Maximum subarray product modulo M

Given an array, arr[] of size N and a positive integer M, the task is to find the maximum subarray product modulo M and the minimum length of the maximum product subarray.

Examples:

Input: arr[] = {2, 3, 4, 2}, N = 4, M = 5
Output:
Maximum subarray product is 4 
Minimum length of the maximum product subarray is 1 
Explanation:
Subarrays of length 1 are {{2}, {3}, {4}, {2}} and their product modulo M(= 5) are {2, 3, 4, 2} respectively. 
Subarrays of length 2 are {{2, 3}, {3, 4}, {4, 2}} and the product modulo M(= 5) are {1, 2, 3} respectively. 
Subarrays of length 3 are {{2, 3, 4}, {3, 4, 2}} and the product modulo M(= 5) are {4, 4, } respectively. 
Subarrays of length 4 is {2, 3, 4, 2} and the product modulo M(= 5) is 3. 
Therefore, the maximum subarray product mod M(= 5) is 4 and smallest possible length is 1.

Input: arr[] = {5, 5, 5}, N = 3, M = 7
Output: 
Maximum subarray product is 6 
Minimum length of the maximum product subarray is 3

Naive Approach: The simplest approach is to generate all possible subarrays and for each subarray, calculate its product modulo M and print the maximum subarray product and the minimum length of such subarray. 

Time Complexity: O(N³)
Auxiliary Space: O(1)

Efficient Approach: The above approach can be optimized by calculating the product of subarray in the range [i, j] by multiplying arr[j] with the precalculated product of subarray in the range [i, j – 1]. Follow the steps below to solve the problem:

• Initialize two variables, say ans and length, to store the maximum subarray product and the minimum length of maximum product subarray.
• Iterate over the range [0, N – 1] and perform the following steps:
  • Initialize a variable, say product, to store the product of subarray {arr[i], …, arr[j]}.
  • Iterate over the range [i, N-1] and update the product by multiplying it by arr[j], i.e. (product * arr[j]) % M.
  • In every iteration, update ans if ans < product and then update length, if length > (j – i + 1).
• Finally, print the maximum subarray product obtained in ans and minimum length of subarray having the maximum product, length.

Below is the implementation of the above approach:

C++

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

// Function to find maximum subarray product
// modulo M and minimum length of the subarray
void maxModProdSubarr(int arr[], int n, int M)
{

    // Stores maximum subarray product modulo
    // M and minimum length of the subarray
    int ans = 0;

    // Stores the minimum length of
    // subarray having maximum product
    int length = n;

    // Traverse the array
    for (int i = 0; i < n; i++) {

        // Stores the product of a subarray
        int product = 1;

        // Calculate Subarray whose start
        // index is i
        for (int j = i; j < n; j++) {

            // Multiply product by arr[i]
            product = (product * arr[i]) % M;

            // If product greater than ans
            if (product > ans) {

                // Update ans
                ans = product;
                if (length > j - i + 1) {

                    // Update length
                    length = j - i + 1;
                }
            }
        }
    }

    // Print maximum subarray product mod M
    cout << "Maximum subarray product is "
         << ans << endl;

    // Print minimum length of subarray
    // having maximum product
    cout << "Minimum length of the maximum product "
         << "subarray is " << length << endl;
}

// Drivers Code
int main()
{
    int arr[] = { 2, 3, 4, 2 };
    int N = sizeof(arr) / sizeof(arr[0]);
    int M = 5;

    maxModProdSubarr(arr, N, M);
    return 0;
}

Java

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

class GFG{

// Function to find maximum subarray product
// modulo M and minimum length of the subarray
static void maxModProdSubarr(int arr[], int n, int M)
{

    // Stores maximum subarray product modulo
    // M and minimum length of the subarray
    int ans = 0;

    // Stores the minimum length of
    // subarray having maximum product
    int length = n;

    // Traverse the array
    for(int i = 0; i < n; i++)
    {

        // Stores the product of a subarray
        int product = 1;

        // Calculate Subarray whose start
        // index is i
        for(int j = i; j < n; j++)
        {

            // Multiply product by arr[i]
            product = (product * arr[i]) % M;

            // If product greater than ans
            if (product > ans)
            {

                // Update ans
                ans = product;

                if (length > j - i + 1)
                {

                    // Update length
                    length = j - i + 1;
                }
            }
        }
    }

    // Print maximum subarray product mod M
    System.out.println(
        "Maximum subarray product is " + ans);

    // Print minimum length of subarray
    // having maximum product
    System.out.println(
        "Minimum length of the maximum " +
        "product subarray is " + length);
}

// Driver Code
public static void main(String[] args)
{
    int arr[] = { 2, 3, 4, 2 };
    int N = arr.length;
    int M = 5;

    maxModProdSubarr(arr, N, M);
}
}

// This code is contributed by Kingash

Python3

# Python3 program for above approach

# Function to find maximum subarray product
# modulo M and minimum length of the subarray
def maxModProdSubarr(arr, n, M):

    # Stores maximum subarray product modulo
    # M and minimum length of the subarray
    ans = 0

    # Stores the minimum length of
    # subarray having maximum product
    length = n

    # Traverse the array
    for i in range(n):

        # Stores the product of a subarray
        product = 1

        # Calculate Subarray whose start
        # index is i
        for j in range(i, n, 1):

            # Multiply product by arr[i]
            product = (product * arr[i]) % M

            # If product greater than ans
            if (product > ans):

                # Update ans
                ans = product
                if (length > j - i + 1):

                    # Update length
                    length = j - i + 1

    # Print maximum subarray product mod M
    print("Maximum subarray product is", ans)

    # Print minimum length of subarray
    # having maximum product
    print("Minimum length of the maximum product subarray is",length)

# Drivers Code
if __name__ == '__main__':
    arr =  [2, 3, 4, 2]
    N = len(arr)
    M = 5
    maxModProdSubarr(arr, N, M)

    # This code is contributed by ipg2016107.

C#

// C# program for above approach
using System;

class GFG{

// Function to find maximum subarray product
// modulo M and minimum length of the subarray
static void maxModProdSubarr(int[] arr, int n,
                             int M)
{

    // Stores maximum subarray product modulo
    // M and minimum length of the subarray
    int ans = 0;

    // Stores the minimum length of
    // subarray having maximum product
    int length = n;

    // Traverse the array
    for(int i = 0; i < n; i++)
    {

        // Stores the product of a subarray
        int product = 1;

        // Calculate Subarray whose start
        // index is i
        for(int j = i; j < n; j++)
        {

            // Multiply product by arr[i]
            product = (product * arr[i]) % M;

            // If product greater than ans
            if (product > ans)
            {

                // Update ans
                ans = product;

                if (length > j - i + 1)
                {

                    // Update length
                    length = j - i + 1;
                }
            }
        }
    }

    // Print maximum subarray product mod M
    Console.WriteLine(
        "Maximum subarray product is " + ans);

    // Print minimum length of subarray
    // having maximum product
    Console.WriteLine(
        "Minimum length of the maximum " +
        "product subarray is " + length);
}

// Driver code
static void Main()
{
    int[] arr = { 2, 3, 4, 2 };
    int N = arr.Length;
    int M = 5;

    maxModProdSubarr(arr, N, M);
}
}

// This code is contributed by code_hunt

Javascript

<script>
// javascript program for the above approach
// Function to find maximum subarray product
    // modulo M and minimum length of the subarray
    function maxModProdSubarr(arr , n , M)
    {

        // Stores maximum subarray product modulo
        // M and minimum length of the subarray
        var ans = 0;

        // Stores the minimum length of
        // subarray having maximum product
        var length = n;

        // Traverse the array
        for (i = 0; i < n; i++) {

            // Stores the product of a subarray
            var product = 1;

            // Calculate Subarray whose start
            // index is i
            for (j = i; j < n; j++) {

                // Multiply product by arr[i]
                product = (product * arr[i]) % M;

                // If product greater than ans
                if (product > ans) {

                    // Update ans
                    ans = product;

                    if (length > j - i + 1) {

                        // Update length
                        length = j - i + 1;
                    }
                }
            }
        }

        // Print maximum subarray product mod M
        document.write("Maximum subarray product is " + ans+"<br/>");

        // Print minimum length of subarray
        // having maximum product
        document.write("Minimum length of the maximum " + "product subarray is " + length);
    }

    // Driver Code
        var arr = [ 2, 3, 4, 2 ];
        var N = arr.length;
        var M = 5;

        maxModProdSubarr(arr, N, M);

// This code is contributed by umadevi9616.
</script>

Output: 

Maximum subarray product is 4
Minimum length of the maximum product subarray is 1

Time Complexity: O(N²)
Auxiliary Space: O(1)