# Minimum product subset of an array

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

Examples:

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

Input : a[] = { -1, 0 }
Output : -1
Explanation : -1(single element) is minimum 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, the result is the product of all except the largest valued negative number.
2. If there are an odd number of negative numbers and no zeros, the result is simply the product of all.
• If there are zeros and positive, no negative, the result is 0. The exceptional case is when there is no negative number and all other elements positive then our result should be the first minimum positive number.                       Below is the implementation of above approach:
#include <bits/stdc++.h>
using namespace std;

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

// Find count of negative numbers, count
// of zeros, maximum valued negative number,
// minimum valued positive number and product
// of non-zero numbers
int max_neg = INT_MIN;
int min_pos = INT_MAX;
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]);
}

// Track minimum positive
// number of array
if (a[i] > 0)
min_pos = min(min_pos, a[i]);

prod = prod * a[i];
}

// If there are all zeros
// or no negative number present
if (count_zero == n ||
(count_neg == 0 && count_zero > 0))
return 0;

// If there are all positive
if (count_neg == 0)
return min_pos;

// If there are even number of
// negative numbers and count_neg not 0
if (!(count_neg & 1) && count_neg != 0) {

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

return prod;
}

int main()
{
int a[] = { -2, -3 ,-1, 2, 0, 4, 3 };
int n = sizeof(a) / sizeof(a[0]);
cout <<"Minimum product of subset="<< minProductSubset(a, n);
return 0;
}

------------------------------------------------------------------
Output:
Minimum product of subset=-144

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