Subset sum problem

From Rosetta Code
Subset sum problem is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

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. |

Table of weighted words
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[edit]

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;
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[edit]

#include <stdio.h>
#include <stdlib.h>
 
typedef struct {
char *word;
int weight;
} item_t;
 
item_t items[] = {
{"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},
};
 
int n = sizeof (items) / sizeof (item_t);
int *set;
 
void subsum (int i, int weight) {
int j;
if (i && !weight) {
for (j = 0; j < i; j++) {
item_t item = items[set[j]];
printf("%s%s", j ? " " : "", items[set[j]].word);
}
printf("\n");
}
for (j = i ? set[i - 1] + 1: 0; j < n; j++) {
set[i] = j;
subsum(i + 1, weight + items[j].weight);
}
}
 
int main () {
set = malloc(n * sizeof (int));
subsum(0, 0);
return 0;
}
Output:
alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate escritoire exorcism fiat filmy flatworm mincemeat plugging speakeasy
...

D[edit]

A simple brute-force solution. This used the module of the third D solution of the Combinations Task.

Translation of: Ruby
void main() {
import std.stdio, std.algorithm, std.typecons, combinations3;
 
alias P = tuple;
immutable items = [
P("alliance", -624), P("archbishop", -915), P("balm", 397),
P("bonnet", 452), P("brute", 870), P("centipede", -658),
P("cobol", 362), P("covariate", 590), P("departure", 952),
P("deploy", 44), P("diophantine", 645), P("efferent", 54),
P("elysee", -326), P("eradicate", 376), P("escritoire", 856),
P("exorcism", -983), P("fiat", 170), P("filmy", -874),
P("flatworm", 503), P("gestapo", 915), P("infra", -847),
P("isis", -982), P("lindholm", 999), P("markham", 475),
P("mincemeat", -880), P("moresby", 756), P("mycenae", 183),
P("plugging", -266), P("smokescreen", 423), P("speakeasy", -745),
P("vein", 813)];
 
foreach (immutable n; 1 .. items.length)
foreach (const comb; items.combinations(n))
if (comb.map!q{ a[1] }.sum == 0)
return writefln("A subset of length %d: %-(%s, %)", n,
comb.map!q{ a[0] });
"No solution found.".writeln;
}
Output:
A subset of length 2: archbishop, gestapo

Alternative Version[edit]

This version prints all the 349_167 solutions in about 1.8 seconds and counts them in about 0.05 seconds.

Translation of: C
import std.stdio, std.algorithm;
 
enum showAllSolutions = true;
 
struct Item { string data; int weight; }
struct Sum { int sum; uint mask; }
 
immutable 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}];
 
Sum[] mkSums(in Item[] p, in size_t n, in size_t shift) {
auto r = new Sum[1 << n];
foreach (immutable i; 0 .. n)
r[1 << i].sum = p[i].weight;
 
foreach (immutable i, ref ri; r) {
immutable size_t b = i & -int(i);
ri = Sum(r[i & ~b].sum + r[b].sum, i << shift);
}
 
return r.sort!q{ a.sum < b.sum }.release;
}
 
void showMask(in uint mask) nothrow {
for (size_t m = 0; (1U << m) <= mask; m++)
if (mask & (1U << m))
// Much faster than writeln.
// The names are all zero-terminated.
printf("%s ", em[m].data.ptr);
if (mask)
putchar('\n');
}
 
int printList(in int i, in int j, in int i1, in int j1,
in Sum[] l, in Sum[] r) nothrow {
int s = (i1 - i) * (j - j1);
if (!l[i].sum)
s--;
 
static if (showAllSolutions)
foreach (immutable x; i .. i1)
foreach_reverse (immutable size_t y; j1 + 1 .. j + 1)
showMask(l[x].mask | r[y].mask);
return s;
}
 
void main() {
immutable N = em.length;
assert(N <= em[0].sizeof * 8, "Not enough bits in the mask");
immutable size_t n1 = N / 2;
immutable size_t n2 = N - n1;
immutable size_t n1p = 1 << n1;
immutable size_t n2p = 1 << n2;
 
auto l = mkSums(em[], n1, 0);
auto r = mkSums(em[n1 .. $], n2, n1);
 
size_t sols = 0;
int i = 0;
int j = n2p - 1;
while (true) {
while (l[i].sum + r[j].sum) {
while (i < n1p && l[i].sum + r[j].sum < 0)
i++;
while (j >= 0 && l[i].sum + r[j].sum > 0)
j--;
if (i >= n1p || j < 0)
break;
}
if (i >= n1p || j < 0)
break;
 
int i1 = i + 1;
while (i1 < n1p && l[i1].sum == l[i].sum)
i1++;
 
int j1 = j - 1;
while (j1 >= 0 && r[j1].sum == r[j].sum)
j1--;
 
sols += printList(i, j, i1, j1, l, r);
i = i1;
j = j1;
}
 
writeln("Zero sums: ", sols);
}
Output:
Zero sums: 349167

EchoLisp[edit]

Dynamic programming[edit]

We use the Pseudo-polynomial time dynamic programming solution, found in the Subset sum problem Wikipedia article. If A and B are the min and max possible sums, the time and memory needed are O((B-A)*N). Q is an array such as Q(i,s) = true if there is a nonempty subset of x0, ..., xi which sums to s.

 
;; 0 <= i < N , A <= s < B , -A = abs(A)
;; mapping two dims Q(i,s) to one-dim Q(qidx(i,s)) :
 
(define-syntax-rule (qidx i s) (+ i (* (+ s -A) N)))
 
;; filling the Q array with true/false values
;; Q(i, s) := Q(i − 1, s) or (xi == s) or Q(i − 1, s − xi), for A ≤ s < B.
 
(define (fillQ xs (ds))
(define N (length xs))
(define A (apply + (filter negative? xs)))
(define B (1+ (apply + (filter positive? xs))))
(define -A (abs A))
(define Q (make-vector (* N (- B A))))
(set! xs (list->vector xs))
 
(printf "Q[%d] allocated." (vector-length Q))
(for ((s (in-range A B)))
(vector-set! Q (qidx 0 s ) (= [xs 0] s)))
 
(for* ([i (in-range 1 N)]
[s (in-range A B)])
 
(set! ds (- s [xs i]))
(vector-set! Q (qidx i s)
(or
[Q (qidx (1- i) s)]
(= [xs i] s)
(and (>= ds A) (< ds B) [Q (qidx (1- i) ds )])))
 
;; stop on first zero-sum found
#:break (and (zero? s) [Q (qidx i s)]) => (solQ Q xs i s -A N)
))
 
;; backtracking to get the list of i's such as sum([xs i]) = 0
;; start from q[i,0] === true
 
(define (solQ Q xs i s -A N (sol null))
(cond
(( = s [xs i]) (cons i sol))
([Q (qidx (1- i ) s)] (solQ Q xs (1- i) s -A N sol))
(else (solQ Q xs (1- i) (- s [xs i]) -A N (cons i sol)))))
 
(define (task input)
(map (lambda(i) (first (list-ref input i))) (fillQ (map rest input))))
 
 
Output:
    
(define input 
    '({"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}))
    
(task input)
Q[587016] allocated.
    → ("archbishop" "balm" "bonnet" "centipede" "cobol" "covariate" 
"deploy" "efferent" "elysee")

;; using Haskell test data
(define items 
    '[-61 1 32 373 311 249 311 32 -92 -185 -433
    -402 -247 156 125 249 32 -464 -278 218 32 -123
    -216 373 -185 -402 156 -402 -61 -31 902 ])
    
(map (lambda(i) (list-ref items i)) (fillQ items))

Q[221185] allocated.
   → (-61 32 373 311 249 311 32 -92 -185 -433 -402 -247 156 125 249 32 -
    464 -278 218 32 -123 -216 373 -185 -402 156 -402 -61 902)

Brute force[edit]

We use the powerset procrastinator which gives in sequence all subsets of the input list.

 
(lib 'sequences) ;; for powerset
 
(define (sum0? xs)
(zero? (apply + (map rest xs))))
 
;; filter the powerset and
;; take first 5 solutions
(for-each writeln (take (filter sum0? (powerset input)) 5))
 
() ;; empty
 
(("archbishop" . -915) ("balm" . 397) ("bonnet" . 452)
("centipede" . -658) ("cobol" . 362) ("covariate" . 590)
("deploy" . 44) ("efferent" . 54) ("elysee" . -326))
 
(("archbishop" . -915) ("balm" . 397) ("bonnet" . 452)
("centipede" . -658) ("departure" . 952) ("deploy" . 44)
("efferent" . 54) ("elysee" . -326))
 
(("alliance" . -624) ("brute" . 870) ("centipede" . -658)
("cobol" . 362) ("elysee" . -326) ("eradicate" . 376))
 
(("alliance" . -624) ("archbishop" . -915) ("bonnet" . 452)
("centipede" . -658) ("cobol" . 362) ("covariate" . 590)
("deploy" . 44) ("diophantine" . 645) ("efferent" . 54)
("elysee" . -326) ("eradicate" . 376))
 

FunL[edit]

def subsetSum( s, w, v ) =
def sumset( a ) = foldl1( (+), map(w, a) )
 
for i <- s.subsets() if i != {}
if sumset( i ) == v
return Some( i )
 
None
 
s = {
('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)
}
 
for i <- 0..5
println( i, subsetSum(s, snd, i).get() )
Output:
0, {(archbishop, -915), (gestapo, 915)}
1, {(fiat, 170), (vein, 813), (isis, -982)}
2, {(alliance, -624), (departure, 952), (elysee, -326)}
3, {(alliance, -624), (archbishop, -915), (departure, 952), (covariate, 590)}
4, {(markham, 475), (infra, -847), (eradicate, 376)}
5, {(flatworm, 503), (eradicate, 376), (filmy, -874)}

Go[edit]

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 := []sss{{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")
}
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[edit]

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,
sum (map weight 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
Output:
["archbishop","gestapo"]

None bruteforce: the list of numbers used here are different, and difficult for a bruteforce method.

subsum w = snd.head.filter ((==w).fst).(++[(w,[])]).foldl s [(0,[])]
where
s a x = merge a $ map f a where f (a,l) = (a+x, l++[x])
 
-- keep list of sums sorted and unique
merge [] a = a
merge a [] = a
merge a@((av,al):as) b@((bv,bl):bs)
| av < bv = (av,al):merge as b
| av == bv = (bv,bl):merge as bs
| otherwise = (bv,bl):merge a bs
 
items = [-61, 1, 32, 373, 311, 249, 311, 32, -92, -185, -433,
-402, -247, 156, 125, 249, 32, -464, -278, 218, 32, -123,
-216, 373, -185, -402, 156, -402, -61, -31, 902 ]
 
main = print $ subsum 0 items
Output:
[-61,32,373,311,249,311,32,-92,-185,-433,-402,-247,156,125,249,32,-464,-278,218,32,-123,-216,373,-185,-402,156,-402,-61,902]

Icon and Unicon[edit]

Translation of: Ruby
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

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

J[edit]

Task data:

text=:0 :0
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=:{.@;:;._2 text
numbs=:+/|:0&".;._2 text

Implementation:

wsum0=:4 :0
p=:(#~ 0&<)y
n=:(#~ 0&>)y
poss=: +/@#~2#.inv 2 i.@^#
P=:poss p
N=:poss -n
choose=:(1{I.P e. N){P
keep=: [ #~ #&2@#@[ #: choose i.~ ]
 ;:inv words #~y e. (p keep P),n keep N
)

Task example:

   words wsum0 numbs
centipede markham mycenae

Note also that there are over 300,000 valid solutions here. More than can be comfortably displayed:

   Ps=: </.~ /:~ (I.P e. N){P
Ns=: </.~ /:~ (I.N e. P){N
+/#@,@{"1 Ps,.Ns
349168

(One of those is the empty solution, but the rest of them are valid.)

Mathematica[edit]

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 = [email protected][ Subsets[a, 7], (Total[#[[;; , 2]]] == 0) &];
Map[ (Print["A zero-sum subset of length ", Length[#], " : ", #[[;; , 1]]])& , result ]
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}
....

The above code uses a brute-force approach, but Mathematica includes several solution schemes that can be used to solve this problem. We can cast it as an integer linear programming problem, and thus find the largest or smallest subset sum, or even sums with specific constraints, such as a sum using three negative values and nine positive values.

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}};
 
desiredValue = 0;
aNames = #[[1]] & /@ a;
aValues = #[[2]] & /@ a;
aOnes = ConstantArray[1, Length[a]];
aZeroOnes = ConstantArray[{0, 1}, Length[a]];
Off[LinearProgramming::lpip];
 
maxSoln =
LinearProgramming[-aOnes, {aValues}, {{desiredValue, 0}}, aZeroOnes, Integers];
 
Print["Maximal solution: ", Select[Transpose[{maxSoln*aValues, aNames}], #[[1]] != 0 &]];
 
minSoln =
LinearProgramming[
aOnes, {aValues, aOnes}, {{desiredValue, 0}, {1, 1}}, aZeroOnes, Integers];
 
Print["Minimal solution: ", Select[Transpose[{minSoln*aValues, aNames}], #[[1]] != 0 &]];
 
threeNineSoln =
LinearProgramming[
aOnes, {aValues,
Boole[# < 0] & /@ aValues,
Boole[# > 0] & /@ aValues},
{{desiredValue, 0}, {3, 0}, {9, 0}}, aZeroOnes, Integers];
 
Print["3 -ves, 9 +ves: ", Select[Transpose[{threeNineSoln*aValues, aNames}], #[[1]] != 0 &]];
 
Maximal solution: {{-624, alliance}, {-915, archbishop}, {397, balm}, 
    {870, brute}, {-658, centipede}, {362, cobol}, {590, covariate}, 
    {44, deploy}, {645, diophantine}, {54, efferent}, {-326, elysee}, 
    {376, eradicate}, {-983, exorcism}, {170, fiat}, {-874, filmy}, 
    {503, flatworm}, {915, gestapo}, {-847, infra}, {-982, isis}, 
    {999, lindholm}, {-880, mincemeat}, {756, moresby}, {183, mycenae}, 
    {-266, plugging}, {423, smokescreen}, {-745, speakeasy}, {813, vein}}

Minimal solution: {{-915, archbishop}, {915, gestapo}}

3 -ves, 9 +ves: {{-915, archbishop}, {397, balm}, {452, bonnet}, 
    {362, cobol}, {44, deploy}, {54, efferent}, {-983, exorcism}, 
    {170, fiat}, {503, flatworm}, {-982, isis}, {475, markham}, 
    {423, smokescreen}}.

OCaml[edit]

Just search randomly until a result is found:

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

Perl[edit]

Library: ntheory
use ntheory qw/:all/;
my $print_all_combinations = 0;
 
my %pairs = (
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 );
my @names = keys(%pairs);
my @weights = values(%pairs);
 
if ($print_all_combinations) {
 
foreach my $n (1 .. @names) {
forcomb {
print "Length $n: @names[@_]\n" unless vecsum(@weights[@_]);
} @names, $n;
}
 
} else {
 
foreach my $n (1 .. @names) {
eval {
forcomb {
if (vecsum(@weights[@_]) == 0) {
print "Length $n: @names[@_]\n";
die;
}
} @names, $n;
};
}
}

Printing just the first one found for each number of elements:

Output:
Length 2: archbishop gestapo
Length 3: exorcism fiat vein
Length 4: efferent plugging brute centipede
Length 5: efferent exorcism cobol fiat balm
Length 6: efferent exorcism isis vein gestapo mycenae
Length 7: efferent exorcism isis cobol covariate gestapo deploy
Length 8: efferent exorcism isis speakeasy covariate vein escritoire balm
Length 9: efferent exorcism isis speakeasy cobol markham smokescreen lindholm balm
... to length 27 ...

We can also use different modules. Assuming the same pairs/names/weights variables set and first combination only, this iterator style is a little cleaner when wanting to exit early:

use List::Util qw/sum/;
use Algorithm::Combinatorics qw/combinations/;
foreach my $n (1 .. @names) {
my $iter = combinations([0..$#weights], $n);
while (my $c = $iter->next) {
next if sum(@weights[@$c]);
print "Length $n: @names[@$c]\n";
last;
}
}

Perl 6[edit]

my @pairs =
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;
my @weights = @pairs».value;
my %name = @pairs.hash.invert;
 
for 1..^@weights -> $n {
given @weights.combinations($n).first({ 0 == [+] @^comb }) {
when .so { say "Length $n: ", .map: {%name{$_}} }
default { say "Length $n: (none)" }
}
}
Output:
Length 1: (none)
Length 2: archbishop gestapo
Length 3: centipede markham mycenae
Length 4: alliance balm deploy mycenae
Length 5: alliance brute covariate deploy mincemeat
Length 6: alliance archbishop balm deploy gestapo mycenae
Length 7: alliance archbishop bonnet cobol departure exorcism moresby
Length 8: alliance archbishop balm bonnet fiat flatworm isis lindholm
Length 9: alliance archbishop balm bonnet brute covariate eradicate mincemeat plugging
Length 10: alliance archbishop balm bonnet brute centipede cobol departure deploy mincemeat
Length 11: alliance archbishop balm bonnet brute centipede cobol departure infra moresby speakeasy
Length 12: alliance archbishop balm bonnet brute centipede cobol covariate diophantine efferent elysee infra
Length 13: alliance archbishop balm bonnet brute centipede cobol covariate departure efferent eradicate filmy isis
Length 14: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy elysee filmy markham speakeasy
Length 15: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy elysee exorcism flatworm infra mycenae
Length 16: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine elysee exorcism filmy gestapo infra
Length 17: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine exorcism isis mincemeat mycenae plugging vein
Length 18: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee exorcism filmy isis mycenae vein
Length 19: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate exorcism fiat infra isis smokescreen
Length 20: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate exorcism gestapo infra isis smokescreen speakeasy
Length 21: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate exorcism flatworm infra lindholm mincemeat plugging speakeasy
Length 22: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate escritoire exorcism fiat filmy flatworm mincemeat plugging speakeasy
Length 23: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate escritoire exorcism infra isis mincemeat moresby mycenae smokescreen speakeasy
Length 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
Length 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
Length 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
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

PicoLisp[edit]

(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) )

Minimal brute force solution:

(load "@lib/simul.l")  # For 'subsets'
 
(pick
'((N)
(find '((L) (=0 (sum cdr L)))
(subsets N *Words) ) )
(range 1 (length *Words)) )
Output:
-> ((archbishop . -915) (gestapo . 915))

Python[edit]

Version 1[edit]

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
Output:

('mycenae', 183) ('speakeasy', -745) ('bonnet', 452) ('lindholm', 999) ('cobol', 362) ('archbishop', -925) ('elysee', -326)

Brute force[edit]

>>> 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)]

Racket[edit]

 
#lang racket
 
(define 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]))
 
;; Simple brute-force solution to find the smallest subset
(define (nsubsets l n)
(cond [(zero? n) '(())] [(null? l) '()]
[else (append (for/list ([l2 (nsubsets (cdr l) (- n 1))])
(cons (car l) l2))
(nsubsets (cdr l) n))]))
(for*/first ([i (sub1 (length words))] [s (nsubsets words (add1 i))]
#:when (zero? (apply + (map cadr s))))
(map car s))
;; => '(archbishop gestapo)
 
;; Alternative: customize the subsets to ones with zero sum, abort early
;; if we're in a hopeless case (using the fact that weights are <1000)
(define (zero-subsets l)
(define (loop l len n r sum)
(cond [(zero? n) (when (zero? sum) (displayln (reverse r)))]
[(and (pair? l) (<= sum (* 1000 n)))
(when (< n len) (loop (cdr l) (sub1 len) n r sum))
(loop (cdr l) (sub1 len) (sub1 n) (cons (caar l) r)
(+ (cadar l) sum))]))
(define len (length l))
(for ([i (sub1 len)]) (loop l len (add1 i) '() 0)))
(zero-subsets words)
 
Output:
'(archbishop gestapo) ; <- the first solution
(archbishop gestapo)
(exorcism fiat vein)
(centipede markham mycenae)
... 43M of printouts ...
(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)

REXX[edit]

This REXX solution isn't limited to integers for the weights.     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 any number, and can be specifiable on the command line.

/*REXX pgm finds some non-null subsets of a weighted list whose sum=zero*/
parse arg target stopAt chunkette . /*get args from the command line*/
if target==''|target==',' then target=0 /*TARGET given? No, use default*/
if stopAt==''|stopAt==',' then stopAt=1 /*Max solutions? " " " */
 
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 esort N /*sort the names with weights.*/
call tellZ 'sorted' /*show the sorted list. */
chunkStart=1 /*the default place to start. */
chunkEnd =N /* " " " " end. */
if chunkette\=='' then do /*solutions just for chunkette*/
chunkStart=chunkette
chunkEnd =chunkette
end
call time 'Reset' /*reset the REXX elapsed time.*/
??=0 /*number of solutions so far. */
 
do chunk=chunkStart to chunkEnd /*traipse through the items. */
call tello center(' doing chunk:' chunk" ",79,'─')
call combN N, chunk /*N items, a CHUNK at a time. */
_=??; if _==0 then _='No' /*Englishise for a zero count.*/
call tello _ 'solution's(??) "found so far and took",
format(time('Elapsed'),,2) 'seconds so far.',.
end /*chunk*/
 
if ??==0 then ??='no' /*Englishise 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 the first combination. */
 
do j=1; _=!.1; s=#._ /*get the first digit and the sum*/
if s>target then leave /*1st dig>target? Then we're done*/
 
do k=2 for ym; _=!.k; s=s+#._ /*Σ the weights; is sum > target?*/
if s>target then do; if .combUp(k-1) then return; iterate j; end
end /*k*/
 
if s==target then call telly /*have we 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: 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
if $==. then do while $.k<$.j; parse value $.j $.k with $.k $.j
if h>=j then leave; j=j-h; k=k-h
end /*while $.k<$.j*/
else 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
/*──────────────────────────────────one─liner subroutines─────────────────────*/
s: if arg(1)==1 then return arg(3); return word(arg(2) 's',1) /*plural*/
tello: parse arg _,e; if e==. then say; say _; call lineout 'SUBSET.'y,_; return
/*──────────────────────────────────TELLY subroutine────────────────────*/
telly: ??=??+1; nameL= /*start with a "null" name list. */
do gi=1 for y; ggg=!.gi /*build dup array (to be sorted).*/
$.gi=@.ggg /*transform from index──► a name.*/
end /*gi*/ /*build dup array (to be sorted).*/
call eSort y,. /*sort the names alphabetically. */
do gs=1 for y; nameL=nameL $.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 | stopAt==0 then return /*see if we reached a (the) limit*/
call tello 'Stopped after finding '  ?? " subset"s(??)'.',.
exit /*a short─timer, we have to quit.*/
/*──────────────────────────────────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 /*j*/
call tello
call tello 'There are ' N " entries in the (above)" arg(1) 'table.'
call tello; return

Output note:   this program also writes the displayed output to file(s):   SUBSET.nnn
──────── where   nnn   is the chunk number.

Output:
when using the input of:   0 12

(The above arguments set the target sum to zero and limits finding of a dozen solutions.)

                           [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 ────────────────────────────────

No solutions found so far  and  took 0.00 seconds so far.
─────────────────────────────── doing chunk: 2 ────────────────────────────────
[2 names] archbishop gestapo

1 solution found so far  and  took 0.01 seconds so far.
─────────────────────────────── doing chunk: 3 ────────────────────────────────
[3 names] exorcism fiat vein
[3 names] centipede markham mycenae

3 solutions found so far  and  took 0.06 seconds so far.
─────────────────────────────── doing chunk: 4 ────────────────────────────────
[4 names] exorcism gestapo speakeasy vein
[4 names] deploy exorcism moresby mycenae
[4 names] bonnet elysee escritoire isis
[4 names] eradicate isis mycenae smokescreen
[4 names] balm efferent filmy smokescreen
[4 names] centipede covariate gestapo infra
[4 names] centipede covariate speakeasy vein
[4 names] brute centipede efferent plugging
[4 names] alliance balm deploy mycenae

Stopped after finding  12  subsets.

Ruby[edit]

a brute force solution:

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
puts "a subset of length #{n} that sums to zero: #{zerosum}"
else
puts "no subsets of length #{n} sum to zero"
end
end
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

Sidef[edit]

var pairs = Hash(
alliance => -624, archbishop => -915,
brute => 870, centipede => -658,
departure => 952, deploy => 44,
elysee => -326, eradicate => 376,
fiat => 170, filmy => -874,
infra => -847, isis => -982,
mincemeat => -880, moresby => 756,
smokescreen => 423, speakeasy => -745,
balm => 397, bonnet => 452,
cobol => 362, covariate => 590,
diophantine => 645, efferent => 54,
escritoire => 856, exorcism => -983,
flatworm => 503, gestapo => 915,
lindholm => 999, markham => 475,
mycenae => 183, plugging => -266,
vein => 813,
)
 
var weights = pairs.keys.sort.map{|k| pairs{k} }
var inverse = pairs.flip
 
for n in (1 .. weights.end) {
var found = false
weights.combinations(n, {|comb|
if (comb.sum == 0) {
say "Length #{n}: "+" ".join(inverse{comb...})
found = true
break
}
})
found || say "Length #{n}: (none)"
}
Output:
Length 1: (none)
Length 2: archbishop gestapo
Length 3: centipede markham mycenae
Length 4: alliance balm deploy mycenae
Length 5: alliance brute covariate deploy mincemeat
Length 6: alliance archbishop balm deploy gestapo mycenae
Length 7: alliance archbishop bonnet cobol departure exorcism moresby
Length 8: alliance archbishop balm bonnet fiat flatworm isis lindholm
Length 9: alliance archbishop balm bonnet brute covariate eradicate mincemeat plugging
Length 10: alliance archbishop balm bonnet brute centipede cobol departure deploy mincemeat
Length 11: alliance archbishop balm bonnet brute centipede cobol departure infra moresby speakeasy
Length 12: alliance archbishop balm bonnet brute centipede cobol covariate diophantine efferent elysee infra
Length 13: alliance archbishop balm bonnet brute centipede cobol covariate departure efferent eradicate filmy isis
Length 14: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy elysee filmy markham speakeasy
Length 15: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy elysee exorcism flatworm infra mycenae
Length 16: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine elysee exorcism filmy gestapo infra
Length 17: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine exorcism isis mincemeat mycenae plugging vein
Length 18: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee exorcism filmy isis mycenae vein
Length 19: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate exorcism fiat infra isis smokescreen
Length 20: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate exorcism gestapo infra isis smokescreen speakeasy
Length 21: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate exorcism flatworm infra lindholm mincemeat plugging speakeasy
Length 22: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate escritoire exorcism fiat filmy flatworm mincemeat plugging speakeasy
Length 23: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate escritoire exorcism infra isis mincemeat moresby mycenae smokescreen speakeasy
Length 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
Length 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
Length 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
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
Length 28: (none)
Length 29: (none)
Length 30: (none)

Tcl[edit]

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.

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"
}

Demonstrating:

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] {, }]"
Output:
Found zero-summing subset: archbishop, gestapo

Ursala[edit]

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.

#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 [email protected] ~&lrmPC
 
#cast %zm
 
main = nullset weights

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\~&r If the partial result contains a zero sum, return it.
  • ^TnK2hS\~&r Concatenate the partial result with the new list of subsets (computed as follows) and delete duplicate sums.
  • ^C/~&lmPlNCX Cons a singleton subset containing the next word to the partial results.
  • *D Distribute the next word in the set to the partial results and do the following to each.
  • [email protected] Add the weight of the new word to the existing sum.
  • ~&lrmPC Cons 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>

zkl[edit]

Translation of: C
var items=T(
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));
 
fcn subSum(set,i,weight){
if(i and not weight){
itms:=i.pump(List,'wrap(n){ items[set[n]][0] });
println(itms.len(),": ",itms.concat(","));
throw(Exception.TheEnd);
}
foreach j in ([i and set[i-1] + 1 or 0 .. items.len()-1]){
set[i]=j;
self.fcn(set, i+1, weight + items[j][1]);
}
}
 
set:=List.createLong(items.len(),0);
try{ subSum(set,0,0); }catch(TheEnd){}
Output:
22: alliance,archbishop,balm,bonnet,brute,centipede,cobol,covariate,departure,deploy,diophantine,efferent,elysee,eradicate,escritoire,exorcism,fiat,filmy,flatworm,mincemeat,plugging,speakeasy