Subset sum problem
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.
- Task
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
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.
11l
V words = [
‘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
]
V neg = 0
V pos = 0
L(w, v) words
I v > 0
pos += v
E
neg += v
V sums = [[String]()] * (pos - neg + 1)
L(w, v) words
V s = copy(sums)
I s[v - neg].empty
s[v - neg] = [w]
L(w2) sums
V i = L.index
I !w2.empty & s[i + v].empty
s[i + v] = w2 [+] [w]
sums = s
I !s[-neg].empty
L(x) s[-neg]
print(x‘ ’words[x])
L.break- Output:
alliance -624 archbishop -925 balm 397 bonnet 452 centipede -658 cobol 362 departure 952 deploy 44
Action!
DEFINE PTR="CARD"
DEFINE PAIR_SIZE="4"
TYPE Pair=[PTR word INT weight]
BYTE ARRAY pairs(200)
BYTE count=[0]
PTR FUNC GetPairAddr(INT index)
RETURN (pairs+index*PAIR_SIZE)
PROC PrintWords(BYTE ARRAY indices BYTE len)
Pair POINTER p
BYTE i
FOR i=0 TO len-1
DO
IF i>0 THEN Put(' ) FI
p=GetPairAddr(indices(i))
Print(p.word)
OD
PutE()
RETURN
PROC Append(CHAR ARRAY wrd INT wght)
Pair POINTER dst
dst=GetPairAddr(count)
dst.word=wrd
dst.weight=wght
count==+1
RETURN
PROC Init()
Append("alliance",-624) Append("archbishop",-915)
Append("balm",397) Append("bonnet",452)
Append("brute",870) Append("centipede",-658)
Append("cobol",362) Append("covariate",590)
Append("departure",952) Append("deploy",44)
Append("diophantine",645) Append("efferent",54)
Append("elysee",-326) Append("eradicate",376)
Append("escritoire",856) Append("exorcism",-983)
Append("fiat",170) Append("filmy",-874)
Append("flatworm",503) Append("gestapo",915)
Append("infra",-847) Append("isis",-982)
Append("lindholm",999) Append("markham",475)
Append("mincemeat",-880) Append("moresby",756)
Append("mycenae",183) Append("plugging",-266)
Append("smokescreen",423) Append("speakeasy",-745)
Append("vein",813)
RETURN
INT FUNC Sum(BYTE ARRAY indices BYTE len)
Pair POINTER p
INT sum
BYTE i
sum=0
FOR i=0 TO len-1
DO
p=GetPairAddr(indices(i))
sum==+p.weight
OD
RETURN (sum)
BYTE FUNC NextSubset(BYTE ARRAY indices BYTE len)
INT i,j
i=len-1
WHILE i>=0
DO
IF indices(i)#i+count-len THEN
indices(i)==+1
FOR j=i+1 TO len-1
DO
indices(j)=indices(j-1)+1
OD
RETURN (1)
FI
i==-1
OD
RETURN (0)
PROC Test(INT len)
BYTE ARRAY indices(100)
BYTE i
PrintF("%I: ",len)
FOR i=0 TO len-1
DO
indices(i)=i
OD
DO
IF Sum(indices,len)=0 THEN
PrintWords(indices,len) PutE()
RETURN
FI
IF NextSubset(indices,len)=0 THEN
PrintE("no subset") PutE()
RETURN
FI
OD
RETURN
PROC Main()
Init()
Test(2)
Test(3)
Test(4)
Test(5)
Test(10)
Test(27)
RETURN- Output:
Screenshot from Atari 8-bit computer
2: archbishop gestapo 3: centipede markham mycenae 4: alliance balm deploy mycenae 5: alliance brute covariate deploy mincemeat 10: alliance archbishop balm bonnet brute centipede cobol departure deploy mincemeat 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
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;
- 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
#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 ...
C#
using System;
using System.Collections.Generic;
namespace SubsetSum {
class Item {
public Item(string word, int weight) {
Word = word;
Weight = weight;
}
public string Word { get; set; }
public int Weight { get; set; }
public override string ToString() {
return string.Format("({0}, {1})", Word, Weight);
}
}
class Program {
private static readonly List<Item> items = new List<Item>() {
new Item("alliance", -624),
new Item("archbishop", -915),
new Item("balm", 397),
new Item("bonnet", 452),
new Item("brute", 870),
new Item("centipede", -658),
new Item("cobol", 362),
new Item("covariate", 590),
new Item("departure", 952),
new Item("deploy", 44),
new Item("diophantine", 645),
new Item("efferent", 54),
new Item("elysee", -326),
new Item("eradicate", 376),
new Item("escritoire", 856),
new Item("exorcism", -983),
new Item("fiat", 170),
new Item("filmy", -874),
new Item("flatworm", 503),
new Item("gestapo", 915),
new Item("infra", -847),
new Item("isis", -982),
new Item("lindholm", 999),
new Item("markham", 475),
new Item("mincemeat", -880),
new Item("moresby", 756),
new Item("mycenae", 183),
new Item("plugging", -266),
new Item("smokescreen", 423),
new Item("speakeasy", -745),
new Item("vein", 813),
};
private static readonly int n = items.Count;
private static readonly int LIMIT = 5;
private static int[] indices = new int[n];
private static int count = 0;
private static void ZeroSum(int i, int w) {
if (i != 0 && w == 0) {
for (int j = 0; j < i; j++) {
Console.Write("{0} ", items[indices[j]]);
}
Console.WriteLine("\n");
if (count < LIMIT) count++;
else return;
}
int k = (i != 0) ? indices[i - 1] + 1 : 0;
for (int j = k; j < n; j++) {
indices[i] = j;
ZeroSum(i + 1, w + items[j].Weight);
if (count == LIMIT) return;
}
}
static void Main(string[] args) {
Console.WriteLine("The weights of the following {0} subsets add up to zero:\n", LIMIT);
ZeroSum(0, 0);
}
}
}
- Output:
The weights of the following 5 subsets add up to zero: (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) (mincemeat, -880) (plugging, -266) (speakeasy, -745) (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) (infra, -847) (isis, -982) (mincemeat, -880) (moresby, 756) (mycenae, 183) (smokescreen, 423) (speakeasy, -745) (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) (fiat, 170) (infra, -847) (isis, -982) (markham, 475) (mincemeat, -880) (plugging, -266) (speakeasy, -745) (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) (exorcism, -983) (fiat, 170) (infra, -847) (isis, -982) (mincemeat, -880) (moresby, 756) (plugging, -266) (vein, 813) (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) (exorcism, -983) (fiat, 170) (infra, -847) (isis, -982) (smokescreen, 423)
C++
#include <iostream>
#include <vector>
std::ostream& operator<<(std::ostream& out, const std::string& str) {
return out << str.c_str();
}
std::vector<std::pair<std::string, int>> 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},
};
std::vector<int> indices;
int count = 0;
const int LIMIT = 5;
void subsum(int i, int weight) {
if (i != 0 && weight == 0) {
for (int j = 0; j < i; ++j) {
auto item = items[indices[j]];
std::cout << (j ? " " : "") << item.first;
}
std::cout << '\n';
if (count < LIMIT) count++;
else return;
}
int k = (i != 0) ? indices[i - 1] + 1 : 0;
for (int j = k; j < items.size(); ++j) {
indices[i] = j;
subsum(i + 1, weight + items[j].second);
if (count == LIMIT) return;
}
}
int main() {
indices.resize(items.size());
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 alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate escritoire exorcism infra isis mincemeat moresby mycenae smokescreen speakeasy alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate escritoire fiat infra isis markham mincemeat plugging speakeasy alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate exorcism fiat infra isis mincemeat moresby plugging vein alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate exorcism fiat infra isis smokescreen
D
A simple brute-force solution. This used the module of the third D solution of the Combinations Task.
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
This version prints all the 349_167 solutions in about 1.8 seconds and counts them in about 0.05 seconds.
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 .. $], n2, n1);
size_t sols = 0;
int i = 0;
int j = n2p - 1;
while (true) {
while (l[i].sum + r[j].sum) {
while (i < n1p && l[i].sum + r[j].sum < 0)
i++;
while (j >= 0 && l[i].sum + r[j].sum > 0)
j--;
if (i >= n1p || j < 0)
break;
}
if (i >= n1p || j < 0)
break;
int i1 = i + 1;
while (i1 < n1p && l[i1].sum == l[i].sum)
i1++;
int j1 = j - 1;
while (j1 >= 0 && r[j1].sum == r[j].sum)
j1--;
sols += printList(i, j, i1, j1, l, r);
i = i1;
j = j1;
}
writeln("Zero sums: ", sols);
}
- Output:
Zero sums: 349167
EasyLang
n$[] = [ "alliance" "archbishop" "balm" "bonnet" "brute" "centipede" "cobol" "covariate" "departure" "deploy" "diophantine" "efferent" "elysee" "eradicate" "escritoire" "exorcism" "fiat" "filmy" "flatworm" "gestapo" "infra" "isis" "lindholm" "markham" "mincemeat" "moresby" "mycenae" "plugging" "smokescreen" "speakeasy" "vein" ]
w[] = [ -624 -915 397 452 870 -658 362 590 952 44 645 54 -326 376 856 -983 170 -874 503 915 -847 -982 999 475 -880 756 183 -266 423 -745 813 ]
#
n = len w[]
len set[] n
done = 0
#
proc subsum i w k . .
if done = 1
return
.
if i > 0 and w = 0
for j = 1 to i
write n$[set[j]] & " "
.
done = 1
.
for j = k + 1 to n
set[i + 1] = j
subsum i + 1 w + w[j] j
.
.
subsum 0 0 1
- Output:
alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate escritoire exorcism fiat filmy flatworm mincemeat plugging speakeasy
EchoLisp
Dynamic programming
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
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))
FreeBASIC
Type Item
As String*11 word
As Integer weight
End Type
Dim Shared As Item items(30) = { ("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) }
Dim Shared indicies(30) As Integer
Dim Shared cnt As Integer = 0
Sub WriteItem(self As Item)
Print self.word; " "; self.weight
End Sub
Sub ZeroSum(i As Integer, w As Integer)
Dim As Integer j, k
If (i <> 0) And (w = 0) Then
For j = 0 To i - 1
WriteItem(items(indicies(j)))
Next j
Print
If cnt < 5 Then
cnt += 1
Else
Exit Sub
End If
End If
k = Iif(i <> 0, indicies(i - 1) + 1, 0)
For j = k To 30
indicies(i) = j
ZeroSum(i + 1, w + items(j).weight)
If cnt = 5 Then Exit Sub
Next j
End Sub
Print !"The weights of the following 5 subsets add up to zero:\n"
ZeroSum(0, 0)
Sleep
- Output:
Similar to Wren entry.
FunL
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
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
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"]
Non brute-force: the list of numbers used here are different, and difficult for a bruteforce method.
subsum :: Int -> [Int] -> [Int]
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 :: [Int]
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 :: IO ()
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
printf.icn provides formatting lists.icn provides lcomb for list combinations
- Output:
No zero-sum subsets of length 1A 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
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.)
Java
public class SubsetSum {
private static class Item {
private String word;
private int weight;
public Item(String word, int weight) {
this.word = word;
this.weight = weight;
}
@Override
public String toString() {
return String.format("(%s, %d)", word, weight);
}
}
private static Item[] items = new Item[]{
new Item("alliance", -624),
new Item("archbishop", -915),
new Item("balm", 397),
new Item("bonnet", 452),
new Item("brute", 870),
new Item("centipede", -658),
new Item("cobol", 362),
new Item("covariate", 590),
new Item("departure", 952),
new Item("deploy", 44),
new Item("diophantine", 645),
new Item("efferent", 54),
new Item("elysee", -326),
new Item("eradicate", 376),
new Item("escritoire", 856),
new Item("exorcism", -983),
new Item("fiat", 170),
new Item("filmy", -874),
new Item("flatworm", 503),
new Item("gestapo", 915),
new Item("infra", -847),
new Item("isis", -982),
new Item("lindholm", 999),
new Item("markham", 475),
new Item("mincemeat", -880),
new Item("moresby", 756),
new Item("mycenae", 183),
new Item("plugging", -266),
new Item("smokescreen", 423),
new Item("speakeasy", -745),
new Item("vein", 813),
};
private static final int n = items.length;
private static final int[] indices = new int[n];
private static int count = 0;
private static final int LIMIT = 5;
private static void zeroSum(int i, int w) {
if (i != 0 && w == 0) {
for (int j = 0; j < i; ++j) {
System.out.printf("%s ", items[indices[j]]);
}
System.out.println("\n");
if (count < LIMIT) count++;
else return;
}
int k = (i != 0) ? indices[i - 1] + 1 : 0;
for (int j = k; j < n; ++j) {
indices[i] = j;
zeroSum(i + 1, w + items[j].weight);
if (count == LIMIT) return;
}
}
public static void main(String[] args) {
System.out.printf("The weights of the following %d subsets add up to zero:\n\n", LIMIT);
zeroSum(0, 0);
}
}
- Output:
The weights of the following 5 subsets add up to zero: (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) (mincemeat, -880) (plugging, -266) (speakeasy, -745) (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) (infra, -847) (isis, -982) (mincemeat, -880) (moresby, 756) (mycenae, 183) (smokescreen, 423) (speakeasy, -745) (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) (fiat, 170) (infra, -847) (isis, -982) (markham, 475) (mincemeat, -880) (plugging, -266) (speakeasy, -745) (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) (exorcism, -983) (fiat, 170) (infra, -847) (isis, -982) (mincemeat, -880) (moresby, 756) (plugging, -266) (vein, 813) (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) (exorcism, -983) (fiat, 170) (infra, -847) (isis, -982) (smokescreen, 423)
jq
Works with gojq, the Go implementation of jq (provided `keys_unsorted` is replaced by `keys`)
# Input: an array of n elements, each of which is either in or out
# Output: a stream of the 2^n possible selections
def selections:
# map( [null, .] ) | combinations | map(select(.));
if length == 0
then .
else .[0] as $x
| .[1:] | selections | ., ([$x] + .)
end ;
# input: a JSON object giving the weights
def zero_sums:
def sum($dict; $sum):
map( $dict[.] ) | add == $sum;
. as $dict
| keys_unsorted | selections | select( sum($dict; 0));An Example
def 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
};
weights | first(zero_sums)- Output:
[ "archbishop", "balm", "bonnet", "centipede", "cobol", "covariate", "deploy", "efferent", "elysee" ]
Julia
using Combinatorics
const 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]
const weights = [v for (k, v) in pairs]
const weightkeyed = Dict(Pair(v, k) for (k, v) in pairs)
function zerosums()
total = 0
for i in 1:length(weights)
print("\nFor length $i: ")
sets = [a for a in combinations(weights, i) if sum(a) == 0]
if (n = length(sets)) == 0
print("None")
else
total += n
print("$n sets, example: ", map(x -> weightkeyed[x], rand(sets)))
end
end
println("\n\nGrand total sets: $total")
end
zerosums()
- Output:
For length 1: None For length 2: 1 sets, example: ["archbishop", "gestapo"] For length 3: 2 sets, example: ["centipede", "markham", "mycenae"] For length 4: 9 sets, example: ["balm", "efferent", "filmy", "smokescreen"] For length 5: 48 sets, example: ["departure", "elysee", "lindholm", "mincemeat", "speakeasy"] For length 6: 178 sets, example: ["alliance", "bonnet", "centipede", "isis", "lindholm", "vein"] For length 7: 629 sets, example: ["balm", "brute", "cobol", "deploy", "filmy", "isis", "mycenae"] For length 8: 1634 sets, example: ["alliance", "brute", "centipede", "departure", "mincemeat", "mycenae", "plugging", "smokescreen"] For length 9: 4040 sets, example: ["alliance", "brute", "deploy", "efferent", "elysee", "fiat", "filmy", "flatworm", "mycenae"] For length 10: 8673 sets, example: ["archbishop", "brute", "escritoire", "filmy", "gestapo", "isis", "lindholm", "mincemeat", "moresby", "speakeasy"] For length 11: 15680 sets, example: ["cobol", "covariate", "deploy", "efferent", "elysee", "eradicate", "exorcism", "filmy", "flatworm", "lindholm", "speakeasy"] For length 12: 25492 sets, example: ["alliance", "balm", "diophantine", "efferent", "eradicate", "fiat", "filmy", "flatworm", "infra", "isis", "lindholm", "mycenae"] For length 13: 35940 sets, example: ["alliance", "archbishop", "bonnet", "departure", "deploy", "efferent", "fiat", "filmy", "gestapo", "infra", "moresby", "mycenae", "plugging"] For length 14: 44920 sets, example: ["archbishop", "centipede", "cobol", "covariate", "deploy", "efferent", "eradicate", "escritoire", "fiat", "gestapo", "isis", "mincemeat", "speakeasy", "vein"] For length 15: 49368 sets, example: ["alliance", "archbishop", "bonnet", "covariate", "departure", "diophantine", "efferent", "elysee", "eradicate", "escritoire", "isis", "mincemeat", "plugging", "speakeasy", "vein"] For length 16: 47835 sets, example: ["alliance", "archbishop", "balm", "bonnet", "centipede", "cobol", "covariate", "departure", "elysee", "flatworm", "infra", "isis", "moresby", "mycenae", "plugging", "smokescreen"] For length 17: 40960 sets, example: ["balm", "bonnet", "centipede", "cobol", "covariate", "departure", "deploy", "diophantine", "efferent", "elysee", "eradicate", "fiat", "filmy", "isis", "mincemeat", "smokescreen", "speakeasy"] For length 18: 31139 sets, example: ["balm", "bonnet", "centipede", "covariate", "departure", "deploy", "diophantine", "efferent", "elysee", "eradicate", "exorcism", "gestapo", "infra", "isis", "mincemeat", "mycenae", "speakeasy", "vein"] For length 19: 20530 sets, example: ["alliance", "archbishop", "balm", "bonnet", "departure", "deploy", "elysee", "exorcism", "fiat", "gestapo", "infra", "isis", "lindholm", "markham", "mincemeat", "mycenae", "plugging", "smokescreen", "vein"] For length 20: 11926 sets, example: ["archbishop", "bonnet", "brute", "centipede", "cobol", "deploy", "diophantine", "elysee", "eradicate", "exorcism", "fiat", "isis", "lindholm", "markham", "mincemeat", "moresby", "mycenae", "plugging", "smokescreen", "speakeasy"] For length 21: 6089 sets, example: ["alliance", "balm", "bonnet", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "elysee", "escritoire", "exorcism", "filmy", "flatworm", "infra", "isis", "markham", "mincemeat", "moresby", "mycenae", "plugging"] For length 22: 2688 sets, example: ["archbishop", "balm", "bonnet", "brute", "centipede", "covariate", "deploy", "diophantine", "elysee", "eradicate", "exorcism", "filmy", "flatworm", "gestapo", "isis", "markham", "mincemeat", "moresby", "mycenae", "plugging", "smokescreen", "speakeasy"] For length 23: 956 sets, example: ["alliance", "archbishop", "balm", "brute", "centipede", "cobol", "departure", "deploy", "efferent", "elysee", "eradicate", "escritoire", "exorcism", "fiat", "gestapo", "infra", "isis", "lindholm", "markham", "mincemeat", "moresby", "plugging", "speakeasy"] For length 24: 341 sets, example: ["alliance", "archbishop", "balm", "brute", "centipede", "cobol", "covariate", "departure", "deploy", "diophantine", "efferent", "elysee", "eradicate", "escritoire", "exorcism", "infra", "isis", "lindholm", "markham", "mincemeat", "mycenae", "plugging", "smokescreen", "speakeasy"] For length 25: 73 sets, example: ["alliance", "archbishop", "balm", "bonnet", "brute", "centipede", "cobol", "departure", "deploy", "diophantine", "efferent", "elysee", "exorcism", "fiat", "filmy", "flatworm", "gestapo", "infra", "isis", "lindholm", "markham", "mincemeat", "mycenae", "speakeasy", "vein"] For length 26: 14 sets, example: ["alliance", "archbishop", "balm", "bonnet", "centipede", "cobol", "departure", "diophantine", "efferent", "elysee", "eradicate", "escritoire", "exorcism", "fiat", "filmy", "flatworm", "gestapo", "infra", "isis", "lindholm", "mincemeat", "mycenae", "plugging", "smokescreen", "speakeasy", "vein"] For length 27: 2 sets, example: ["alliance", "archbishop", "balm", "brute", "centipede", "cobol", "covariate", "deploy", "diophantine", "efferent", "elysee", "eradicate", "exorcism", "fiat", "filmy", "flatworm", "gestapo", "infra", "isis", "lindholm", "mincemeat", "moresby", "mycenae", "plugging", "smokescreen", "speakeasy", "vein"] For length 28: None For length 29: None For length 30: None For length 31: None Grand total sets: 349167
Kotlin
// version 1.1.2
class Item(val word: String, val weight: Int) {
override fun toString() = "($word $weight)"
}
val items = arrayOf(
Item("alliance", -624),
Item("archbishop", -915),
Item("balm", 397),
Item("bonnet", 452),
Item("brute", 870),
Item("centipede", -658),
Item("cobol", 362),
Item("covariate", 590),
Item("departure", 952),
Item("deploy", 44),
Item("diophantine", 645),
Item("efferent", 54),
Item("elysee", -326),
Item("eradicate", 376),
Item("escritoire", 856),
Item("exorcism", -983),
Item("fiat", 170),
Item("filmy", -874),
Item("flatworm", 503),
Item("gestapo", 915),
Item("infra", -847),
Item("isis", -982),
Item("lindholm", 999),
Item("markham", 475),
Item("mincemeat", -880),
Item("moresby", 756),
Item("mycenae", 183),
Item("plugging", -266),
Item("smokescreen", 423),
Item("speakeasy", -745),
Item("vein", 813)
)
val n = items.size
val indices = IntArray(n)
var count = 0
const val LIMIT = 5
fun zeroSum(i: Int, w: Int) {
if (i != 0 && w == 0) {
for (j in 0 until i) print("${items[indices[j]]} ")
println("\n")
if (count < LIMIT) count++ else return
}
val k = if (i != 0) indices[i - 1] + 1 else 0
for (j in k until n) {
indices[i] = j
zeroSum(i + 1, w + items[j].weight)
if (count == LIMIT) return
}
}
fun main(args: Array<String>) {
println("The weights of the following $LIMIT subsets add up to zero:\n")
zeroSum(0, 0)
}
- Output:
The weights of the following 5 subsets add up to zero: (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) (mincemeat -880) (plugging -266) (speakeasy -745) (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) (infra -847) (isis -982) (mincemeat -880) (moresby 756) (mycenae 183) (smokescreen 423) (speakeasy -745) (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) (fiat 170) (infra -847) (isis -982) (markham 475) (mincemeat -880) (plugging -266) (speakeasy -745) (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) (exorcism -983) (fiat 170) (infra -847) (isis -982) (mincemeat -880) (moresby 756) (plugging -266) (vein 813) (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) (exorcism -983) (fiat 170) (infra -847) (isis -982) (smokescreen 423)
Mathematica /Wolfram Language
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 ]
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}}.
MiniZinc
%Subset sum. Nigel Galloway: January 6th., 2021.
enum Items={alliance,archbishop,balm,bonnet,brute,centipede,cobol,covariate,departure,deploy,diophantine,efferent,elysee,eradicate,escritoire,exorcism,fiat,filmy,flatworm,gestapo,infra,isis,lindholm,markham,mincemeat,moresby,mycenae,plugging,smokescreen,speakeasy,vein};
array[Items] of int: weight=[-624,-915,397,452,870,-658,362,590,952,44,645,54,-326,376,856,-983,170,-874,503,915,-847,-982,999,475,-880,756,183,-266,423,-745,813];
var set of Items: selected;
var int: wSelected=sum(n in selected)(weight[n]);
constraint wSelected=0;- Output:
selected = {alliance, archbishop, balm, bonnet, brute, centipede, cobol, covariate, departure, deploy, diophantine, efferent, elysee, eradicate, escritoire, exorcism, fiat, filmy, flatworm, mincemeat, plugging, speakeasy};
----------
Finished in 185msec
Modula-2
MODULE SubsetSum;
FROM FormatString IMPORT FormatString;
FROM Terminal IMPORT WriteString,WriteLn,ReadChar;
TYPE
String = ARRAY[0..63] OF CHAR;
Item = RECORD
word : String;
weight : INTEGER;
END;
PROCEDURE WriteItem(self : Item);
VAR buf : String;
BEGIN
FormatString("(%s, %i)", buf, self.word, self.weight);
WriteString(buf);
END WriteItem;
CONST N = 31;
VAR
items : ARRAY[0..N] OF Item;
indicies : ARRAY[0..N] OF INTEGER;
count : INTEGER;
PROCEDURE Init;
VAR i : INTEGER;
BEGIN
items[0] := Item{"alliance", -624};
items[1] := Item{"archbishop", -915};
items[2] := Item{"balm", 397};
items[3] := Item{"bonnet", 452};
items[4] := Item{"brute", 870};
items[5] := Item{"centipede", -658};
items[6] := Item{"cobol", 362};
items[7] := Item{"covariate", 590};
items[8] := Item{"departure", 952};
items[9] := Item{"deploy", 44};
items[10] := Item{"diophantine", 645};
items[11] := Item{"efferent", 54};
items[12] := Item{"elysee", -326};
items[13] := Item{"eradicate", 376};
items[14] := Item{"escritoire", 856};
items[15] := Item{"exorcism", -983};
items[16] := Item{"fiat", 170};
items[17] := Item{"filmy", -874};
items[18] := Item{"flatworm", 503};
items[19] := Item{"gestapo", 915};
items[20] := Item{"infra", -847};
items[21] := Item{"isis", -982};
items[22] := Item{"lindholm", 999};
items[23] := Item{"markham", 475};
items[24] := Item{"mincemeat", -880};
items[25] := Item{"moresby", 756};
items[26] := Item{"mycenae", 183};
items[27] := Item{"plugging", -266};
items[28] := Item{"smokescreen", 423};
items[29] := Item{"speakeasy", -745};
items[30] := Item{"vein", 813};
count := 0;
END Init;
CONST LIMIT = 5;
PROCEDURE ZeroSum(i,w : INTEGER);
VAR j,k : INTEGER;
BEGIN
IF (i#0) AND (w=0) THEN
FOR j:=0 TO i-1 DO
WriteItem(items[indicies[j]]);
WriteString(" ");
END;
WriteLn;
WriteString("---------------");
WriteLn;
IF count<LIMIT THEN
INC(count)
ELSE
RETURN;
END;
END;
IF i#0 THEN
k := indicies[i-1]+1;
ELSE
k := 0;
END;
FOR j:=k TO N-1 DO
indicies[i] := j;
ZeroSum(i+1,w+items[j].weight);
IF count=LIMIT THEN RETURN; END;
END;
END ZeroSum;
VAR buf : ARRAY[0..63] OF CHAR;
VAR d : INTEGER;
BEGIN
Init;
d := LIMIT;
FormatString("The weights of the following %i subsets add up to zero:\n\n", buf, d);
WriteString(buf);
ZeroSum(0,0);
ReadChar;
END SubsetSum.
Nim
We need the third party module "itertools" to compute the combinations.
import sequtils, strformat, strutils
import itertools
const 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}
var words: seq[string]
var values: seq[int]
for (word, value) in Words:
words.add word
values.add value
for lg in 1..words.len:
block checkCombs:
for comb in combinations(words.len, lg):
var sum = 0
for idx in comb: sum += values[idx]
if sum == 0:
echo &"For length {lg}, found for example: ", comb.mapIt(words[it]).join(" ")
break checkCombs
echo &"For length {lg}, no set found."
- Output:
For length 1, no set found. For length 2, found for example: archbishop gestapo For length 3, found for example: centipede markham mycenae For length 4, found for example: alliance balm deploy mycenae For length 5, found for example: alliance brute covariate deploy mincemeat For length 6, found for example: alliance archbishop balm deploy gestapo mycenae For length 7, found for example: alliance archbishop bonnet cobol departure exorcism moresby For length 8, found for example: alliance archbishop balm bonnet fiat flatworm isis lindholm For length 9, found for example: alliance archbishop balm bonnet brute covariate eradicate mincemeat plugging For length 10, found for example: alliance archbishop balm bonnet brute centipede cobol departure deploy mincemeat For length 11, found for example: alliance archbishop balm bonnet brute centipede cobol departure infra moresby speakeasy For length 12, found for example: alliance archbishop balm bonnet brute centipede cobol covariate diophantine efferent elysee infra For length 13, found for example: alliance archbishop balm bonnet brute centipede cobol covariate departure efferent eradicate filmy isis For length 14, found for example: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy elysee filmy markham speakeasy For length 15, found for example: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy elysee exorcism flatworm infra mycenae For length 16, found for example: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine elysee exorcism filmy gestapo infra For length 17, found for example: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine exorcism isis mincemeat mycenae plugging vein For length 18, found for example: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee exorcism filmy isis mycenae vein For length 19, found for example: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate exorcism fiat infra isis smokescreen For length 20, found for example: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate exorcism gestapo infra isis smokescreen speakeasy For length 21, found for example: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate exorcism flatworm infra lindholm mincemeat plugging speakeasy For length 22, found for example: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate escritoire exorcism fiat filmy flatworm mincemeat plugging speakeasy For length 23, found for example: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate escritoire exorcism infra isis mincemeat moresby mycenae smokescreen speakeasy For length 24, found for example: alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee exorcism filmy gestapo infra markham mincemeat moresby mycenae plugging smokescreen speakeasy For length 25, found for example: 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 For length 26, found for example: 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 For length 27, found for example: 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 For length 28, no set found. For length 29, no set found. For length 30, no set found. For length 31, no set found.
OCaml
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
use ntheory qw/:all/;
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 );
# sort so we get the same order each time
my @names = sort keys(%pairs);
my @weights = @pairs{@names}; # hash slice gives all values in same order
foreach my $n (1 .. @names) {
forcomb {
# Remove the "lastfor, " to get all combinations
lastfor, print "Length $n: @names[@_]\n" if vecsum(@weights[@_]) == 0;
} @names, $n;
}
Printing just the first one found for each number of elements:
- Output:
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 ... to length 27 ...
We can also use different modules for this brute force method. Assuming the same pairs/names/weights variables:
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;
}
}
Phix
Simple Brute force
with javascript_semantics constant {words,weights} = columnize({{"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}}) function comb(sequence pool, integer needed, done=0, sequence chosen={}) if needed=0 then -- got a full set if sum(extract(weights,chosen))=0 then sequence cw = extract(words,chosen), scw = shorten(cw,"items",4) printf(1,"%d: %s\n",{length(chosen),join(scw,",")}) return true end if elsif done+needed<=length(pool) then -- get all combinations with and without the next item: done += 1 if comb(pool,needed-1,done,append(deep_copy(chosen),pool[done])) or comb(pool,needed ,done,chosen) then return true end if end if return false end function integer n = length(weights) for i=1 to n do if not comb(tagset(n),i) then printf(1,"%d: No zero-sum subsets of that length\n",{i}) end if end for
- Output:
1: No zero-sum subsets of that length 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,...,departure,infra,moresby,speakeasy, (11 items) 12: alliance,archbishop,balm,bonnet,...,diophantine,efferent,elysee,infra, (12 items) 13: alliance,archbishop,balm,bonnet,...,efferent,eradicate,filmy,isis, (13 items) 14: alliance,archbishop,balm,bonnet,...,elysee,filmy,markham,speakeasy, (14 items) 15: alliance,archbishop,balm,bonnet,...,exorcism,flatworm,infra,mycenae, (15 items) 16: alliance,archbishop,balm,bonnet,...,exorcism,filmy,gestapo,infra, (16 items) 17: alliance,archbishop,balm,bonnet,...,mincemeat,mycenae,plugging,vein, (17 items) 18: alliance,archbishop,balm,bonnet,...,filmy,isis,mycenae,vein, (18 items) 19: alliance,archbishop,balm,bonnet,...,fiat,infra,isis,smokescreen, (19 items) 20: alliance,archbishop,balm,bonnet,...,infra,isis,smokescreen,speakeasy, (20 items) 21: alliance,archbishop,balm,bonnet,...,lindholm,mincemeat,plugging,speakeasy, (21 items) 22: alliance,archbishop,balm,bonnet,...,flatworm,mincemeat,plugging,speakeasy, (22 items) 23: alliance,archbishop,balm,bonnet,...,moresby,mycenae,smokescreen,speakeasy, (23 items) 24: alliance,archbishop,balm,bonnet,...,mycenae,plugging,smokescreen,speakeasy, (24 items) 25: alliance,archbishop,balm,bonnet,...,moresby,mycenae,plugging,speakeasy, (25 items) 26: alliance,archbishop,balm,bonnet,...,mycenae,plugging,speakeasy,vein, (26 items) 27: alliance,archbishop,balm,bonnet,...,mycenae,plugging,smokescreen,speakeasy, (27 items) 28: No zero-sum subsets of that length 29: No zero-sum subsets of that length 30: No zero-sum subsets of that length 31: No zero-sum subsets of that length
Alternative
Using the harder set of weights from Go, and the version 1 approach of Python, modified to omit words and
using a dictionary so that fractional weights can be accomodated.
Note that fractional weights may trigger all the usual IEEE-754 issues such as 0.3+0.6-0.9 != 0, so you may need to use sprintf()'d keys with matching to_number()'s.
This is significantly faster (near instant, in fact) than an "all possible combinations" approach.
Shows the first zero-sum subset found, only.
with javascript_semantics constant weights = {-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 } integer sums = new_dict() for w=1 to length(weights) do -- make a separate modifiable copy of sums, otherwise -- it c/would mark sums[weights[w]*{2,3,etc}] as valid, -- ie there cannot be any w in the getd_by_index(node). integer s = new_dict(sums) atom v = weights[w] if getd_index(v,s)=NULL then setd(v,{w},s) end if sequence sk = getd_all_keys(s) for i=1 to length(sk) do integer node = getd_index(sk[i],sums) if node!=NULL and getd_index(sk[i]+v,s)=NULL then setd(sk[i]+v,deep_copy(getd_by_index(node,sums))&w,s) end if end for destroy_dict(sums) sums = s integer node = getd_index(0,sums) if node!=NULL then sequence t = extract(weights,getd_by_index(node,sums)) printf(1,"Total %d for %v\n",{sum(t),t}) exit end if end for
- Output:
Total 0 for {-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}
Picat
Using constraint modelling.
import cp.
%
% Get any solution.
%
go =>
things(Things),
subset_sum(Things, X, Z),
println(z=Z),
println(x=X),
println([Things[I] : I in 1..Things.length, X[I] == 1]),
nl.
%
% Solve subset problem for any length (wrapper).
%
subset_sum(Things, X, Z) =>
subset_sum(Things,_,X,Z).
%
% Solve for a specific length (if Len2 is nonvar).
%
subset_sum(Things, Len2, X, Z) =>
Weights = [ T[2] : T in Things],
Len = Things.length,
X = new_list(Len),
X :: 0..1,
Z #= sum(X),
if nonvar(Len2) then
Z #= Len2
end,
Z #> 0, % require at least one thing
sum([X[I] * Weights[I] : I in 1..Len]) #= 0,
solve($[ff], X).
things(Things) =>
Things = [
[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 ]
].- Output:
z = 8 x = [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0,0,1,1,1,1] [[flatworm,503],[infra,-847],[lindholm,999],[mincemeat,-880],[plugging,-266],[smokescreen,423],[speakeasy,-745],[vein,813]]
Some other experiments.
Get a solution - if possible - for a specific number of selected things.
go2 =>
things(Things),
foreach(Len in 1..Things.length)
println(len=Len),
(subset_sum(Things, Len, X, _Z) ->
println([Things[I,1] : I in 1..Things.length, X[I] == 1])
;
true
),
nl
end,
nl.- Output:
len = 1 len = 2 [archbishop,gestapo] len = 3 [exorcism,fiat,vein] len = 4 [exorcism,gestapo,speakeasy,vein] len = 5 [fiat,infra,lindholm,smokescreen,speakeasy] len = 6 [exorcism,flatworm,gestapo,isis,plugging,vein] len = 7 [fiat,flatworm,infra,isis,lindholm,plugging,smokescreen] len = 8 [flatworm,infra,lindholm,mincemeat,plugging,smokescreen,speakeasy,vein] len = 9 [exorcism,filmy,flatworm,infra,markham,moresby,plugging,smokescreen,vein] len = 10 [fiat,filmy,flatworm,gestapo,isis,markham,mincemeat,moresby,mycenae,plugging] len = 11 [exorcism,fiat,flatworm,gestapo,infra,isis,lindholm,plugging,smokescreen,speakeasy,vein] len = 12 [eradicate,fiat,infra,isis,lindholm,mincemeat,moresby,mycenae,plugging,smokescreen,speakeasy,vein] len = 13 [escritoire,fiat,filmy,gestapo,infra,isis,lindholm,markham,mincemeat,moresby,plugging,smokescreen,speakeasy] len = 14 [escritoire,exorcism,fiat,flatworm,infra,isis,lindholm,mincemeat,moresby,mycenae,plugging,smokescreen,speakeasy,vein] len = 15 [efferent,elysee,escritoire,exorcism,fiat,filmy,flatworm,infra,isis,lindholm,moresby,mycenae,smokescreen,speakeasy,vein] len = 16 [diophantine,elysee,eradicate,fiat,filmy,flatworm,infra,isis,lindholm,markham,mincemeat,moresby,mycenae,plugging,speakeasy,vein] len = 17 [deploy,diophantine,elysee,eradicate,escritoire,exorcism,fiat,filmy,flatworm,infra,isis,lindholm,markham,mincemeat,moresby,speakeasy,vein] len = 18 [deploy,diophantine,efferent,escritoire,exorcism,fiat,filmy,gestapo,infra,isis,lindholm,markham,mincemeat,mycenae,plugging,smokescreen,speakeasy,vein] len = 19 [departure,deploy,diophantine,efferent,elysee,eradicate,exorcism,fiat,filmy,flatworm,infra,isis,lindholm,markham,mincemeat,mycenae,smokescreen,speakeasy,vein] len = 20 [covariate,diophantine,efferent,elysee,eradicate,exorcism,fiat,filmy,flatworm,gestapo,infra,isis,markham,mincemeat,moresby,mycenae,plugging,smokescreen,speakeasy,vein] len = 21 [cobol,deploy,diophantine,efferent,elysee,eradicate,escritoire,exorcism,fiat,filmy,flatworm,infra,isis,lindholm,markham,mincemeat,mycenae,plugging,smokescreen,speakeasy,vein] len = 22 [brute,centipede,cobol,covariate,deploy,diophantine,efferent,elysee,escritoire,exorcism,fiat,filmy,flatworm,infra,isis,markham,mincemeat,moresby,plugging,smokescreen,speakeasy,vein] len = 23 [bonnet,centipede,cobol,departure,deploy,diophantine,efferent,elysee,eradicate,escritoire,exorcism,fiat,filmy,infra,isis,markham,mincemeat,moresby,mycenae,plugging,smokescreen,speakeasy,vein] len = 24 [archbishop,brute,centipede,cobol,covariate,deploy,diophantine,efferent,elysee,escritoire,exorcism,fiat,filmy,flatworm,gestapo,infra,isis,markham,mincemeat,moresby,plugging,smokescreen,speakeasy,vein] len = 25 [archbishop,bonnet,centipede,cobol,departure,deploy,diophantine,efferent,elysee,eradicate,escritoire,exorcism,fiat,filmy,gestapo,infra,isis,markham,mincemeat,moresby,mycenae,plugging,smokescreen,speakeasy,vein] len = 26 [alliance,archbishop,bonnet,brute,centipede,covariate,departure,deploy,efferent,elysee,eradicate,exorcism,fiat,filmy,flatworm,gestapo,infra,isis,lindholm,mincemeat,moresby,mycenae,plugging,smokescreen,speakeasy,vein] len = 27 [alliance,archbishop,balm,brute,centipede,cobol,covariate,deploy,diophantine,efferent,elysee,eradicate,exorcism,fiat,filmy,flatworm,gestapo,infra,isis,lindholm,mincemeat,moresby,mycenae,plugging,smokescreen,speakeasy,vein] len = 28 len = 29 len = 30 len = 31
There are in total 349167 solutions of any number of items (the empty set is not a solution in our book).
go3 =>
things(Things),
Count = count_all(subset_sum(Things, _X, _Z)),
println(count=Count),
nl.
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) )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
Version 1
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
>>> 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
#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)
Raku
(formerly Perl 6)
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;
.say for (1..27).hyper(:3batch).map: -> $n {
given @weights.combinations($n).first({ 0 == [+] @^comb }) {
when .so { "Length $n: ({.map: {%name{$_}}})" }
default { "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)
REXX
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).
Added was "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 program finds some non─null subsets of a weighted list whose sum eqals zero.*/
parse arg target stopAt chunkette . /*option optional arguments from the CL*/
if target=='' | target=="," then target= 0 /*Not specified? Then use the default.*/
if stopAt=='' | stopAt=="," then stopAt= 1 /* " " " " " " */
y= 0
zz= 'archbishop -915 covariate 590 mycenae 183 brute 870 balm 397 fiat 170' ,
'smokescreen 423 eradicate 376 efferent 54 bonnet 452 vein 813 ' ,
'diophantine 645 departure 952 lindholm 999 moresby 756 isis -982 ' ,
'mincemeat -880 alliance -624 flatworm 503 elysee -326 cobol 362 ' ,
'centipede -658 exorcism -983 gestapo 915 filmy -874 deploy 44 ' ,
'speakeasy -745 plugging -266 markham 475 infra -847 escritoire 856 '
@.= 0
do N=1 until zz='' /*construct an array from the ZZ list. */
parse var zz @.N #.N zz /*pick from the list like a nose. */
end /*N*/
call eSort N /*sort the names with weights. */
call tellZ 'sorted' /*display the sorted list. */
chunkStart= 1 /*the default place to start. */
chunkEnd = N /* " " " " end. */
if chunkette\=='' then do /*solutions just for a chunkette. */
chunkStart= chunkette
chunkEnd = chunkette
end
call time 'Reset' /*reset the REXX elapsed time. */
??= 0 /*the 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. */
_= ?? /*set a temporary variable to ??. */
if _==0 then _= 'No' /*Englishise _ for a zero count. */
call tello _ 'solution's(??) "found so far."
end /*chunk*/
if ??==0 then ??= 'no' /*Englishise the solutions number. */
call tello 'Found' ?? "subset"s(??) 'whose summed weight's(??) "=" target
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
combN: procedure expose @. #. ?? stopAt target; parse arg x,y; !.= @.0
base= x + 1; bbase= base - y /*!.n are the combination digits. */
do n=1 for y; !.n=n /*construct the first combination. */
end /*n*/
ym= y - 1
do j=1; _= !.1; s= #._ /*obtain the first digit and the sum. */
if s>target then leave /*Is 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 digit.*/
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 one to digit we're pointing at. */
if !.u >= bbase+u then return .combUp(u-1)
p= !.u /*P will be used for the next digt. */
end /*u*/; return 0 /*go back and sum this combination. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
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
/*──────────────────────────────────────────────────────────────────────────────────────*/
s: if arg(1)==1 then return arg(3); return word(arg(2) 's', 1) /*simple pluralizer*/
tello: parse arg _,e; if e==. then say; say _; call lineout 'SUBSET.'y, _; return
/*──────────────────────────────────────────────────────────────────────────────────────*/
telly: ??= ?? + 1; nameL= /*start with a "null" name list. */
do gi=1 for y; ggg= !.gi /*build duplicate array (to be sorted).*/
$.gi= @.ggg /*transform from index ──► a name. */
end /*gi*/ /*build duplicate array (to be sorted).*/
call eSort y, . /*sort the names alphabetically. */
do gs=1 for y; nameL= nameL $.gs /*build a list of names whose sum = 0 */
end /*gs*/ /*the 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 (or the) limit.*/
call tello 'Stopped after finding ' ?? " subset"s(??)'.', .
exit /*a short─timer, we should quit then. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
tellz: do j=1 for N /*show a 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 the 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.
─────────────────────────────── doing chunk: 2 ────────────────────────────────
[2 names] archbishop gestapo
1 solution found so far.
─────────────────────────────── doing chunk: 3 ────────────────────────────────
[3 names] exorcism fiat vein
[3 names] centipede markham mycenae
3 solutions found 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.
Ring
# Project : Subset sum problem
knap = [["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]]
knapsack = createDimList([pow(2, len(knap)),len(knap)+2])
knapweight = createDimList([pow(2, len(knap)),len(knap)+2])
lenknap = list(pow(2, len(knap)))
powerset(knap)
func powerset(list)
n1 = 0
num = 0
for i = 2 to (2 << len(list)) - 1 step 2
n2 = 0
n1 = n1 + 1
weight = 0
for j = 1 to len(list)
if i & (1 << j)
n2 = n2 + 1
knapsack[n1][n2] = list[j][1]
weight = weight + list[j][2]
knapweight[n1][n2] = list[j][2]
ok
next
lenknap[n1] = n2+1
if weight = 0
see "" + num + ": "
for p = 1 to lenknap[n1]-1
see "{" + knapsack[n1][p] + " " + knapweight[n1][p]+ "}"
next
see nl
num = num + 1
ok
next
func createDimList(dimArray)
sizeList = len(dimArray)
newParms = []
for i = 2 to sizeList
Add(newParms, dimArray[i])
next
alist = list(dimArray[1])
if sizeList = 1
return aList
ok
for t in alist
t = createDimList(newParms)
next
return alistOutput:
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)}
Ruby
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
Scala
- Output:
Best seen running in your browser by Scastie (remote JVM).
object SubsetSum extends App {
private val LIMIT = 5
private val n = items.length
private val indices = new Array[Int](n)
private var count = 0
private def items = Seq(
Item("alliance", -624),
Item("archbishop", -915),
Item("balm", 397),
Item("bonnet", 452),
Item("brute", 870),
Item("centipede", -658),
Item("cobol", 362),
Item("covariate", 590),
Item("departure", 952),
Item("deploy", 44),
Item("diophantine", 645),
Item("efferent", 54),
Item("elysee", -326),
Item("eradicate", 376),
Item("escritoire", 856),
Item("exorcism", -983),
Item("fiat", 170),
Item("filmy", -874),
Item("flatworm", 503),
Item("gestapo", 915),
Item("infra", -847),
Item("isis", -982),
Item("lindholm", 999),
Item("markham", 475),
Item("mincemeat", -880),
Item("moresby", 756),
Item("mycenae", 183),
Item("plugging", -266),
Item("smokescreen", 423),
Item("speakeasy", -745),
Item("vein", 813)
)
private def zeroSum(i: Int, w: Int): Unit = {
if (count < LIMIT) {
if (i != 0 && w == 0) {
for (j <- 0 until i) print(f"${items(indices(j))}%s ")
println
count += 1
} else
for (j <- (if (i != 0) indices(i - 1) + 1 else 0) until n) {
indices(i) = j
zeroSum(i + 1, w + items(j).weight)
}
}
}
// Not optimized
private case class Item(word: String, weight: Int) {
override def toString: String = f"($word%s, $weight%d)"
}
println(f"The weights of the following $LIMIT%d subsets add up to zero:")
zeroSum(0, 0)
}
Sidef
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
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
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 ^A/sum@lmPrnPX ~&lrmPC
#cast %zm
main = nullset weightsThe 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>
Wren
import "./fmt" for Fmt
class Item {
construct new(word, weight) {
_word = word
_weight = weight
}
word { _word }
weight { _weight }
toString {Fmt.swrite("$-11s $4d", _word, _weight) }
}
var items = [
Item.new("alliance", -624),
Item.new("archbishop", -915),
Item.new("balm", 397),
Item.new("bonnet", 452),
Item.new("brute", 870),
Item.new("centipede", -658),
Item.new("cobol", 362),
Item.new("covariate", 590),
Item.new("departure", 952),
Item.new("deploy", 44),
Item.new("diophantine", 645),
Item.new("efferent", 54),
Item.new("elysee", -326),
Item.new("eradicate", 376),
Item.new("escritoire", 856),
Item.new("exorcism", -983),
Item.new("fiat", 170),
Item.new("filmy", -874),
Item.new("flatworm", 503),
Item.new("gestapo", 915),
Item.new("infra", -847),
Item.new("isis", -982),
Item.new("lindholm", 999),
Item.new("markham", 475),
Item.new("mincemeat", -880),
Item.new("moresby", 756),
Item.new("mycenae", 183),
Item.new("plugging", -266),
Item.new("smokescreen", 423),
Item.new("speakeasy", -745),
Item.new("vein", 813)
]
var n = items.count
var indices = List.filled(n, 0)
var count = 0
var LIMIT = 5
var zeroSum // recursive
zeroSum = Fn.new { |i, w|
if (i != 0 && w == 0) {
for (j in 0...i) System.print("%(items[indices[j]]) ")
System.print()
if (count < LIMIT) {
count = count + 1
} else {
return
}
}
var k = (i != 0) ? indices[i-1] + 1 : 0
var j = k
while (j < n) {
indices[i] = j
zeroSum.call(i + 1, w + items[j].weight)
if (count == LIMIT) return
j = j + 1
}
}
System.print("The weights of the following %(LIMIT) subsets add up to zero:\n")
zeroSum.call(0, 0)
- Output:
The weights of the following 5 subsets add up to zero: 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 mincemeat -880 plugging -266 speakeasy -745 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 infra -847 isis -982 mincemeat -880 moresby 756 mycenae 183 smokescreen 423 speakeasy -745 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 fiat 170 infra -847 isis -982 markham 475 mincemeat -880 plugging -266 speakeasy -745 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 exorcism -983 fiat 170 infra -847 isis -982 mincemeat -880 moresby 756 plugging -266 vein 813 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 exorcism -983 fiat 170 infra -847 isis -982 smokescreen 423
XPL0
include xpllib; \for Print
def \Items\ Wd, Wt;
def N = 31;
int Set(N), Items;
proc SubSum(I, Weight);
int I, Weight, J;
[if I#0 & Weight=0 then
[for J:= 0 to I-1 do
Print("%s%s", if J then " " else "", Items(Set(J),Wd));
Print("\n");
exit;
];
for J:= if I#0 then Set(I-1)+1 else 0 to N-1 do
[Set(I):= J;
SubSum(I+1, Weight+Items(J,Wt));
];
];
[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] ];
SubSum(0, 0);
]- Output:
alliance archbishop balm bonnet brute centipede cobol covariate departure deploy diophantine efferent elysee eradicate escritoire exorcism fiat filmy flatworm mincemeat plugging speakeasy
zkl
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