if iw<=ww then
w(17)=iw: v(17)=iv
for i17=0 to 1:m(17)=i17:j=16
if i17=1 then
iw=w(17)+data(j*3+1):iv=v(17)+data(j*3+2)
if iv>xv and iw<=ww then xw=iw:xv=iv:l=m
end if 'i17
if iw<=ww then
w(18)=iw: v(18)=iv
for i18=0 to 1:m(18)=i18:j=17
if i18=1 then
iw=w(18)+data(j*3+1):iv=v(18)+data(j*3+2)
if iv>xv and iw<=ww then xw=iw:xv=iv:l=m
end if 'i18
if iw<=ww then
w(19)=iw: v(19)=iv
for i19=0 to 1:m(19)=i19:j=18
if i19=1 then
iw=w(19)+data(j*3+1):iv=v(19)+data(j*3+2)
if iv>xv and iw<=ww then xw=iw:xv=iv:l=m
end if 'i19
if iw<=ww then
w(20)=iw: v(20)=iv
for i20=0 to 1:m(20)=i20:j=19
if i20=1 then
iw=w(20)+data(j*3+1):iv=v(20)+data(j*3+2)
if iv>xv and iw<=ww then xw=iw:xv=iv:l=m
end if 'i20
if iw<=ww then
w(21)=iw: v(21)=iv
for i21=0 to 1:m(21)=i21:j=20
if i21=1 then
iw=w(21)+data(j*3+1):iv=v(21)+data(j*3+2)
if iv>xv and iw<=ww then xw=iw:xv=iv:l=m
end if 'i21
if iw<=ww then
w(22)=iw: v(22)=iv
for i22=0 to 1:m(22)=i22:j=21
nn=nn+1
if i22=1 then
iw=w(22)+data(j*3+1):iv=v(22)+data(j*3+2)
if iv>xv and iw<=ww then xw=iw:xv=iv:l=m
end if 'i22
if iw<=ww then
end if 'i22
next:m(22)=0
end if 'i21
next:m(21)=0
end if 'i20
next:m(20)=0
end if 'i19
next:m(19)=0
end if 'i18
next:m(18)=0
end if 'i17
next:m(17)=0
end if 'i16
next:m(16)=0
end if 'i15
next:m(15)=0
end if 'i14
next:m(14)=0
end if 'i13
next:m(13)=0
end if 'i12
next:m(12)=0
end if 'i11
next:m(11)=0
end if 'i10
next:m(10)=0
end if 'i9
next:m(9)=0
end if 'i8
next:m(8)=0
end if 'i7
next:m(7)=0
end if 'i6
next:m(6)=0
end if 'i5
next:m(5)=0
end if 'i4
next:m(4)=0
end if 'i3
next:m(3)=0
end if 'i2
next:m(2)=0
end if 'i1
next:m(1)=0 for i=1 to 22
if l(i)=1 then wlist=wlist&vbCrlf&data((i-1)*3)
next Msgbox mid(wlist,3)&vbCrlf&vbCrlf&"weight="&xw&vbCrlf&"value="&xv,,"Knapsack - nn="&nn</lang>
- Output:
map compass water sandwich glucose banana suntan cream waterproof trousers waterproof overclothes note-case sunglasses socks weight=396 value=1030
Visual Basic
<lang vb>'Knapsack problem/0-1 - 12/02/2017 Option Explicit Const maxWeight = 400 Dim DataList As Variant Dim xList(64, 3) As Variant Dim nItems As Integer Dim s As String, xss As String Dim xwei As Integer, xval As Integer, nn As Integer
Private Sub Form_Load()
Dim i As Integer, j As Integer
DataList = Array("map", 9, 150, "compass", 13, 35, "water", 153, 200, "sandwich", 50, 160, _
"glucose", 15, 60, "tin", 68, 45, "banana", 27, 60, "apple", 39, 40, _
"cheese", 23, 30, "beer", 52, 10, "suntan cream", 11, 70, "camera", 32, 30, _
"T-shirt", 24, 15, "trousers", 48, 10, "umbrella", 73, 40, "book", 30, 10, _
"waterproof trousers", 42, 70, "waterproof overclothes", 43, 75, _
"note-case", 22, 80, "sunglasses", 7, 20, "towel", 18, 12, "socks", 4, 50)
nItems = (UBound(DataList) + 1) / 3
j = 0
For i = 1 To nItems
xList(i, 1) = DataList(j)
xList(i, 2) = DataList(j + 1)
xList(i, 3) = DataList(j + 2)
j = j + 3
Next i
For i = 1 To nItems
xListBox.AddItem xList(i, 1)
Next i
End Sub
Private Sub cmdOK_Click()
Dim i As Integer, j As Integer
For i = 1 To xListBox.ListCount
xListBox.RemoveItem 0
Next i
s = ""
For i = 1 To nItems
s = s & Chr(i)
Next
nn = 0
Call ChoiceBin(1, "")
For i = 1 To Len(xss)
j = Asc(Mid(xss, i, 1))
xListBox.AddItem xList(j, 1)
Next i
xListBox.AddItem "*Total* " & xwei & " " & xval
End Sub
Private Sub ChoiceBin(n As String, ss As String)
Dim r As String
Dim i As Integer, j As Integer, iwei As Integer, ival As Integer
Dim ipct As Integer
If n = Len(s) + 1 Then
iwei = 0: ival = 0
For i = 1 To Len(ss)
j = Asc(Mid(ss, i, 1))
iwei = iwei + xList(j, 2)
ival = ival + xList(j, 3)
Next
If iwei <= maxWeight And ival > xval Then
xss = ss: xwei = iwei: xval = ival
End If
Else
r = Mid(s, n, 1)
Call ChoiceBin(n + 1, ss & r)
Call ChoiceBin(n + 1, ss)
End If
End Sub 'ChoiceBin</lang>
- Output:
map compass water sandwich glucose banana suntan cream waterproof trousers waterproof overclothes note-case sunglasses socks *Total* weight=396 val=1030
Visual Basic .NET
<lang vbnet>'Knapsack problem/0-1 - 12/02/2017 Public Class KnapsackBin
Const knam = 0, kwei = 1, kval = 2
Const maxWeight = 400
Dim xList(,) As Object = { _
{"map", 9, 150}, _
{"compass", 13, 35}, _
{"water", 153, 200}, _
{"sandwich", 50, 160}, _
{"glucose", 15, 60}, _
{"tin", 68, 45}, _
{"banana", 27, 60}, _
{"ChoiceBinle", 39, 40}, _
{"cheese", 23, 30}, _
{"beer", 52, 10}, _
{"suntan cream", 11, 70}, _
{"camera", 32, 30}, _
{"T-shirt", 24, 15}, _
{"trousers", 48, 10}, _
{"umbrella", 73, 40}, _
{"waterproof trousers", 42, 70}, _
{"waterproof overclothes", 43, 75}, _
{"note-case", 22, 80}, _
{"sunglasses", 7, 20}, _
{"towel", 18, 12}, _
{"socks", 4, 50}, _
{"book", 30, 10}}
Dim s, xss As String, xwei, xval, nn As Integer
Private Sub KnapsackBin_Load(sender As Object, e As EventArgs) Handles MyBase.Load
Dim i As Integer
xListView.View = View.Details
xListView.Columns.Add("item", 120, HorizontalAlignment.Left)
xListView.Columns.Add("weight", 50, HorizontalAlignment.Right)
xListView.Columns.Add("value", 50, HorizontalAlignment.Right)
For i = 0 To UBound(xList, 1)
xListView.Items.Add(New ListViewItem(New String() {xList(i, 0), _
xList(i, 1).ToString, xList(i, 2).ToString}))
Next i
End Sub 'KnapsackBin_Load
Private Sub cmdOK_Click(sender As Object, e As EventArgs) Handles cmdOK.Click
Dim i, j, nItems As Integer
For i = xListView.Items.Count - 1 To 0 Step -1
xListView.Items.RemoveAt(i)
Next i
Me.Refresh()
nItems = UBound(xList, 1) + 1
s = ""
For i = 1 To nItems
s = s & Chr(i - 1)
Next
nn = 0
Call ChoiceBin(1, "")
For i = 1 To Len(xss)
j = Asc(Mid(xss, i, 1))
xListView.Items.Add(New ListViewItem(New String() {xList(j, 0), _
xList(j, 1).ToString, xList(j, 2).ToString}))
Next i
xListView.Items.Add(New ListViewItem(New String() {"*Total*", xwei, xval}))
End Sub 'cmdOK_Click
Private Sub ChoiceBin(n As String, ss As String)
Dim r As String, i, j, iwei, ival As Integer
Dim ipct As Integer
If n = Len(s) + 1 Then
iwei = 0 : ival = 0
For i = 1 To Len(ss)
j = Asc(Mid(ss, i, 1))
iwei = iwei + xList(j, 1)
ival = ival + xList(j, 2)
Next
If iwei <= maxWeight And ival > xval Then
xss = ss : xwei = iwei : xval = ival
End If
Else
r = Mid(s, n, 1)
Call ChoiceBin(n + 1, ss & r)
Call ChoiceBin(n + 1, ss)
End If
End Sub 'ChoiceBin
End Class 'KnapsackBin </lang>
- Output:
KnapsackBin_Load cmdOK_Click map compass water sandwich glucose banana suntan cream waterproof trousers waterproof overclothes note-case sunglasses socks *Total* weight=396 val=1030
Vlang
<lang go>struct Item {
name string w int v int
}
const wants = [
Item{'map', 9, 150},
Item{'compass', 13, 35},
Item{'water', 153, 200},
Item{'sandwich', 50, 60},
Item{'glucose', 15, 60},
Item{'tin', 68, 45},
Item{'banana', 27, 60},
Item{'apple', 39, 40},
Item{'cheese', 23, 30},
Item{'beer', 52, 10},
Item{'suntancream', 11, 70},
Item{'camera', 32, 30},
Item{'T-shirt', 24, 15},
Item{'trousers', 48, 10},
Item{'umbrella', 73, 40},
Item{'w-trousers', 42, 70},
Item{'w-overclothes', 43, 75},
Item{'note-case', 22, 80},
Item{'sunglasses', 7, 20},
Item{'towel', 18, 12},
Item{'socks', 4, 50},
Item{'book', 30, 10}
]
const max_wt = 400
fn main(){
items, w, v := m(wants.len-1, max_wt)
println(items)
println('weight: $w')
println('value: $v')
}
fn m(i int, w int) ([]string, int, int) {
if i<0 || w==0{
return []string{}, 0, 0
} else if wants[i].w > w {
return m(i-1, w)
}
i0, w0, v0 := m(i-1, w)
mut i1, w1, mut v1 := m(i-1, w-wants[i].w)
v1 += wants[i].v
if v1 > v0 {
i1 << wants[i].name
return i1, w1+wants[i].w, v1
}
return i0, w0, v0
}</lang>
- Output:
['map', 'compass', 'water', 'sandwich', 'glucose', 'banana', 'suntancream', 'w-trousers', 'w-overclothes', 'note-case', 'sunglasses', 'socks'] weight: 396 value: 930
Wren
Based on the Go example, though modified to give output in tabular form. <lang ecmascript>import "/fmt" for Fmt
var wants = [
["map", 9, 150], ["compass", 13, 35], ["water", 153, 200], ["sandwich", 50, 160], ["glucose", 15, 60], ["tin", 68, 45], ["banana", 27, 60], ["apple", 39, 40], ["cheese", 23, 30], ["beer", 52, 10], ["suntan cream", 11, 70], ["camera", 32, 30], ["T-shirt", 24, 15], ["trousers", 48, 10], ["umbrella", 73, 40], ["waterproof trousers", 42, 70], ["waterproof overclothes", 43, 75], ["note-case", 22, 80], ["sunglasses", 7, 20], ["towel", 18, 12], ["socks", 4, 50], ["book", 30, 10]
]
var m m = Fn.new { |i, w|
if (i < 0 || w == 0) return [[], 0, 0]
if (wants[i][1] > w) return m.call(i-1, w)
System.write("") // guard against VM recursion bug
var res = m.call(i-1, w)
var i0 = res[0]
var w0 = res[1]
var v0 = res[2]
res = m.call(i-1, w - wants[i][1])
var i1 = res[0]
var w1 = res[1]
var v1 = res[2] + wants[i][2]
if (v1 > v0) {
i1.add(wants[i])
return [i1, w1 + wants[i][1], v1]
}
return [i0, w0, v0]
}
var maxWt = 400 var res = m.call(wants.count-1, maxWt) var items = res[0] var tw = res[1] var tv = res[2] System.print("Max weight: %(maxWt)\n") System.print("Item Weight Value") System.print("------------------------------------") for (i in 0...items.count) {
Fmt.print("$-22s $3d $4s", items[i][0], items[i][1], items[i][2])
} System.print(" --- ----") Fmt.print("totals $3d $4d", tw, tv)</lang>
- Output:
Max weight: 400
Item Weight Value
------------------------------------
map 9 150
compass 13 35
water 153 200
sandwich 50 160
glucose 15 60
banana 27 60
suntan cream 11 70
waterproof trousers 42 70
waterproof overclothes 43 75
note-case 22 80
sunglasses 7 20
socks 4 50
--- ----
Totals 396 1030
XPL0
<lang XPL0>include c:\cxpl\codes; \include 'code' declarations
int Item, Items, Weights, Values,
BestItems, BestValues,
I, W, V, N;
def Tab=9; def Name, Weight, Value; [Item:= [["map ", 9, 150],
["compass ", 13, 35],
["water ", 153, 200],
["sandwich ", 50, 160],
["glucose ", 15, 60],
["tin ", 68, 45],
["banana ", 27, 60],
["apple ", 39, 40],
["cheese ", 23, 30],
["beer ", 52, 10],
["suntan cream ", 11, 70],
["camera ", 32, 30],
["T-shirt ", 24, 15],
["trousers ", 48, 10],
["umbrella ", 73, 40],
["waterproof trousers ", 42, 70],
["waterproof overclothes", 43, 75],
["note-case ", 22, 80],
["sunglasses ", 7, 20],
["towel ", 18, 12],
["socks ", 4, 50],
["book ", 30, 10]];
BestValues:= 0; for Items:= 0 to 1<<22-1 do \for all possible combinations of Items...
[I:= Items; W:= 0; V:= 0; N:= 0;
while I do \add weights and values for each item (bit in I)
[if I&1 then
[W:= W + Item(N,Weight); V:= V + Item(N,Value)];
I:= I>>1; N:= N+1;
];
if V>BestValues & W<=400 then \save best combination found so far
[BestValues:= V; BestItems:= Items];
];
I:= BestItems; W:= 0; V:= 0; N:= 0; \show best combination of items while I do
[if I&1 then
[Text(0, " "); Text(0, Item(N,Name)); ChOut(0, Tab);
IntOut(0, Item(N,Weight)); ChOut(0, Tab);
IntOut(0, Item(N,Value)); CrLf(0);
W:= W + Item(N,Weight);
V:= V + Item(N,Value);
];
I:= I>>1; N:= N+1;
];
Text(0, "Totals: "); IntOut(0, W); ChOut(0, Tab); IntOut(0, V); CrLf(0); ]</lang>
- Output:
map 9 150 compass 13 35 water 153 200 sandwich 50 160 glucose 15 60 banana 27 60 suntan cream 11 70 waterproof trousers 42 70 waterproof overclothes 43 75 note-case 22 80 sunglasses 7 20 socks 4 50 Totals: 396 1030
zkl
<lang zkl>fcn addItem(pairs,it){ // pairs is list of (cost of:,names), it is (name,w,v)
w,left,right:=it[1],pairs[0,w],pairs[w,*];
left.extend(right.zipWith(
fcn([(t1,_)]a,[(t2,_)]b){ t1>t2 and a or b },
pairs.apply('wrap([(tot,names)]){ T(tot + it[2], names + it[0]) })))
}//--> new list of pairs</lang> <lang zkl>items:=T(T("apple", 39, 40),T("banana", 27,60), // item: (name,weight,value)
T("beer", 52, 10),T("book", 30,10),T("camera", 32, 30),
T("cheese", 23, 30),T("compass", 13,35),T("glucose", 15, 60), T("map", 9,150),T("note-case",22,80),T("sandwich", 50,160), T("socks", 4, 50),T("sunglasses",7,20),T("suntan cream",11, 70), T("t-shirt", 24, 15),T("tin", 68,45),T("towel", 18, 12), T("trousers", 48, 10),T("umbrella", 73,40),T("water", 153,200), T("overclothes",43, 75),T("waterproof trousers",42,70) ); const MAX_WEIGHT=400; knapsack:=items.reduce(addItem,
(MAX_WEIGHT).pump(List,T(0,T).copy))[-1]; // nearest to max weight
weight:=items.apply('wrap(it){ knapsack[1].holds(it[0]) and it[1] }).sum(0); knapsack.println(weight);</lang>
- Output:
L(1030,L("banana","compass","glucose","map","note-case","sandwich","socks","sunglasses","suntan cream","water","overclothes","waterproof trousers"))396
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