Subset sum problem
Implement a function/procedure/method/subroutine that takes a set/array/list/stream/table/collection of words with integer weights, and identifies a non-empty subset of them whose weights sum to zero (cf. the Dropbox Diet candidate screening exercise and the Subset sum problem Wikipedia article).
For example, for this set of weighted words, one solution would be the set of words {elysee, efferent, deploy, departure, centipede, bonnet, balm, archbishop}, because their respective weights of -326, 54, 44, 952, -658, 452, 397, and -915 sum to zero.
| word | weight |
|---|---|
| alliance | -624 |
| archbishop | -915 |
| balm | 397 |
| bonnet | 452 |
| brute | 870 |
| centipede | -658 |
| cobol | 362 |
| covariate | 590 |
| departure | 952 |
| deploy | 44 |
| diophantine | 645 |
| efferent | 54 |
| elysee | -326 |
| eradicate | 376 |
| escritoire | 856 |
| exorcism | -983 |
| fiat | 170 |
| filmy | -874 |
| flatworm | 503 |
| gestapo | 915 |
| infra | -847 |
| isis | -982 |
| lindholm | 999 |
| markham | 475 |
| mincemeat | -880 |
| moresby | 756 |
| mycenae | 183 |
| plugging | -266 |
| smokescreen | 423 |
| speakeasy | -745 |
| vein | 813 |
Another solution would be the set of words {flatworm, gestapo, infra, isis, lindholm, plugging, smokescreen, speakeasy}, because their respective weights of 503, 915, -847, -982, 999, -266, 423, and -745 also sum to zero.
You may assume the weights range from -1000 to 1000. If there are multiple solutions, only one needs to be found. Use any algorithm you want and demonstrate it on a set of at least 30 weighted words with the results shown in a human readable form. Note that an implementation that depends on enumerating all possible subsets is likely to be infeasible.
Ada
<lang Ada>with Ada.Text_IO; use Ada.Text_IO; with Ada.Strings.Unbounded; use Ada.Strings.Unbounded; procedure SubsetSum is
function "+"(S:String) return Unbounded_String renames To_Unbounded_String;
type Point is record
str : Unbounded_String;
num : Integer;
end record;
type Points is array (Natural range <>) of Point;
type Indices is array (Natural range <>) of Natural;
procedure Print (data : Points; list : Indices; len : Positive) is begin
Put (len'Img & ":");
for i in 0..len-1 loop
Put (" "& To_String(data(list(i)).str));
end loop; New_Line;
end Print;
function Check (data : Points; list : Indices; len : Positive) return Boolean is
sum : Integer := 0;
begin
for i in 0..len-1 loop sum := sum + data(list(i)).num; end loop;
return sum = 0;
end Check;
procedure Next (list : in out Indices; n, r : Positive ) is begin
for i in reverse 0..r-1 loop
if list(i)/=i+n-r then list(i):=list(i)+1;
for j in i+1..r-1 loop list(j):=list(j-1)+1; end loop; exit;
end if;
end loop;
end Next;
data : constant Points := ((+"alliance", -624), (+"archbishop", -915),
(+"balm", 397), (+"bonnet", 452), (+"brute", 870),
(+"centipede", -658), (+"cobol", 362), (+"covariate", 590),
(+"departure", 952), (+"deploy", 44), (+"diophantine", 645),
(+"efferent", 54), (+"elysee", -326), (+"eradicate", 376),
(+"escritoire", 856), (+"exorcism", -983), (+"fiat", 170),
(+"filmy", -874), (+"flatworm", 503), (+"gestapo", 915),
(+"infra", -847), (+"isis", -982), (+"lindholm", 999),
(+"markham", 475), (+"mincemeat", -880), (+"moresby", 756),
(+"mycenae", 183), (+"plugging", -266), (+"smokescreen", 423),
(+"speakeasy", -745), (+"vein", 813));
list, last : Indices (data'Range);
begin
for len in 2..data'Length loop
for i in 0..len-1 loop list(i):=i; end loop;
loop
if Check(data, list, len) then Print(data, list, len); exit; end if;
last := list;
Next(list, data'Length, len);
exit when last=list;
end loop;
end loop;
end SubsetSum;</lang>
- Output:
2: archbishop gestapo 3: centipede markham mycenae 4: alliance balm deploy mycenae 5: alliance brute covariate deploy mincemeat 6: alliance archbishop balm deploy gestapo mycenae 7: alliance archbishop bonnet cobol departure exorcism moresby 8: alliance archbishop balm bonnet fiat flatworm isis lindholm 9: alliance archbishop balm bonnet brute covariate eradicate mincemeat plugging 10: alliance archbishop balm bonnet brute centipede cobol departure deploy mincemeat 11: alliance archbishop balm bonnet brute centipede cobol departure infra moresby speakeasy 12: alliance archbishop balm bonnet brute centipede cobol covariate diophantine efferent elysee infra 13: alliance archbishop balm bonnet brute centipede cobol covariate departure efferent eradicate filmy isis 14: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy elysee filmy markham speakeasy 15: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy elysee exorcism flatworm infra mycenae 16: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine elysee exorcism filmy gestapo infra 17: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine exorcism isis mincemeat mycenae plugging vein 18: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee exorcism filmy isis mycenae vein 19: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate exorcism fiat infra isis smokescreen 20: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate exorcism gestapo infra isis smokescreen speakeasy 21: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate exorcism flatworm infra lindholm mincemeat plugging speakeasy 22: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate escritoire exorcism fiat filmy flatworm mincemeat plugging speakeasy 23: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate escritoire exorcism infra isis mincemeat moresby mycenae smokescreen speakeasy 24: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee exorcism filmy gestapo infra markham mincemeat moresby mycenae plugging smokescreen speakeasy 25: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine eradicate exorcism fiat filmy flatworm infra isis lindholm markham mincemeat moresby mycenae plugging speakeasy 26: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine elysee eradicate escritoire exorcism fiat filmy gestapo infra isis markham mincemeat mycenae plugging speakeasy vein 27: alliance archbishop balm bonnet brute centipede covariate departure deploy efferent elysee eradicate escritoire exorcism fiat filmy flatworm infra isis lindholm markham mincemeat moresby mycenae plugging smokescreen speakeasy
C
<lang c>#include <stdio.h>
- include <stdlib.h>
- include <stdint.h>
- include <string.h>
typedef struct { void* data; int weight; } item;
uint64_t subsum(item *x, int n) { int i, j, w, from, to, step, pos = 0, neg = 0; uint64_t bit, *buf;
for (i = 0; i < n; i++) if (x[i].weight >= 0) pos += x[i].weight; else neg += x[i].weight;
buf = calloc(pos - neg + 1, sizeof(uint64_t)); buf -= neg;
for (i = 0; i < n; i++) { w = x[i].weight; bit = (uint64_t)1 << i;
if (w < 0) from = neg, to = pos + 1, step = 1; else from = pos, to = neg - 1, step = -1;
for (j = from; j != to; j += step) if (buf[j]) buf[j + w] = buf[j] | bit; buf[w] = bit;
if (buf[0]) break; }
bit = buf[0]; free(buf + neg);
return bit; }
int main(void) { item em[] = { {"alliance", -624}, {"archbishop", -915}, {"balm", 397}, {"bonnet", 452}, {"brute", 870}, {"centipede", -658}, {"cobol", 362}, {"covariate", 590}, {"departure", 952}, {"deploy", 44}, {"diophantine", 645}, {"efferent", 54}, {"elysee", -326}, {"eradicate", 376}, {"escritoire", 856}, {"exorcism", -983}, {"fiat", 170}, {"filmy", -874}, {"flatworm", 503}, {"gestapo", 915}, {"infra", -847}, {"isis", -982}, {"lindholm", 999}, {"markham", 475}, {"mincemeat", -880}, {"moresby", 756}, {"mycenae", 183}, {"plugging", -266}, {"smokescreen", 423}, {"speakeasy", -745}, {"vein", 813} };
uint64_t i, v, ret = subsum(em, sizeof(em)/sizeof(em[0])); if (!ret) { puts("no zero sums\n"); return 1; }
puts("Found zero sum:"); for (i = 0; i < 64; i++) { v = (uint64_t)1 << i; if (ret & v) printf("%2llu | %5d %s\n", i, em[i].weight, (char*)em[i].data); }
return 0;
}</lang>
- Output:
Found zero sum: 1 | -915 archbishop 2 | 397 balm 3 | 452 bonnet 5 | -658 centipede 8 | 952 departure 9 | 44 deploy 11 | 54 efferent 12 | -326 elysee
D
A simple brute-force solution.
<lang d>import std.stdio, std.algorithm, std.string, std.typecons;
T[][] combinations(T)(T[] arr, int k) {
if (k == 0) return [[]];
T[][] result;
foreach (i, x; arr)
foreach (suffix; combinations(arr[i+1 .. $], k-1))
result ~= x ~ suffix;
return result;
}
void main() {
alias tuple T; immutable items = [
T("alliance", -624), T("archbishop", -915), T("balm", 397), T("bonnet", 452), T("brute", 870), T("centipede", -658), T("cobol", 362), T("covariate", 590), T("departure", 952), T("deploy", 44), T("diophantine", 645), T("efferent", 54), T("elysee", -326), T("eradicate", 376), T("escritoire", 856), T("exorcism", -983), T("fiat", 170), T("filmy", -874), T("flatworm", 503), T("gestapo", 915), T("infra", -847), T("isis", -982), T("lindholm", 999), T("markham", 475), T("mincemeat", -880), T("moresby", 756), T("mycenae", 183), T("plugging", -266), T("smokescreen", 423), T("speakeasy", -745), T("vein", 813)];
foreach (n; 1 .. items.length)
foreach (comb; combinations(items, n))
if (reduce!q{ a + b[1] }(0, comb) == 0) {
writefln("A subset of length %d: %s", n,
//comb.map!q{ a[0] }.join(", "));
comb.map!q{ cast()a[0] }.join(", "));
return;
}
writefln("No solution found.");
}</lang> Output:
A subset of length 2: archbishop, gestapo
Go
<lang go>package main
import "fmt"
type ww struct {
word string weight int
}
var input = []*ww{
{"alliance", -624},
{"archbishop", -915},
{"balm", 397},
{"bonnet", 452},
{"brute", 870},
{"centipede", -658},
{"cobol", 362},
{"covariate", 590},
{"departure", 952},
{"deploy", 44},
{"diophantine", 645},
{"efferent", 54},
{"elysee", -326},
{"eradicate", 376},
{"escritoire", 856},
{"exorcism", -983},
{"fiat", 170},
{"filmy", -874},
{"flatworm", 503},
{"gestapo", 915},
{"infra", -847},
{"isis", -982},
{"lindholm", 999},
{"markham", 475},
{"mincemeat", -880},
{"moresby", 756},
{"mycenae", 183},
{"plugging", -266},
{"smokescreen", 423},
{"speakeasy", -745},
{"vein", 813},
}
type sss struct {
subset []*ww sum int
}
func main() {
ps := []sssTemplate:Nil, 0 for _, i := range input { pl := len(ps) for j := 0; j < pl; j++ { subset := append([]*ww{i}, ps[j].subset...) sum := i.weight + ps[j].sum if sum == 0 { fmt.Println("this subset sums to 0:") for _, i := range subset { fmt.Println(*i) } return } ps = append(ps, sss{subset, sum}) } } fmt.Println("no subset sums to 0")
}</lang>
- Output:
this subset sums to 0:
{elysee -326}
{efferent 54}
{deploy 44}
{covariate 590}
{cobol 362}
{centipede -658}
{bonnet 452}
{balm 397}
{archbishop -915}
Haskell
<lang haskell>combinations :: Int -> [a] -> a combinations 0 _ = [[]] combinations _ [] = [] combinations k (x:xs) = map (x:) (combinations (k - 1) xs) ++
combinations k xs
data W = W { word :: String,
weight :: Int }
solver :: [W] -> W solver it = [comb | n <- [1 .. length it],
comb <- combinations n it,
foldr (\a acc -> weight a + acc) 0 comb == 0]
items = [W "alliance" (-624), W "archbishop" (-915),
W "balm" 397, W "bonnet" 452,
W "brute" 870, W "centipede" (-658),
W "cobol" 362, W "covariate" 590,
W "departure" 952, W "deploy" 44,
W "diophantine" 645, W "efferent" 54,
W "elysee" (-326), W "eradicate" 376,
W "escritoire" 856, W "exorcism" (-983),
W "fiat" 170, W "filmy" (-874),
W "flatworm" 503, W "gestapo" 915,
W "infra" (-847), W "isis" (-982),
W "lindholm" 999, W "markham" 475,
W "mincemeat" (-880), W "moresby" 756,
W "mycenae" 183, W "plugging" (-266),
W "smokescreen" 423, W "speakeasy" (-745),
W "vein" 813]
main = print $ map word $ head $ solver items</lang>
- Output:
["archbishop","gestapo"]
Icon and Unicon
<lang Icon>link printf,lists
procedure main()
BruteZeroSubset(string2table(
"alliance/-624/archbishop/-915/balm/397/bonnet/452/brute/870/_
centipede/-658/cobol/362/covariate/590/departure/952/deploy/44/_
diophantine/645/efferent/54/elysee/-326/eradicate/376/escritoire/856/_
exorcism/-983/fiat/170/filmy/-874/flatworm/503/gestapo/915/infra/-847/_
isis/-982/lindholm/999/markham/475/mincemeat/-880/moresby/756/_
mycenae/183/plugging/-266/smokescreen/423/speakeasy/-745/vein/813/"))
end
procedure BruteZeroSubset(words) # brute force 1 of each length
every n := 1 to *words do {
every t := tcomb(words,n) do { # generate combination
every (sum := 0) +:= words[!t] # sum combination
if sum = 0 then {
printf("A zero-sum subset of length %d : %s\n",n,list2string(sort(t)))
break next # found one
}
}
printf("No zero-sum subsets of length %d\n",n)
}
end
- helper procedures
procedure tcomb(T, i) #: Table (key) combinations
local K every put(K := [],key(T)) # list of keys every suspend lcomb(K,i) # return list combs
end
procedure list2string(L) #: format list as a string
every (s := "[ ") ||:= !L || " " # reformat as string return s || "]"
end
procedure string2table(s,d) #: format string "k1/v1/.../kn/vn" as table
T := table()
/d := "/"
s ? until pos(0) do
T[1(tab(find(d)),=d)] := numeric(1(tab(find(d)),=d))
return T
end</lang>
printf.icn provides formatting lists.icn provides lcomb for list combinations
Output:
No zero-sum subsets of length 1 A zero-sum subset of length 2 : [ archbishop gestapo ] A zero-sum subset of length 3 : [ centipede markham mycenae ] A zero-sum subset of length 4 : [ alliance balm deploy mycenae ] A zero-sum subset of length 5 : [ balm eradicate isis markham plugging ] A zero-sum subset of length 6 : [ archbishop balm escritoire exorcism fiat markham ] A zero-sum subset of length 7 : [ balm bonnet cobol fiat filmy isis markham ] A zero-sum subset of length 8 : [ balm bonnet cobol filmy markham mincemeat speakeasy vein ] A zero-sum subset of length 9 : [ alliance archbishop balm bonnet cobol lindholm markham mincemeat plugging ] A zero-sum subset of length 10 : [ archbishop balm bonnet cobol filmy gestapo markham mincemeat speakeasy vein ] A zero-sum subset of length 11 : [ alliance archbishop balm bonnet cobol deploy gestapo isis markham mincemeat moresby ] A zero-sum subset of length 12 : [ alliance archbishop balm bonnet cobol exorcism fiat lindholm markham mincemeat plugging vein ] A zero-sum subset of length 13 : [ alliance archbishop balm bonnet brute cobol deploy diophantine exorcism markham mincemeat plugging smokescreen ] A zero-sum subset of length 14 : [ alliance archbishop balm bonnet centipede cobol diophantine exorcism lindholm markham mincemeat mycenae plugging vein ] A zero-sum subset of length 15 : [ alliance archbishop balm bonnet cobol diophantine fiat gestapo isis markham mincemeat mycenae plugging speakeasy vein ] A zero-sum subset of length 16 : [ alliance archbishop balm bonnet brute cobol diophantine eradicate exorcism filmy infra lindholm markham mincemeat plugging vein ] A zero-sum subset of length 17 : [ alliance archbishop balm bonnet centipede cobol covariate deploy diophantine exorcism filmy lindholm markham mincemeat plugging smokescreen vein ] A zero-sum subset of length 18 : [ alliance archbishop balm bonnet centipede cobol diophantine eradicate escritoire exorcism filmy gestapo infra markham mincemeat moresby plugging vein ] A zero-sum subset of length 19 : [ alliance archbishop balm bonnet cobol diophantine efferent exorcism filmy flatworm gestapo infra isis lindholm markham mincemeat moresby plugging vein ] A zero-sum subset of length 20 : [ alliance archbishop balm bonnet centipede cobol deploy diophantine efferent escritoire exorcism fiat filmy gestapo isis lindholm markham mincemeat plugging vein ] A zero-sum subset of length 21 : [ alliance archbishop balm bonnet brute centipede cobol covariate deploy diophantine efferent elysee exorcism filmy gestapo infra markham mincemeat moresby plugging vein ] A zero-sum subset of length 22 : [ alliance archbishop balm bonnet centipede cobol deploy diophantine eradicate escritoire exorcism fiat filmy gestapo isis lindholm markham mincemeat plugging smokescreen speakeasy vein ] A zero-sum subset of length 23 : [ alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine exorcism filmy flatworm gestapo infra isis markham mincemeat moresby plugging speakeasy vein ] A zero-sum subset of length 24 : [ alliance archbishop balm bonnet brute centipede cobol departure deploy diophantine efferent escritoire exorcism filmy gestapo infra isis markham mincemeat moresby mycenae plugging speakeasy vein ] A zero-sum subset of length 25 : [ alliance archbishop balm bonnet brute centipede cobol covariate deploy diophantine efferent elysee eradicate exorcism filmy gestapo infra isis markham mincemeat moresby mycenae plugging smokescreen vein ] A zero-sum subset of length 26 : [ alliance archbishop balm bonnet centipede cobol covariate departure deploy diophantine efferent elysee eradicate escritoire exorcism fiat filmy gestapo infra isis lindholm markham mincemeat plugging speakeasy vein ] A zero-sum subset of length 27 : [ alliance archbishop balm bonnet brute centipede covariate departure deploy efferent elysee eradicate escritoire exorcism fiat filmy flatworm infra isis lindholm markham mincemeat moresby mycenae plugging smokescreen speakeasy ] No zero-sum subsets of length 28 No zero-sum subsets of length 29 No zero-sum subsets of length 30 No zero-sum subsets of length 31
Mathematica
<lang Mathematica>a = {{"alliance", -624}, {"archbishop", -915}, {"balm", 397}, {"bonnet", 452}, {"brute", 870}, {"centipede", -658}, {"cobol", 362}, {"covariate", 590},{"departure", 952}, {"deploy", 44}, {"diophantine", 645}, {"efferent", 54}, {"elysee", -326}, {"eradicate", 376}, {"escritoire", 856}, {"exorcism", -983}, {"fiat", 170}, {"filmy", -874}, {"flatworm", 503}, {"gestapo", 915}, {"infra", -847}, {"isis", -982}, {"lindholm", 999}, {"markham", 475}, {"mincemeat", -880}, {"moresby", 756}, {"mycenae", 183}, {"plugging", -266}, {"smokescreen", 423}, {"speakeasy", -745}, {"vein", 813}};
result = Rest@Select[ Subsets[a, 7], (Total[#;; , 2] == 0) &]; Map[ (Print["A zero-sum subset of length ", Length[#], " : ", #;; , 1])& , result ]</lang>
A zero-sum subset of length 2 : {archbishop,gestapo}
A zero-sum subset of length 3 : {centipede,markham,mycenae}
A zero-sum subset of length 3 : {exorcism,fiat,vein}
A zero-sum subset of length 4 : {alliance,balm,deploy,mycenae}
A zero-sum subset of length 4 : {balm,efferent,filmy,smokescreen}
A zero-sum subset of length 4 : {bonnet,elysee,escritoire,isis}
A zero-sum subset of length 4 : {brute,centipede,efferent,plugging}
....
OCaml
Just search randomly until a result is found:
<lang ocaml>let d =
[ "alliance", -624; "archbishop", -915; "balm", 397; "bonnet", 452; "brute", 870; "centipede", -658; "cobol", 362; "covariate", 590; "departure", 952; "deploy", 44; "diophantine", 645; "efferent", 54; "elysee", -326; "eradicate", 376; "escritoire", 856; "exorcism", -983; "fiat", 170; "filmy", -874; "flatworm", 503; "gestapo", 915; "infra", -847; "isis", -982; "lindholm", 999; "markham", 475; "mincemeat", -880; "moresby", 756; "mycenae", 183; "plugging", -266; "smokescreen", 423; "speakeasy", -745; "vein", 813; ]
let sum = List.fold_left (fun sum (_,w) -> sum + w) 0 let p = function [] -> false | lst -> (sum lst) = 0
let take lst set =
let x = List.nth set (Random.int (List.length set)) in (x::lst, List.filter (fun y -> y <> x) set)
let swap (a, b) = (b, a) let pop lst set = swap (take set lst)
let () =
Random.self_init ();
let rec aux lst set =
let f =
match lst, set with
| [], _ -> take
| _, [] -> pop
| _ -> if Random.bool () then take else pop
in
let lst, set = f lst set in
if p lst then lst
else aux lst set
in
let res = aux [] d in
List.iter (fun (n,w) -> Printf.printf " %4d\t%s\n" w n) res</lang>
PicoLisp
<lang PicoLisp>(de *Words
(alliance . -624) (archbishop . -915) (balm . 397) (bonnet . 452) (brute . 870) (centipede . -658) (cobol . 362) (covariate . 590) (departure . 952) (deploy . 44) (diophantine . 645) (efferent . 54) (elysee . -326) (eradicate . 376) (escritoire . 856) (exorcism . -983) (fiat . 170) (filmy . -874) (flatworm . 503) (gestapo . 915) (infra . -847) (isis . -982) (lindholm . 999) (markham . 475) (mincemeat . -880) (moresby . 756) (mycenae . 183) (plugging . -266) (smokescreen . 423) (speakeasy . -745) (vein . 813) )</lang>
Minimal brute force solution: <lang PicoLisp>(load "@lib/simul.l") # For 'subsets'
(pick
'((N)
(find '((L) (=0 (sum cdr L)))
(subsets N *Words) ) )
(range 1 (length *Words)) )</lang>
Output:
-> ((archbishop . -915) (gestapo . 915))
Python
Version 1
<lang python>words = { # some values are different from example "alliance": -624, "archbishop": -925, "balm": 397, "bonnet": 452, "brute": 870, "centipede": -658, "cobol": 362, "covariate": 590, "departure": 952, "deploy": 44, "diophantine": 645, "efferent": 54, "elysee": -326, "eradicate": 376, "escritoire": 856, "exorcism": -983, "fiat": 170, "filmy": -874, "flatworm": 503, "gestapo": 915, "infra": -847, "isis": -982, "lindholm": 999, "markham": 475, "mincemeat": -880, "moresby": 756, "mycenae": 183, "plugging": -266, "smokescreen": 423, "speakeasy": -745, "vein": 813 }
neg = 0 pos = 0 for (w,v) in words.iteritems(): if v > 0: pos += v else: neg += v
sums = [0] * (pos - neg + 1)
for (w,v) in words.iteritems(): s = sums[:] if not s[v - neg]: s[v - neg] = (w,)
for (i, w2) in enumerate(sums): if w2 and not s[i + v]: s[i + v] = w2 + (w,)
sums = s if s[-neg]: for x in s[-neg]: print(x, words[x]) break</lang>
- Output:
('mycenae', 183) ('speakeasy', -745) ('bonnet', 452) ('lindholm', 999) ('cobol', 362) ('archbishop', -925) ('elysee', -326)
Brute force
<lang python>>>> from itertools import combinations >>> >>> word2weight = {"alliance": -624, "archbishop": -915, "balm": 397, "bonnet": 452,
"brute": 870, "centipede": -658, "cobol": 362, "covariate": 590, "departure": 952, "deploy": 44, "diophantine": 645, "efferent": 54, "elysee": -326, "eradicate": 376, "escritoire": 856, "exorcism": -983, "fiat": 170, "filmy": -874, "flatworm": 503, "gestapo": 915, "infra": -847, "isis": -982, "lindholm": 999, "markham": 475, "mincemeat": -880, "moresby": 756, "mycenae": 183, "plugging": -266, "smokescreen": 423, "speakeasy": -745, "vein": 813}
>>> answer = None >>> for r in range(1, len(word2weight)+1): if not answer: for comb in combinations(word2weight, r): if sum(word2weight[w] for w in comb) == 0: answer = [(w, word2weight[w]) for w in comb] break
>>> answer
[('archbishop', -915), ('gestapo', 915)]</lang>
REXX
This REXX solution isn't limited to integers.
This isn't a brute force solution.
While optimizing the original program, it was found that sorting the names by weight could
yield a vastly improved algorithm (by an order of magnitude), so the extra code to sort the list
was included, as well as another sort to show the solutions in alphabetical order.
Support was also added to allow specification of which "chunk" to search for solutions (that is,
out of the 31 names, take a "chunk" at a time).
Showing of the timing (elapsed time) was also added, as well as "que pasa" informational messages.
The sum (which is zero for this task) can be anything, and can be specifiable on the command line.
<lang rexx>/*REXX pgm finds some non-null subsets of a weighted list whose sum = 0.*/
arg target stopAt chunkette . /*get args from the command line*/
if target==|target==',' then target=0 /*No TARGET given? Use default*/
if stopAt==|stopAt==',' then stopAt=1 /*No max sols given? Use default*/
zzz= 'alliance -624 archbishop -915 balm 397' ,
'bonnet 452 brute 870 centipede -658' ,
'cobol 362 covariate 590 departure 952' ,
'deploy 44 diophantine 645 efferent 54' ,
'elysee -326 eradicate 376 escritoire 856' ,
'exorcism -983 fiat 170 filmy -874' ,
'flatworm 503 gestapo 915 infra -847' ,
'isis -982 lindholm 999 markham 475' ,
'mincemeat -880 moresby 756 mycenae 183' ,
'plugging -266 smokescreen 423 speakeasy -745' ,
'vein 813'
@.=0; y=0; do N=1 until zzz= /*build an array from a list. */
parse var zzz @.N #.N zzz /*pick from list like a nose. */
end /*N*/
call tellZ 'unsorted' /*show the un-sorted list. */ call esort N /*sort the names with weights.*/ call tellZ 'sorted' /*show the sorted list. */ ??=0 /*number of solutions so far. */ chunkStart=1 /*default place to start. */ chunkEnd =N /* " " " end. */ if chunkette\== then do /*solutions just for chunkette*/
chunkStart=chunkette
chunkEnd =chunkette
end
call time 'R' /*reset the REXX elapsed time.*/
do chunk=chunkStart to chunkEnd /*traipse through the items. */
call tello 'doing chunk:' chunk /*inform what's happening. */
call combN N,chunk /*N items, a CHUNK at a time. */
call tello 'chunk' chunk "took" format(time('E'),,2) 'seconds.'
end /*chunk*/
if ??==0 then ??='no' /*Englishize solutions number.*/ call tello 'Found' ?? "subset"s(??) 'whose summed weight's(??) "=" target exit /*stick a fork in it, we done.*/ /*──────────────────────────────────combN subroutine────────────────────*/ combN: procedure expose @. #. ?? stopAt target; parse arg x,y; !.=0 base=x+1; bbase=base-y; ym=y-1 /*!.n are the combination digits*/
do n=1 for y; !.n=n; end /*build 1st combination*/
do j=1; _=!.1; s=#._ /*get 1st dig & the sum*/ if s>target then leave /*1st dig>target? Then we're done*/
do k=2 for ym; _=!.k; s=s+#._; /*Σ weights; >target? */
if s>target then do; if .combUp(k-1) then return; iterate j; end
end /*k*/
if s==target then call telly /*found a pot of gold? */ !.y=!.y+1; if !.y==base then if .combUp(ym) then leave /*bump dig*/ end /*j*/
return /*done with this combination set.*/ /*──────────────────────────────────.combUp subroutine──────────────────*/ .combUp: procedure expose !. y bbase; parse arg d; if d==0 then return 1 p=!.d; do u=d to y; !.u=p+1 /*add 1 to dig we're pointing at.*/
if !.u>=bbase+u then return .combUp(u-1)
p=!.u /*P will be used for the next dig*/
end /*u*/
return 0 /*go back & sum this combination.*/ /*──────────────────────────────────ESORT subroutine────────────────────*/ esort: procedure expose #. @.; parse arg N; h=N
do while h>1; h=h%2
do i=1 for N-h; j=i; k=h+i
do while #.k<#.j
parse value @.j @.k #.j #.k with @.k @.j #.k #.j
if h>=j then leave; j=j-h; k=k-h
end /*while #.k<#.j*/
end /*i*/
end /*while h>1*/
return /*──────────────────────────────────nSort subroutine────────────────────*/ nSort: procedure expose names.; parse arg many; h=many
do while h>1; h=h%2
do i=1 for many-h; j=i; k=h+i
do while names.k<names.j
parse value names.j names.k with names.k names.j
if h>=j then leave; j=j-h; k=k-h
end /*while names.k<names.j*/
end /*i*/
end /*while h>1*/
return /*──────────────────────────────────S subroutine────────────────────────*/ s: if arg(1)==1 then return arg(3); return word(arg(2) 's',1) /*──────────────────────────────────telly subroutine────────────────────*/ telly: ??=??+1; nameL= /*start with a "null" name list. */
do gi=1 for y; ggg=!.gi /*build dup array (to be sorted).*/
names.gi=@.ggg /*transform from index──> a name.*/
end /*gi*/ /*build dup array (to be sorted).*/
/*at this point, the names are */
/* in order of their weight.*/
call nSort y /*sort the names alphabetically. */
do gs=1 for y; nameL=nameL names.gs /*build list of names whose sum=0*/ end /*gs*/ /*list of names could be sorted */
call tello '['y" name"s(y)']' space(nameL) if ??>=stopAt &, /*see if we reached a (the) limit*/
stopAt\==0 then do
call tello 'Stopped after finding' ?? "subset"s(??)'.'
exit /*a short-timer, we have to quit.*/
end
return /*go back and keep on truckin'. */ /*──────────────────────────────────tello subroutine────────────────────*/ tello: say arg(1); call lineout 'SUBSET.'y,arg(1) /*write to file*/; return /*──────────────────────────────────tellz subroutine────────────────────*/ tellz: do J=1 for N /*show list of names and weights.*/
call tello right('['j']',30) right(@.j,11) right(#.j,5)
end
call tello; call tello 'There are' N "entries in the (above)" arg(1) 'table.'
call tello
return</lang>
output when using the default input of: 0 1
(The above arguments set the target sum to zero and limits finding of solutions to one.)
[1] alliance -624
[2] archbishop -915
[3] balm 397
[4] bonnet 452
[5] brute 870
[6] centipede -658
[7] cobol 362
[8] covariate 590
[9] departure 952
[10] deploy 44
[11] diophantine 645
[12] efferent 54
[13] elysee -326
[14] eradicate 376
[15] escritoire 856
[16] exorcism -983
[17] fiat 170
[18] filmy -874
[19] flatworm 503
[20] gestapo 915
[21] infra -847
[22] isis -982
[23] lindholm 999
[24] markham 475
[25] mincemeat -880
[26] moresby 756
[27] mycenae 183
[28] plugging -266
[29] smokescreen 423
[30] speakeasy -745
[31] vein 813
There are 31 entries in the (above) unsorted table.
[1] exorcism -983
[2] isis -982
[3] archbishop -915
[4] mincemeat -880
[5] filmy -874
[6] infra -847
[7] speakeasy -745
[8] centipede -658
[9] alliance -624
[10] elysee -326
[11] plugging -266
[12] deploy 44
[13] efferent 54
[14] fiat 170
[15] mycenae 183
[16] cobol 362
[17] eradicate 376
[18] balm 397
[19] smokescreen 423
[20] bonnet 452
[21] markham 475
[22] flatworm 503
[23] covariate 590
[24] diophantine 645
[25] moresby 756
[26] vein 813
[27] escritoire 856
[28] brute 870
[29] gestapo 915
[30] departure 952
[31] lindholm 999
There are 31 entries in the (above) sorted table.
doing chunk: 1
chunk 1 took 0.00 seconds.
doing chunk: 2
[2 names] archbishop gestapo
Stopped after finding 1 subset.
Ruby
a brute force solution: <lang ruby>weights = {
'alliance' => -624, 'archbishop' => -915, 'balm' => 397, 'bonnet' => 452, 'brute' => 870, 'centipede' => -658, 'cobol' => 362, 'covariate' => 590, 'departure' => 952, 'deploy' => 44, 'diophantine' => 645, 'efferent' => 54, 'elysee' => -326, 'eradicate' => 376, 'escritoire' => 856, 'exorcism' => -983, 'fiat' => 170, 'filmy' => -874, 'flatworm' => 503, 'gestapo' => 915, 'infra' => -847, 'isis' => -982, 'lindholm' => 999, 'markham' => 475, 'mincemeat' => -880, 'moresby' => 756, 'mycenae' => 183, 'plugging' => -266, 'smokescreen' => 423, 'speakeasy' => -745, 'vein' => 813,
}
words = weights.keys 1.upto(words.length) do |n|
zerosum = words.combination(n).find do |subset|
subset.reduce(0) {|sum, word| sum += weights[word]} == 0
end
if zerosum.nil?
puts "no subsets of length #{n} sum to zero"
else
puts "a subset of length #{n} that sums to zero: #{zerosum}"
end
end</lang>
output:
no subsets of length 1 sum to zero a subset of length 2 that sums to zero: ["archbishop", "gestapo"] a subset of length 3 that sums to zero: ["centipede", "markham", "mycenae"] a subset of length 4 that sums to zero: ["alliance", "balm", "deploy", "mycenae"] a subset of length 5 that sums to zero: ["alliance", "brute", "covariate", "deploy", "mincemeat"] a subset of length 6 that sums to zero: ["alliance", "archbishop", "balm", "deploy", "gestapo", "mycenae"] a subset of length 7 that sums to zero: ["alliance", "archbishop", "bonnet", "cobol", "departure", "exorcism", "moresby"] a subset of length 8 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "fiat", "flatworm", "isis", "lindholm"] a subset of length 9 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "covariate", "eradicate", "mincemeat", "plugging"] a subset of length 10 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "departure", "deploy", "mincemeat"] a subset of length 11 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "departure", "infra", "moresby", "speakeasy"] a subset of length 12 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "diophantine", "efferent", "elysee", "infra"] a subset of length 13 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "efferent", "eradicate", "filmy", "isis"] a subset of length 14 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "elysee", "filmy", "markham", "speakeasy"] a subset of length 15 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "elysee", "exorcism", "flatworm", "infra", "mycenae"] a subset of length 16 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "diophantine", "elysee", "exorcism", "filmy", "gestapo", "infra"] a subset of length 17 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "diophantine", "exorcism", "isis", "mincemeat", "mycenae", "plugging", "vein"] a subset of length 18 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "diophantine", "efferent", "elysee", "exorcism", "filmy", "isis", "mycenae", "vein"] a subset of length 19 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "diophantine", "efferent", "elysee", "eradicate", "exorcism", "fiat", "infra", "isis", "smokescreen"] a subset of length 20 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "diophantine", "efferent", "elysee", "eradicate", "exorcism", "gestapo", "infra", "isis", "smokescreen", "speakeasy"] a subset of length 21 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "diophantine", "efferent", "elysee", "eradicate", "exorcism", "flatworm", "infra", "lindholm", "mincemeat", "plugging", "speakeasy"] a subset of length 22 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "diophantine", "efferent", "elysee", "eradicate", "escritoire", "exorcism", "fiat", "filmy", "flatworm", "mincemeat", "plugging", "speakeasy"] a subset of length 23 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "diophantine", "efferent", "elysee", "eradicate", "escritoire", "exorcism", "infra", "isis", "mincemeat", "moresby", "mycenae", "smokescreen", "speakeasy"] a subset of length 24 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "diophantine", "efferent", "elysee", "exorcism", "filmy", "gestapo", "infra", "markham", "mincemeat", "moresby", "mycenae", "plugging", "smokescreen", "speakeasy"] a subset of length 25 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "diophantine", "eradicate", "exorcism", "fiat", "filmy", "flatworm", "infra", "isis", "lindholm", "markham", "mincemeat", "moresby", "mycenae", "plugging", "speakeasy"] a subset of length 26 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "diophantine", "elysee", "eradicate", "escritoire", "exorcism", "fiat", "filmy", "gestapo", "infra", "isis", "markham", "mincemeat", "mycenae", "plugging", "speakeasy", "vein"] a subset of length 27 that sums to zero: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "covariate", "departure", "deploy", "efferent", "elysee", "eradicate", "escritoire", "exorcism", "fiat", "filmy", "flatworm", "infra", "isis", "lindholm", "markham", "mincemeat", "moresby", "mycenae", "plugging", "smokescreen", "speakeasy"] no subsets of length 28 sum to zero no subsets of length 29 sum to zero no subsets of length 30 sum to zero no subsets of length 31 sum to zero
Tcl
As it turns out that the problem space has small subsets that sum to zero, it is more efficient to enumerate subsets in order of their size rather than doing a simple combination search. This is not true of all possible input data sets though; the problem is known to be NP-complete after all. <lang tcl>proc subsetsOfSize {set size} {
if {$size <= 0} {
return
} elseif {$size == 1} {
foreach elem $set {lappend result [list $elem]}
} else {
incr size [set i -1] foreach elem $set { foreach sub [subsetsOfSize [lreplace $set [incr i] $i] $size] { lappend result [lappend sub $elem] } }
} return $result
} proc searchForSubset {wordweights {minsize 1}} {
set words [dict keys $wordweights]
for {set i $minsize} {$i < [llength $words]} {incr i} {
foreach subset [subsetsOfSize $words $i] { set w 0 foreach elem $subset {incr w [dict get $wordweights $elem]} if {!$w} {return $subset} }
} # Nothing was found return -code error "no subset sums to zero"
}</lang> Demonstrating: <lang tcl>set wordweights {
alliance -624 archbishop -915 balm 397 bonnet 452 brute 870 centipede -658 cobol 362 covariate 590 departure 952 deploy 44 diophantine 645 efferent 54 elysee -326 eradicate 376 escritoire 856 exorcism -983 fiat 170 filmy -874 flatworm 503 gestapo 915 infra -847 isis -982 lindholm 999 markham 475 mincemeat -880 moresby 756 mycenae 183 plugging -266 smokescreen 423 speakeasy -745 vein 813
} set zsss [searchForSubset $wordweights] puts "Found zero-summing subset: [join [lsort $zsss] {, }]"</lang> Output:
Found zero-summing subset: archbishop, gestapo
Ursala
This solution scans the set sequentially while maintaining a record of all distinct sums obtainable by words encountered thus far, and stops when a zero sum is found. <lang Ursala>#import std
- import int
weights =
{
'alliance': -624, 'archbishop': -915, 'balm': 397, 'bonnet': 452, 'brute': 870, 'centipede': -658, 'cobol': 362, 'covariate': 590, 'departure': 952, 'deploy': 44, 'diophantine': 645, 'efferent': 54, 'elysee': -326, 'eradicate': 376, 'escritoire': 856, 'exorcism': -983, 'fiat': 170, 'filmy': -874, 'flatworm': 503, 'gestapo': 915, 'infra': -847, 'isis': -982, 'lindholm': 999, 'markham': 475, 'mincemeat': -880, 'moresby': 756, 'mycenae': 183, 'plugging': -266, 'smokescreen': 423, 'speakeasy': -745, 'vein': 813}
nullset = ~&nZFihmPB+ =><> ~&ng?r\~&r ^TnK2hS\~&r ^C/~&lmPlNCX *D ^A/sum@lmPrnPX ~&lrmPC
- cast %zm
main = nullset weights</lang>
The name of the function that takes the weighted set is nullset. It manipulates a partial result represented as a list of pairs, each containing a subset of weighted words and the sum of their weights. Here is a rough translation:
=><>fold right combinator with the empty list as the vacuuous case~&ng?r\~&rIf the partial result contains a zero sum, return it.^TnK2hS\~&rConcatenate the partial result with the new list of subsets (computed as follows) and delete duplicate sums.^C/~&lmPlNCXCons a singleton subset containing the next word to the partial results.*DDistribute the next word in the set to the partial results and do the following to each.sum@lmPrnPXAdd the weight of the new word to the existing sum.~&lrmPCCons the new word to the list of existing ones.~&nZFihmPB+To conclude, search for a result with a zero sum, if any, and return its associated subset of weighted words.
output:
< 'flatworm': 503, 'gestapo': 915, 'infra': -847, 'isis': -982, 'lindholm': 999, 'plugging': -266, 'smokescreen': 423, 'speakeasy': -745>