minimum difference between sum of two subsets

Question

Folks,

came across a problem... found this intersting... am modifying it a little bit just tu pep it up.

Given a set of integers (range 0-500), find the minimum difference between the sum of two subsets that can be formed by splitting them almost equally. (say count of integers is n, if n is even, each set must have n/2 elements and if n is odd, one set has (n-1)/2 elements and other has (n+1)/2 elements)

sample imput : 1 2 3 4 5 6

minimal difference = 1 (subsets being 1 4 6 and 2 3 5 )

sample input 2 : [ 1 1 1 1 2 2 2 2 ]

minimal difference = 0 (subsets being 1 1 2 2 and 1 1 2 2 )

is there DP approach for this problem.

Thanks guys...

raj...

Answer 1

This problem looks almost like the "balanced partition".

You can use a DP approach to build a pseudo-polynomial time algorithm that solves the balanced partition. See problem 7 at http://people.csail.mit.edu/bdean/6.046/dp/

It sounds like you could have a similar approach.

Answer 2

I've solved this problem recently using Dynamic Programming in c++. I have not modified the code to answer your question. But changing some constants and little code should do.

The code below reads and solves N problems.Each problem has some people (in your case number of integers) and their weights (integer values). This code tries to split the set into 2 groups with difference being minimum.

#include <iostream>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#define MAX_PEOPLE 100
#define MAX_WEIGHT 450
#define MAX_WEIGHT_SUM MAX_PEOPLE*MAX_WEIGHT
using namespace std;

int weights[MAX_PEOPLE];
//bool table[MAX_PEOPLE + 1][MAX_WEIGHT_SUM + 1]; 

bool** create2D(int x, int y) {
    bool **array = new bool*[x];
    for (int i = 0; i < x; ++i) {
        array[i] = new bool[y];
        memset(array[i], 0, sizeof(bool)*y);
    }
    return array;
}

void delete2D(int x, int y, bool **array) {
    for (int i = 0; i < x; ++i) {
        delete[] array[i];
    }
    delete[] array;
}

void memset2D(int x, int y, bool **array) {
    for(int i = 0; i < x; ++i)
        memset(array[i], 0, sizeof(bool)*y);
}

int main(void) {
    int n, N, W, maxDiff, teamWeight, temp;
    int minWeight = MAX_WEIGHT, maxWeight = -1;
    cin >> N;
    while(N--) {
        cin >> n;
        W = 0;
        for(int i = 0; i < n; ++i) {
            cin >> weights[i];
            if(weights[i] < minWeight)
                minWeight = weights[i];
            if(weights[i] > maxWeight)
                maxWeight = weights[i];

            W += weights[i];
        }
        int maxW = maxWeight + (W>>1);
        int maxn = n>>1;
        int index = 0;
    /* 
       table[j][i] = 1 if a team of j people can form i weight 
                        from K people, where k is implicit in loop
       table[j][i] = table[j-1][i-weight[j]] if i-weight[j] >=0
     */
        bool **table = create2D(maxn+1, maxW+1);
        //memset2D(maxn+1, maxW+1, table);
        //memset(table, 0, sizeof(table));
        table[0][0] = true;
        /* for k people what can be formed?*/
        for(int k = 0; k < n; ++k) {
            /* forming team of size j out of k people*/
            for(int j = min(k, maxn) ; j >= 1; --j) { 
                /* using j people out of k, can I make weight i?*/
                for(int i = maxW; i >=minWeight ; --i) {
                    if (table[j][i] == false) {
                        /*do not consider k if more than allowable*/
                        index = i - weights[k];
                        if (index < 0) break;
                        /*if without adding k, we can make the weight
                          limit with less than one person then one can
                          also make weight limit by adding k.*/
                        table[j][i] = table[j-1][index];
                    } /*outer if ends here*/
                } /* ith loop */
            } /* jth loop */
        } /* kth loop */

        maxDiff = MAX_WEIGHT_SUM ;
        teamWeight = 0;
        for(int i = 0; i <= maxW; ++i) {
            if (table[n/2][i]) {
                temp = abs(abs(W - i) - i);
                if (temp < maxDiff) {
                    maxDiff = temp;
                    teamWeight = i;
                }
            }
        }
        delete2D(n+1, maxW+1, table);
        teamWeight = min(teamWeight, W-teamWeight);
            cout << teamWeight << " " << W - teamWeight << endl;
        if(N)
            cout << endl;
    }
        return 0;
}