Split the Array in two parts such that maximum product is minimized

Given an array of integers, arr[] of size N (<=16), the task is to partition the array into 2 parts such that the maximum product of the 2 halves is minimized. Find the minimized maximum product of the half. If the array is empty, print -1.

Examples:

Input: arr[] = {3, 5, 7}
Output: 15
Explanation: The possible partitions are – 
-> {5, 7} , {3}  – here the products are 35 and 3 and maximum product is 35
-> {3, 7} , {5}  – here the products are 21 and 5 and maximum product is 21
-> {5, 3} , {7}  – here the products are 15 and 7 and maximum product is 15
-> {5, 7, 3} , {}  – here the products are 105 and 0 and maximum product is 105

Out of the maximum product obtained i.e. from 105, 35, 21 and 15, the minimum value is 15 and therefore our required answer is 15. 

Input: arr[] = { 10 }
Output: 10
Explanation: Since the array contains single element, the array cannot be further divided. Hence the first half contains 10 and the other half contains no element. Therefore, the answer is 10.

Approach: Since the value of N is less than 16 the problem can be solved using bit masking as multiply all the numbers which are at set bits position and put it into one side similarly multiply all the unset bits position and store it in another half find the maximum of those and store it in a set and at last return first element of the set. 

Follow the steps below to solve the problem:

• Initialize a set st[] to store the possible answers in order.
• Iterate over the range [0, 2^N) using the variable i and perform the following tasks:
  • Initialize the variables product1 and product2 as 1.
  • Iterate over the range [0, N) using the variable j and perform the following tasks:
    • If i & (1 << j) is true, then multiply the value arr[j] to the variable product1 else to the variable product2.
  • Insert the maximum of product1 or product2 in the set st[].
• After performing the above steps, print the first value of the set as the answer.

Below is the implementation of the above approach.

C++

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to find the minimum of the
// maximum product of the 2 halves
void findMinimum(vector<int>& arr, int N)
{

    // Set to store the possible answers
    // in ascending order.
    set<int> st;

    // Traverse over the all possible
    for (int i = 0; i < (1 << N); i++) {

        // Variables to find the product
        // of set bits at set positions
        // and unset bits at unset positions
        int product1 = 1, product2 = 1;

        // Traverse over the array
        for (int j = 0; j < N; j++) {

            // Check the condition
            if (i & (1 << j)) {
                product1 = product1 * arr[j];
            }
            else {
                product2 = product2 * arr[j];
            }
        }

        // Insert the maximum one
        st.insert(max(product1, product2));
    }

    cout << *st.begin() << "\n";
}

// Driver Code
int main()
{
    vector<int> arr = { 3, 5, 7 };
    int N = arr.size();

    findMinimum(arr, N);

    return 0;
}

Java

// Java program for the above approach
import java.util.*;

public class GFG
{
// Function to find the minimum of the
// maximum product of the 2 halves
static void findMinimum(int []arr, int N)
{

    // Set to store the possible answers
    // in ascending order.
    HashSet<Integer> st = new HashSet<Integer>();

    // Traverse over the all possible
    for (int i = 0; i < (1 << N); i++) {

        // Variables to find the product
        // of set bits at set positions
        // and unset bits at unset positions
        int product1 = 1, product2 = 1;

        // Traverse over the array
        for (int j = 0; j < N; j++) {

            // Check the condition
            if ((i & (1 << j)) == 0) {
                product1 = product1 * arr[j];
            }
            else {
                product2 = product2 * arr[j];
            }
        }

        // Insert the maximum one
        st.add(Math.max(product1, product2));
    }
    int ans = 0;
    for (int x : st) {
        ans = x;
    }
    System.out.print(ans);
}

// Driver Code
public static void main(String args[])
{
    int []arr = { 3, 5, 7 };
    int N = arr.length;

    findMinimum(arr, N);

}
}
// This code is contributed by Samim Hossain Mondal.

Python3

# Python3 program for the above approach

# Function to find the minimum of the
# maximum product of the 2 halves
def findMinimum(arr, N):

    # Set to store the possible answers
    # in ascending order.
    st = set([])

    # Traverse over the all possible
    for i in range((1 << N)):

        # Variables to find the product
        # of set bits at set positions
        # and unset bits at unset positions
        product1 = 1
        product2 = 1

        # Traverse over the array
        for j in range(N):

            # Check the condition
            if (i & (1 << j)):
                product1 = product1 * arr[j]
            else:
                product2 = product2 * arr[j]

        # Insert the maximum one
        st.add(max(product1, product2))

    print(list(st)[-1])

# Driver Code
if __name__ == "__main__":

    arr = [ 3, 5, 7 ]
    N = len(arr)

    findMinimum(arr, N)

# This code is contributed by ukasp

C#

// C# program for the above approach
using System;
using System.Collections;
using System.Collections.Generic;

class GFG
{

// Function to find the minimum of the
// maximum product of the 2 halves
static void findMinimum(int []arr, int N)
{

    // Set to store the possible answers
    // in ascending order.
    HashSet<int> st = new HashSet<int>();

    // Traverse over the all possible
    for (int i = 0; i < (1 << N); i++) {

        // Variables to find the product
        // of set bits at set positions
        // and unset bits at unset positions
        int product1 = 1, product2 = 1;

        // Traverse over the array
        for (int j = 0; j < N; j++) {

            // Check the condition
            if ((i & (1 << j)) == 0) {
                product1 = product1 * arr[j];
            }
            else {
                product2 = product2 * arr[j];
            }
        }

        // Insert the maximum one
        st.Add(Math.Max(product1, product2));
    }
    int ans = 0;
    foreach (int x in st) {
        ans = x;
    }
    Console.Write(ans);
}

// Driver Code
public static void Main()
{
    int []arr = { 3, 5, 7 };
    int N = arr.Length;

    findMinimum(arr, N);

}
}
// This code is contributed by Samim Hossain Mondal.

Javascript

<script>

        // JavaScript Program to implement
        // the above approach

        // Function to find the minimum of the
        // maximum product of the 2 halves
        function findMinimum(arr, N) {

            // Set to store the possible answers
            // in ascending order.
            let st = new Set();

            // Traverse over the all possible
            for (let i = 0; i < (1 << N); i++) {

                // Variables to find the product
                // of set bits at set positions
                // and unset bits at unset positions
                let product1 = 1;
                let product2 = 1;

                // Traverse over the array
                for (let j = 0; j < N; j++) {

                    // Check the condition
                    if (i & (1 << j)) {
                        product1 = product1 * arr[j];
                    }
                    else {
                        product2 = product2 * arr[j];
                    }
                }

                // Insert the maximum one
                st.add(Math.max(product1, product2));
            }
            const last = [...st][st.size - 1];
            document.write(last);
        }

        // Driver Code
        let arr = [3, 5, 7];
        let N = arr.length;

        findMinimum(arr, N);

    // This code is contributed by Potta Lokesh
    </script>

Output

15

Time complexity: O((2^N)*(N*log(N)))
Auxiliary Space: O(N)