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:
| 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.
Which items does the tourist carry in his knapsack so that their total weight does not exceed 400 dag [4 kg], and their total value is maximised?
[dag = decagram = 10 grams]
- See also
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
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
Brute force takes 0.2 seconds-ish, so not motivated to do caching. <lang c>#include <stdio.h>
- include <stdlib.h>
- include <stdint.h>
typedef struct {
char * name;
int weight, value;
} item_t;
item_t 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},
{"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}
};
- define n_items (sizeof(item)/sizeof(item_t))
typedef struct {
uint32_t bits; /* 32 bits, can solve up to 32 items */
int value;
} solution;
void optimal(int weight, int idx, solution *s)
{
solution v1, v2;
if (idx < 0) {
s->bits = s->value = 0;
return;
}
if (weight < item[idx].weight)
return optimal(weight, idx - 1, s);
optimal(weight, idx - 1, &v1);
optimal(weight - item[idx].weight, idx - 1, &v2);
v2.value += item[idx].value;
v2.bits |= (1 << idx);
*s = (v1.value >= v2.value) ? v1 : v2;
}
int main() {
int i = 0, w = 0;
solution s = {0, 0};
optimal(400, n_items - 1, &s);
for (i = 0; i < n_items; i++) {
if (s.bits & (1 << i)) {
printf("%s\n", item[i].name);
w += item[i].weight;
}
}
printf("Total value: %d; weight: %d\n", s.value, w);
return 0;
}</lang>
- Output:
map compass water sandwich glucose banana suntancream waterproof trousers waterproof overclothes note-case sunglasses socks Total value: 1030; weight: 396
C++
<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 !
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.)
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)))
D
<lang d>import std.stdio, std.algorithm, std.typecons, std.array;
struct Item {
string name; int weight, value;
}
Item[] knapsack01DinamicProg(in Item[] items, in int limit) pure nothrow {
auto tab = new int[][](items.length + 1, limit + 1);
foreach (i, it; items)
foreach (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);
Item[] result;
int w = limit;
foreach_reverse (i, it; items)
if (tab[i + 1][w] != tab[i][w]) {
w -= it.weight;
result ~= it;
}
return result;
}
void main() {
enum int limit = 400;
immutable Item[] 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}];
auto bagged = knapsack01DinamicProg(items, limit);
writeln("Items to pack:");
bagged.map!q{ a.name }().array().sort().join("\n").writeln();
const t = reduce!q{ a[] += [b.weight, b.value][] }([0,0], bagged);
const tot_wv = (t[0] <= limit) ? t : [0, 0];
writefln("\nFor a total weight of %d and a total value of %d",
tot_wv[0], tot_wv[1]);
}</lang>
- Output:
Items to pack: banana compass glucose map note-case sandwich socks sunglasses suntan cream water waterproof overclothes waterproof trousers For a total weight of 396 and a total value of 1030
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
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
Go
Another WP punch and run. <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},
}
func main() {
items, w, v := m(len(wants)-1, 400)
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
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>import Data.List import Data.Ord import Control.Monad import Control.Arrow
inv :: [(String,Int,Int)] inv = map ((\[i,w,v] -> (i,read w, read v)).words) $ lines $
"map\t9\t150\ncompass\t13\t35\nwater\t153\t200\nsandwich\t50\t160\n" ++ "glucose\t15\t60\ntin\t68\t45\nbanana\t27\t60\napple\t39\t40\ncheese\t23\t30\n" ++ "beer\t52\t10\nsuntan_cream\t11\t70\ncamera\t32\t30\ntshirt\t24\t15\n" ++ "trousers\t48\t10\numbrella\t73\t40\nwaterproof_trousers\t42\t70\n" ++ "waterproof_overclothes\t43\t75\nnotecase\t22\t80\nsunglasses\t7\t20\n" ++ "towel\t18\t12\nsocks\t4\t50\nbook\t30\t10"
int2bin :: Int -> [Int] int2bin = unfoldr(\x -> if x==0 then Nothing
else Just (uncurry(flip(,)) (divMod x 2)))
main = do
let fwv = foldl (\(wc,vc) (i,w,v) ->(wc+w,vc+v))(0,0). map (inv!!). elemIndices 1
fmt wb ib (n,w,v) = ns ++ f w ++ f v where
ns = (n ++) $ replicate (wb-length n) ' ' f n = reverse. take ib $ reverse (show n) ++ repeat ' '
((wc,vc),ii) =
foldl' ((.(fwv.int2bin &&& id)). (\o@((_,vo),_) n@((wn,vn),_) -> if wn<=400 && vn>vo then n else o) ) ((1000,0),0) [1..(2^length inv)-1]
mapM_ (putStrLn.fmt 30 5 .(inv!!)). elemIndices 1 $ int2bin ii
putStrLn $ fmt 30 5 ("Total",wc,vc)</lang>
Compile and execute in GHCi:
*Main> :! ghc --make ./Rosetta/knapsack01.hs [1 of 1] Compiling Main ( Rosetta/knapsack01.hs, Rosetta/knapsack01.o ) Linking Rosetta/knapsack01 ... *Main> :! ./Rosetta/knapsack01 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 notecase 22 80 sunglasses 7 20 socks 4 50 Total 396 1030
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>
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)
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.
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.
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([[0, []], map undef, 0 .. $max_weight], 0 .. $#name),
[ map([0, []], 0 .. $max_weight + 1)]);
sub optimal {
my ($i, $w) = @_;
if (!defined $p[$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 $sol = optimal($#name, $max_weight); print "$sol->[0]: @{$sol->[1]}\n";
- prints:
- 1030: map compass water sandwich glucose banana suntancream waterproof trousers
- waterproof overclothes note-case sunglasses socks</lang>
Perl 6
<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 = 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:
% perl6 knapsack1.p6 Value = 1030 Tourist put in the bag: socks sunglasses note-case waterproof overclothes waterproof trousers suntan cream banana glucose sandwich water compass map
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
- It works uncorrectly! For examle if $aw=4. Max value is true, but no "Array Indices" and its parameters are displayed
-
function knapSolveFast2($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,$pickedItems);
if ($w[$i] > $aW) { // Does it return too many?
$m[$i][$aW] = $without_i; // Memo without including this one $m['picked'][$i][$aW] = array(); // and a blank array entry... return array($without_i,array()); // 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]),$m,$pickedItems); $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,$m,$pickedItems);
- Display Result
echo "Items:
".join(", ",$items4)."
";
echo "Max Value Found:
$m4 (in $numcalls calls)
";
echo "Array Indices:
".join(",",$pickedItems)."
";
echo "Chosen Items:
";
echo "
"; echo "";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)
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 (pack (length Lst) ":" W)
(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
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>
REXX
Originally, the combination generator/checker subroutine (SIM) was recursive and made the program solution generic (and very 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 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. <lang rexx>/*REXX pgm solves a knapsack problem (22 items with weight restriction).*/ @.=
@.1 ='map 9 150' @.2 ='compass 13 35' @.3 ='water 153 200' @.4 ='sandwich 50 160' @.5 ='glucose 15 60' @.6 ='tin 68 45' @.7 ='banana 27 60' @.8 ='apple 39 40' @.9 ='cheese 23 30' @.10='beer 52 10' @.11='suntan_cream 11 70' @.12='camera 32 30' @.13='T-shirt 24 15' @.14='trousers 48 10' @.15='umbrella 73 40' @.16='waterproof_trousers 42 70' @.17='waterproof_overclothes 43 75' @.18='note-case 22 80' @.19='sunglasses 7 20' @.20='towel 18 12' @.21='socks 4 50' @.22='book 30 10'
maxWeight=400 /*the maximum weight for knapsack*/ say 'maximum weight allowed for a knapsack:' comma(maxWeight); say maxL=length('item') /*maximum width for table names. */ maxL=length('knapsack items') /*maximum width for table names. */ maxW=length('weight') /* " " " " weights*/ maxV=length('value') /* " " " " values.*/ maxQ=length('pieces') /* " " " " quant. */ highQ=0 /*max quantity specified (if any)*/ items=0; i.=; w.=0; v.=0; q.=0; Tw=0; Tv=0; Tq=0 /*initialize stuff.*/
/*────────────────────────────────sort the choices by decreasing weight.*/
do j=1 while @.j\== /*process each choice and sort. */
_=@.j; _wt=word(_,2) /*choose first item (arbitrary). */
_wt=word(_,2)
do k=j+1 while @.k\== /*find a possible heavier item. */
?wt=word(@.k,2)
if ?wt>_wt then do; _=@.k; @.k=@.j; @.j=_; _wt=?wt; end
end /*k*/
end /*j*/
obj=j-1 /*────────────────────────────────build list of choices.────────────────*/
do j=1 for obj /*build a list of choices. */
_=space(@.j) /*remove superflous blanks. */
parse var _ item w v q . /*parse original choice for table*/
Tw=Tw+w; Tv=Tv+v; Tq=Tq+1 /*add totals up (for alignment). */
if w>maxWeight then iterate /*ignore if the weight > maximum.*/
maxL=max(maxL,length(item)) /*find maximum width for item. */
if q== then q=1
highQ=max(highQ,q)
items=items+1 /*bump the item counter. */
@@.items=translate(_,','," ") /*make: item,weight,value */
i.items=item; w.items=w; v.items=v; q.items=q
do k=2 to q
items=items+1 /*bump the item counter. */
i.items=item; w.items=w; v.items=v; q.items=q
Tw=Tw+w; Tv=Tv+v; Tq=Tq+1
_=item# w v
@@.items=translate(_,','," ")
end
end
maxW=max(maxW,length(comma(Tw))) /*find maximum width for weight. */ maxV=max(maxV,length(comma(Tv))) /* " " " " value. */ maxQ=max(maxQ,length(comma(Tq))) /* " " " " quantity*/ maxL=maxL+maxL%4+4 /*extend width of name for table.*/
/*────────────────────────────────show the list of choices.─────────────*/ call hdr
do j=1 for obj /*show all choices, nice format. */
parse var @.j item weight value q .
if highq==1 then q=
else if q== then q=1
call show item,weight,value,q
end
say say 'number of items:' items say
/*─────────────────────────────────────examine the possible choices. */ h=items; ho=h+1; m=maxWeight; $=0; call sim22
/*─────────────────────────────────────show the best choice (weidht,val)*/
do h-1; ?=strip(strip(?),"L",0); end;
bestC=? bestW=0 bestV=$ highQ=0 call hdr 'best choice' totP=words(bestC)
do j=1 to totP /*J is modified within DO loop. */
_=word(bestC,j); _w=w._; _v=v._; q=1
do k=j+1 to totP
__=word(bestC,k); if i._\==i.__ then leave
j=j+1; w._=w._+_w; v._=v._+_v; q=q+1
end
call show i._,w._,v._,q
bestW=bestw+w._
end
call hdr2; say call show 'best weight' ,bestW /*show a nicely formatted winnerW*/ call show 'best value' ,,bestV /*show a nicely formatted winnerV*/ call show 'knapsack items',,,totP /*show a nicely formatted pieces.*/ exit
/*────────────────────────────────COMMA subroutine──────────────────────*/ comma: procedure; parse arg _,c,p,t;arg ,cu;c=word(c ",",1);
if cu=='BLANK' then c=' ';o=word(p 3,1);p=abs(o);t=word(t 999999999,1);
if \datatype(p,'W')|\datatype(t,'W')|p==0|arg()>4 then return _;n=_'.9';
#=123456789;k=0;if o<0 then do;b=verify(_,' ');if b==0 then return _;
e=length(_)-verify(reverse(_),' ')+1;end;else do;b=verify(n,#,"M");
e=verify(n,#'0',,verify(n,#"0.",'M'))-p-1;end;
do j=e to b by -p while k<t;_=insert(c,_,j);k=k+1;end;return _
/*────────────────────────────────SHOW subroutine───────────────────────*/ show: parse arg _item,_weight,_value,_quant say translate(left(_item,maxL,'─'),,'_'),
right(comma(_weight),maxW),
right(comma(_value ),maxV),
right(comma(_quant ),maxQ)
return
/*────────────────────────────────HDR subroutine────────────────────────*/ hdr: say; _=; if highq\==1 then _=center('pieces',maxq) _item_=arg(1); if _item_== then _item_='item' call show center( _item_ ,maxL),,
center('weight',maxW),,
center('value' ,maxV),,
center(_ ,maxQ)
call hdr2 return
/*────────────────────────────────HDR2 subroutine───────────────────────*/ hdr2: _=maxQ; if highq==1 then _=0 call show copies('=',maxL),,
copies('=',maxW),,
copies('=',maxV),,
copies('=',_ )
return
/*────────────────────────────────J? subroutine─────────────────────────*/ j?: parse arg _,?; $=value('V'_); do j=1 for _; ?=? value('J'j); end; return
/*────────────────────────────────SIM22 subroutine──────────────────────*/ sim22: do j1=0 for h+1 w1=w.j1;v1=v.j1;if v1>$ then call j? 1 do j2=j1+(j1\==0) to h if w.j2+w1>m then iterate j1;w2=w1+w.j2;v2=v1+v.j2;if v2>$ then call j? 2 do j3=j2+(j2\==0) to h if w.j3+w2>m then iterate j2;w3=w2+w.j3;v3=v2+v.j3;if v3>$ then call j? 3 do j4=j3+(j3\==0) to h if w.j4+w3>m then iterate j3;w4=w3+w.j4;v4=v3+v.j4;if v4>$ then call j? 4 do j5=j4+(j4\==0) to h if w.j5+w4>m then iterate j4;w5=w4+w.j5;v5=v4+v.j5;if v5>$ then call j? 5 do j6=j5+(j5\==0) to h if w.j6+w5>m then iterate j5;w6=w5+w.j6;v6=v5+v.j6;if v6>$ then call j? 6 do j7=j6+(j6\==0) to h if w.j7+w6>m then iterate j6;w7=w6+w.j7;v7=v6+v.j7;if v7>$ then call j? 7 do j8=j7+(j7\==0) to h if w.j8+w7>m then iterate j7;w8=w7+w.j8;v8=v7+v.j8;if v8>$ then call j? 8 do j9=j8+(j8\==0) to h if w.j9+w8>m then iterate j8;w9=w8+w.j9;v9=v8+v.j9;if v9>$ then call j? 9 do j10=j9+(j9\==0) to h if w.j10+w9>m then iterate j9;w10=w9+w.j10;v10=v9+v.j10;if v10>$ then call j? 10 do j11=j10+(j10\==0) to h if w.j11+w10>m then iterate j10;w11=w10+w.j11;v11=v10+v.j11;if v11>$ then call j? 11 do j12=j11+(j11\==0) to h if w.j12+w11>m then iterate j11;w12=w11+w.j12;v12=v11+v.j12;if v12>$ then call j? 12 do j13=j12+(j12\==0) to h if w.j13+w12>m then iterate j12;w13=w12+w.j13;v13=v12+v.j13;if v13>$ then call j? 13 do j14=j13+(j13\==0) to h if w.j14+w13>m then iterate j13;w14=w13+w.j14;v14=v13+v.j14;if v14>$ then call j? 14 do j15=j14+(j14\==0) to h if w.j15+w14>m then iterate j14;w15=w14+w.j15;v15=v14+v.j15;if v15>$ then call j? 15 do j16=j15+(j15\==0) to h if w.j16+w15>m then iterate j15;w16=w15+w.j16;v16=v15+v.j16;if v16>$ then call j? 16 do j17=j16+(j16\==0) to h if w.j17+w16>m then iterate j16;w17=w16+w.j17;v17=v16+v.j17;if v17>$ then call j? 17 do j18=j17+(j17\==0) to h if w.j18+w17>m then iterate j17;w18=w17+w.j18;v18=v17+v.j18;if v18>$ then call j? 18 do j19=j18+(j18\==0) to h if w.j19+w18>m then iterate j18;w19=w18+w.j19;v19=v18+v.j19;if v19>$ then call j? 19 do j20=j19+(j19\==0) to h if w.j20+w19>m then iterate j19;w20=w19+w.j20;v20=v19+v.j20;if v20>$ then call j? 20 do j21=j20+(j20\==0) to h if w.j21+w20>m then iterate j20;w21=w20+w.j21;v21=v20+v.j21;if v21>$ then call j? 21 do j22=j21+(j21\==0) to h if w.j22+w21>m then iterate j21;w22=w21+w.j22;v22=v21+v.j22;if v22>$ then call j? 22 end;end;end;end;end;end;end;end;end;end;end;end;end;end;end;end;end;end;end;end;end;end return</lang>
- Output:
maximum weight allowed for a knapsack: 400
item 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 items: 22
best choice weight value pieces
=============================== ====== ===== ======
water────────────────────────── 153 200 1
sandwich─────────────────────── 50 160 1
waterproof overclothes───────── 43 75 1
waterproof trousers──────────── 42 70 1
banana───────────────────────── 27 60 1
note-case────────────────────── 22 80 1
glucose──────────────────────── 15 60 1
compass──────────────────────── 13 35 1
suntan cream─────────────────── 11 70 1
map──────────────────────────── 9 150 1
sunglasses───────────────────── 7 20 1
socks────────────────────────── 4 50 1
=============================== ====== ===== ======
best weight──────────────────── 396
best value───────────────────── 1,030
knapsack items───────────────── 12
Ruby
Brute force
<lang ruby>KnapsackItem = Struct.new(:name, :weight, :value) potential_items = [
KnapsackItem.new('map', 9, 150), KnapsackItem.new('compass', 13, 35),
KnapsackItem.new('water', 153, 200), KnapsackItem.new('sandwich', 50, 160),
KnapsackItem.new('glucose', 15, 60), KnapsackItem.new('tin', 68, 45),
KnapsackItem.new('banana', 27, 60), KnapsackItem.new('apple', 39, 40),
KnapsackItem.new('cheese', 23, 30), KnapsackItem.new('beer', 52, 10),
KnapsackItem.new('suntan cream', 11, 70), KnapsackItem.new('camera', 32, 30),
KnapsackItem.new('t-shirt', 24, 15), KnapsackItem.new('trousers', 48, 10),
KnapsackItem.new('umbrella', 73, 40), KnapsackItem.new('waterproof trousers', 42, 70),
KnapsackItem.new('waterproof overclothes', 43, 75), KnapsackItem.new('note-case', 22, 80),
KnapsackItem.new('sunglasses', 7, 20), KnapsackItem.new('towel', 18, 12),
KnapsackItem.new('socks', 4, 50), KnapsackItem.new('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|
r = []
ps.each do |i|
r << i
new_subset = i + [elem]
yield new_subset if block_given?
r << new_subset
end
r
end
end
end
maxval = 0 solutions = []
potential_items.power_set do |subset|
weight = subset.inject(0) {|w, elem| w += elem.weight}
next if weight > knapsack_capacity
value = subset.inject(0) {|v, elem| v += elem.value}
if value == maxval
solutions << subset
elsif value > maxval
maxval = value
solutions = [subset]
end
end
puts "value: #{maxval}" solutions.each do |set|
items = []
wt = 0
set.each {|elem| wt += elem.weight; items << elem.name}
puts "weight: #{wt}"
puts "items: #{items.sort.join(',')}"
end</lang>
- Output:
value: 1030 weight: 396 items: banana,compass,glucose,map,note-case,sandwich,socks,sunglasses,suntan cream,water,waterproof overclothes,waterproof trousers
Dynamic Programming
Translated from http://sites.google.com/site/mikescoderama/Home/0-1-knapsack-problem-in-p <lang ruby>KnapsackItem = Struct.new(:name, :cost, :value) KnapsackProblem = Struct.new(:items, :max_cost)
def dynamic_programming_knapsack(problem)
num_items = problem.items.size items = problem.items max_cost = problem.max_cost
cost_matrix = zeros(num_items, max_cost+1)
num_items.times do |i|
(max_cost + 1).times do |j|
if(items[i].cost > 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].cost]].max
end
end
end
cost_matrix
end
def get_used_items(problem, cost_matrix)
i = cost_matrix.size - 1 currentCost = cost_matrix[0].size - 1 marked = Array.new(cost_matrix.size, 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 -= problem.items[i].cost
end
i -= 1
end
marked
end
def get_list_of_used_items_names(problem, cost_matrix)
items = problem.items used_items = get_used_items(problem, cost_matrix)
result = []
used_items.each_with_index do |item,i|
if item > 0
result << items[i].name
end
end
result.sort.join(', ')
end
def zeros(rows, cols)
Array.new(rows) do |row| Array.new(cols, 0) end
end
if $0 == __FILE__
items = [
KnapsackItem.new('map' , 9 , 150) , KnapsackItem.new('compass' , 13 , 35) ,
KnapsackItem.new('water' , 153 , 200) , KnapsackItem.new('sandwich' , 50 , 160) ,
KnapsackItem.new('glucose' , 15 , 60) , KnapsackItem.new('tin' , 68 , 45) ,
KnapsackItem.new('banana' , 27 , 60) , KnapsackItem.new('apple' , 39 , 40) ,
KnapsackItem.new('cheese' , 23 , 30) , KnapsackItem.new('beer' , 52 , 10) ,
KnapsackItem.new('suntan cream' , 11 , 70) , KnapsackItem.new('camera' , 32 , 30) ,
KnapsackItem.new('t-shirt' , 24 , 15) , KnapsackItem.new('trousers' , 48 , 10) ,
KnapsackItem.new('umbrella' , 73 , 40) , KnapsackItem.new('waterproof trousers' , 42 , 70) ,
KnapsackItem.new('waterproof overclothes' , 43 , 75) , KnapsackItem.new('note-case' , 22 , 80) ,
KnapsackItem.new('sunglasses' , 7 , 20) , KnapsackItem.new('towel' , 18 , 12) ,
KnapsackItem.new('socks' , 4 , 50) , KnapsackItem.new('book' , 30 , 10)
]
problem = KnapsackProblem.new(items, 400)
cost_matrix = dynamic_programming_knapsack problem puts puts 'Dynamic Programming:' puts puts 'Found solution: ' + get_list_of_used_items_names(problem, cost_matrix) puts 'With value: ' + cost_matrix.last.last.to_s
end</lang>
Dynamic Programming: Found solution: banana, compass, glucose, map, note-case, sandwich, socks, sunglasses, suntan cream, water, waterproof overclothes, waterproof trousers With value: 1030
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