Maximum product subset of an array

Given an array a, we have to find maximum product possible with the subset of elements present in the array. The maximum product can be single element also.

Examples:

Input : a[] = { -1, -1, -2, 4, 3 }
Output : 24
Explanation : Maximum product will be ( -2 * -1 * 4 * 3 ) = 24

Input : a[] = { -1, 0 }
Output : 0
Explanation : 0(single element) is maximum product possible
 
Input : a[] = { 0, 0, 0 }
Output : 0


A simple solution is to generate all subsets, find product of every subset and return maximum product.

A better solution is to use the below facts.

  1. If there are even number of negative numbers and no zeros, result is simply product of all
  2. If there are odd number of negative numbers and no zeros, result is product of all except the largest valued negative number.
  3. If there are zeros, result is product of all except these zeros with one exceptional case. The exceptional case is when there is one negative number and all other elements are 0. In this case, result is 0.
// CPP program to find maximum product of
// a subset.
#include <bits/stdc++.h>
using namespace std;

int maxProductSubset(int a[], int n)
{
    if (n == 1)
        return a[0];

    // Find count of negative numbers, count 
    // of zeros, maximum valued negative number
    // and product of non-zero numbers
    int max_neg = INT_MIN;
    int count_neg = 0, count_zero = 0;
    int prod = 1;
    for (int i = 0; i < n; i++) {

        // If number is 0, we don't
        // multiply it with product.
        if (a[i] == 0) {
            count_zero++;
            continue;
        }

        // Count negatives and keep
        // track of maximum valued negative.
        if (a[i] < 0) {
            count_neg++;
            max_neg = max(max_neg, a[i]);
        }

        prod = prod * a[i];
    }

    // If there are all zeros
    if (count_zero == n)
        return 0;

    // If there are odd number of
    // negative numbers
    if (count_neg & 1) {

        // Exceptional case: There is only
        // negative and all other are zeros
        if (count_neg == 1 && 
            count_zero > 0 && 
            count_zero + count_neg == n)
            return 0;

        // Otherwise result is product of 
        // all non-zeros divided by maximum
        // valued negative.
        prod = prod / max_neg;
    }

    return prod;
}

int main()
{
    int a[] = {  -1, -1, -2, 4, 3  };
    int n = sizeof(a) / sizeof(a[0]);
    cout << maxProductSubset(a, n);
    return 0;
}
Output:

24

Time Complexity : O(n)
Auxiliary Space : O(1)



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