How to find all matching numbers, that sums to 'N' in a given array

Question

My goal here is to find all possible combinations that sums to a given total. For example, if the array is 2 59 3 43 5 9 8 62 10 4 and if the total is 12, then possible combinations are

2 10
3 9
8 4
5 3 4

Here is the first set of code, that I've written. Wondering the best improvements that can be done on this.

   int find_numbers_matching_sum(int *number_coll, int total)
{

    int *search_till = lower_bound(number_coll,number_coll+TOTAL_SIZE, total);
    int location = search_till - number_coll;
    if (*search_till > total && location > 0 )
    {
        --location;
    }

    while ( location >= 0 )
    {
        find_totals(number_coll,total,location);
        --location;
    }
    return 1;
}

int find_totals(int *number_coll, int total, int end_location)
{
    int left_ones = total - number_coll[end_location];
    int curloc = end_location;
    int *search_till = 0;
    int location ;
    int all_numbers[10];
    int i = 0;

    all_numbers[i] = number_coll[end_location];
    while ( left_ones && curloc >= 0 )
    {
        search_till = lower_bound(number_coll,number_coll+end_location, left_ones);
        location = search_till - number_coll;
        if (*search_till > left_ones && location > 0 )
        {
            --location;
        }
        else if ( left_ones < *search_till )
        {
            break;
        }
        curloc=location;
        left_ones = left_ones - number_coll[curloc];
        all_numbers[++i] = number_coll[curloc];
        end_location = curloc - 1;
    }

    if ( !left_ones )
    {
        while ( i>=0)
        {
            cout << all_numbers[i--] << ' ';
        }
    }
    cout << endl;
    return 1;


}

Answer 1

This is a variation of the NP-complete Knapsack Problem, the subset sum problem. The fullblown Knapsack Problem can be reduced linearly to your problem by succesively lowering N. You will not find an exact algorithm for your problem that runs faster than exponentially in N if P != NP holds.

Polynomial time approximations are known however.

Answer 2

The problem You describe is also known as the Subset Sum Problem which is NP-complete. The best You can achieve is an exponential time algorithm that tries all possible subsets of Your array/set.

Answer 3

This is Subset-sum problem (NP-Complete), and until P ?= NP, there is only exponential solution.

From xkcd

Answer 4

If the values are not big, say your sum is bounded by M, you can use dynamic programming. Suppose there are N items.

Imagine you have a matrix DP[M][N]. The cell DP[m][n] means: how many combinations of the first n elements sum exactly m?

Analyzing each item, you may or may not include it in some combination. Then you get the recurrence (taking care of the out-of-bound values)

DP[m][n] = DP[m][n-1] + DP[m - v[n]][n - 1]

the first term of rhs means you are considering all sums that do not use the nth item and the second term all sums that do use the nth item. You start with the basis DP[0][0] = 1, since the empty set is a valid combination which sums 0. The desired value is in DP[M][N].

This is pseudo-polynomial though, O(MN).

Answer 5

This is related to Partition in Number Theory, and it can be solved using dynamic programming.

Let n be the total. Let parts be the list of elements. We assume they are positive integers.

if parts == []
then f(n,parts) = [] 
else let parts = x::queue and f(n,parts) = union(L1, L2)

where:

L1 = f(n, queue)

if n-x>0
then let result = f(n-x, queue) and L2 = concatenation([x], result)
else if n-x==0, L2 = [x]
else L2 = []

This is a typical homework.

Answer 6

#include <iostream>
#include <vector>

using namespace std;

struct State {
    int v;
    const State *rest;
    void dump() const {
        if(rest) {
            cout << ' ' << v;
            rest->dump();
        } else {
            cout << endl;
        }
    }
    State() : v(0), rest(0) {}
    State(int _v, const State &_rest) : v(_v), rest(&_rest) {}
};

void ss(int *ip, int *end, int target, const State &state) {
    if(target < 0) return; // assuming we don't allow any negatives
    if(ip==end && target==0) {
        state.dump();
        return;
    }
    if(ip==end)
        return;
    { // without the first one
        ss(ip+1, end, target, state);
    }
    { // with the first one
        int first = *ip;
        ss(ip+1, end, target-first, State(first, state));
    }
}

int main() {
    int a[] = { 2,59,3,43,5,9,8,62,10,4 };
    int * start = &a[0];
    int * end = start + sizeof(a) / sizeof(a[0]);
    ss(start, end, 12, State());
}