Maximum Product Subarray

Given an array that contains both positive and negative integers, find the product of the maximum product subarray. Expected Time complexity is O(n) and only O(1) extra space can be used.

Examples:

Input: arr[] = {6, -3, -10, 0, 2}
Output:   180  // The subarray is {6, -3, -10}

Input: arr[] = {-1, -3, -10, 0, 60}
Output:   60  // The subarray is {60}

Input: arr[] = {-2, -3, 0, -2, -40}
Output:   80  // The subarray is {-2, -40}

The following solution assumes that the given input array always has a positive output. The solution works for all cases mentioned above. It doesn’t work for arrays like {0, 0, -20, 0}, {0, 0, 0}.. etc. The solution can be easily modified to handle this case.
It is similar to Largest Sum Contiguous Subarray problem. The only thing to note here is, maximum product can also be obtained by minimum (negative) product ending with the previous element multiplied by this element. For example, in array {12, 2, -3, -5, -6, -2}, when we are at element -2, the maximum product is multiplication of, minimum product ending with -6 and -2.

C/C++

// C program to find Maximum Product Subarray
#include <stdio.h>

// Utility functions to get minimum of two integers
int min (int x, int y) {return x < y? x : y; }

// Utility functions to get maximum of two integers
int max (int x, int y) {return x > y? x : y; }

/* Returns the product of max product subarray. 
   Assumes that the given array always has a subarray
   with product more than 1 */
int maxSubarrayProduct(int arr[], int n)
{
    // max positive product ending at the current position
    int max_ending_here = 1;

    // min negative product ending at the current position
    int min_ending_here = 1;

    // Initialize overall max product
    int max_so_far = 1;

    /* Traverse through the array. Following values are
       maintained after the i'th iteration:
       max_ending_here is always 1 or some positive product
                       ending with arr[i]
       min_ending_here is always 1 or some negative product 
                       ending with arr[i] */
    for (int i = 0; i < n; i++)
    {
        /* If this element is positive, update max_ending_here. 
           Update min_ending_here only if min_ending_here is 
           negative */
        if (arr[i] > 0)
        {
            max_ending_here = max_ending_here*arr[i];
            min_ending_here = min (min_ending_here * arr[i], 1);
        }

        /* If this element is 0, then the maximum product 
           cannot end here, make both max_ending_here and 
           min_ending_here 0
           Assumption: Output is alway greater than or equal 
                       to 1. */
        else if (arr[i] == 0)
        {
            max_ending_here = 1;
            min_ending_here = 1;
        }

        /* If element is negative. This is tricky
           max_ending_here can either be 1 or positive. 
           min_ending_here can either be 1 or negative.
           next min_ending_here will always be prev. 
           max_ending_here * arr[i] next max_ending_here
           will be 1 if prev min_ending_here is 1, otherwise 
           next max_ending_here will be prev min_ending_here *
           arr[i] */
        else
        {
            int temp = max_ending_here;
            max_ending_here = max (min_ending_here * arr[i], 1);
            min_ending_here = temp * arr[i];
        }

        // update max_so_far, if needed
        if (max_so_far <  max_ending_here)
          max_so_far  =  max_ending_here;
    }

    return max_so_far;
}

// Driver Program to test above function
int main()
{
    int arr[] = {1, -2, -3, 0, 7, -8, -2};
    int n = sizeof(arr)/sizeof(arr[0]);
    printf("Maximum Sub array product is %d", 
            maxSubarrayProduct(arr, n));
    return 0;
}

Java

// Java program to find maximum product subarray
import java.io.*;

class ProductSubarray {

    // Utility functions to get minimum of two integers
    static int min (int x, int y) {return x < y? x : y; }

    // Utility functions to get maximum of two integers
    static int max (int x, int y) {return x > y? x : y; }

    /* Returns the product of max product subarray.
       Assumes that the given array always has a subarray
       with product more than 1 */
    static int maxSubarrayProduct(int arr[])
    {
        int n = arr.length;
        // max positive product ending at the current position
        int max_ending_here = 1;

        // min negative product ending at the current position
        int min_ending_here = 1;

        // Initialize overall max product
        int max_so_far = 1;

        /* Traverse through the array. Following
           values are maintained after the ith iteration:
           max_ending_here is always 1 or some positive product
                           ending with arr[i]
           min_ending_here is always 1 or some negative product
                           ending with arr[i] */
        for (int i = 0; i < n; i++)
        {
            /* If this element is positive, update max_ending_here.
                Update min_ending_here only if min_ending_here is
                negative */
            if (arr[i] > 0)
            {
                max_ending_here = max_ending_here*arr[i];
                min_ending_here = min (min_ending_here * arr[i], 1);
            }

            /* If this element is 0, then the maximum product cannot
               end here, make both max_ending_here and min_ending
              _here 0
               Assumption: Output is alway greater than or equal to 1. */
            else if (arr[i] == 0)
            {
                max_ending_here = 1;
                min_ending_here = 1;
            }

            /* If element is negative. This is tricky
               max_ending_here can either be 1 or positive.
               min_ending_here can either be 1 or negative.
               next min_ending_here will always be prev.
               max_ending_here * arr[i]
               next max_ending_here will be 1 if prev
               min_ending_here is 1, otherwise
               next max_ending_here will be 
                           prev min_ending_here * arr[i] */
            else
            {
                int temp = max_ending_here;
                max_ending_here = max (min_ending_here * arr[i], 1);
                min_ending_here = temp * arr[i];
            }

            // update max_so_far, if needed
            if (max_so_far <  max_ending_here)
                max_so_far  =  max_ending_here;
        }

        return max_so_far;
    }

    public static void main (String[] args) {

        int arr[] = {1, -2, -3, 0, 7, -8, -2};
        System.out.println("Maximum Sub array product is "+
                            maxSubarrayProduct(arr));
    }
}/*This code is contributed by Devesh Agrawal*/

Python

# Python program to find maximum product subarray

# Returns the product of max product subarray.
# Assumes that the given array always has a subarray
# with product more than 1
def maxsubarrayproduct(arr):

    n = len(arr)

    # max positive product ending at the current position
    max_ending_here = 1

    # min positive product ending at the current position
    min_ending_here = 1

    # Initialize maximum so far
    max_so_far = 1

    # Traverse throughout the array. Following values
    # are maintained after the ith iteration:
    # max_ending_here is always 1 or some positive product
    # ending with arr[i]
    # min_ending_here is always 1 or some negative product
    # ending with arr[i]
    for i in range(0,n):

        # If this element is positive, update max_ending_here.
        # Update min_ending_here only if min_ending_here is
        # negative
        if arr[i] > 0:
            max_ending_here = max_ending_here*arr[i]
            min_ending_here = min (min_ending_here * arr[i], 1)

        # If this element is 0, then the maximum product cannot
        # end here, make both max_ending_here and min_ending_here 0
        # Assumption: Output is alway greater than or equal to 1.
        elif arr[i] == 0:
            max_ending_here = 1
            min_ending_here = 1

        # If element is negative. This is tricky
        # max_ending_here can either be 1 or positive.
        # min_ending_here can either be 1 or negative.
        # next min_ending_here will always be prev.
        # max_ending_here * arr[i]
        # next max_ending_here will be 1 if prev
        # min_ending_here is 1, otherwise
        # next max_ending_here will be prev min_ending_here * arr[i]
        else:
            temp = max_ending_here
            max_ending_here = max (min_ending_here * arr[i], 1)
            min_ending_here = temp * arr[i]
        if (max_so_far <  max_ending_here):
            max_so_far  =  max_ending_here
    return max_so_far

# Driver function to test above function
arr = [1, -2, -3, 0, 7, -8, -2]
print "Maximum product subarray is",maxsubarrayproduct(arr)

# This code is contributed by Devesh Agrawal

C#

// C# program to find maximum product subarray
using System;

class GFG {

    // Utility functions to get minimum of two integers
    static int min (int x, int y) {return x < y? x : y; }

    // Utility functions to get maximum of two integers
    static int max (int x, int y) {return x > y? x : y; }

    /* Returns the product of max product subarray.
    Assumes that the given array always has a subarray
    with product more than 1 */
    static int maxSubarrayProduct(int []arr)
    {
        int n = arr.Length;
        // max positive product ending at the current 
        // position
        int max_ending_here = 1;

        // min negative product ending at the current
        // position
        int min_ending_here = 1;

        // Initialize overall max product
        int max_so_far = 1;

        /* Traverse through the array. Following
        values are maintained after the ith iteration:
        max_ending_here is always 1 or some positive
        product ending with arr[i] min_ending_here is
        always 1 or some negative product ending 
        with arr[i] */
        for (int i = 0; i < n; i++)
        {
            /* If this element is positive, update 
            max_ending_here. Update min_ending_here 
            only if min_ending_here is negative */
            if (arr[i] > 0)
            {
                max_ending_here = max_ending_here*arr[i];
                min_ending_here = min (min_ending_here 
                                            * arr[i], 1);
            }

            /* If this element is 0, then the maximum 
            product cannot end here, make both 
            max_ending_here and min_ending_here 0
            Assumption: Output is alway greater than or
            equal to 1. */
            else if (arr[i] == 0)
            {
                max_ending_here = 1;
                min_ending_here = 1;
            }

            /* If element is negative. This is tricky
            max_ending_here can either be 1 or positive.
            min_ending_here can either be 1 or negative.
            next min_ending_here will always be prev.
            max_ending_here * arr[i]
            next max_ending_here will be 1 if prev
            min_ending_here is 1, otherwise
            next max_ending_here will be 
            prev min_ending_here * arr[i] */
            else
            {
                int temp = max_ending_here;
                max_ending_here = max (min_ending_here
                                          * arr[i], 1);
                min_ending_here = temp * arr[i];
            }

            // update max_so_far, if needed
            if (max_so_far < max_ending_here)
                max_so_far = max_ending_here;
        }

        return max_so_far;
    }

    public static void Main () {

        int []arr = {1, -2, -3, 0, 7, -8, -2};
        
        Console.WriteLine("Maximum Sub array product is "+
                            maxSubarrayProduct(arr));
    }
}

/*This code is contributed by vt_m*/

PHP

<?php
// php program to find Maximum Product
// Subarray

// Utility functions to get minimum of
// two integers
function minn ($x, $y) 
{
    return $x < $y? $x : $y;
}

// Utility functions to get maximum of
// two integers
function maxx ($x, $y) 
{
    return $x > $y? $x : $y; 
}

/* Returns the product of max product
subarray. Assumes that the given array
always has a subarray with product
more than 1 */
function maxSubarrayProduct($arr, $n)
{
    
    // max positive product ending at 
    // the current position
    $max_ending_here = 1;

    // min negative product ending at
    // the current position
    $min_ending_here = 1;

    // Initialize overall max product
    $max_so_far = 1;

    /* Traverse through the array.
    Following values are maintained 
    after the i'th iteration: 
    max_ending_here is always 1 or
    some positive product ending with
    arr[i] min_ending_here is always
    1 or some negative product ending
    with arr[i] */
    for ($i = 0; $i < $n; $i++)
    {
        
        /* If this element is positive,
        update max_ending_here. Update
        min_ending_here only if 
        min_ending_here is negative */
        if ($arr[$i] > 0)
        {
            $max_ending_here = 
              $max_ending_here * $arr[$i];
              
            $min_ending_here = 
                min ($min_ending_here
                           * $arr[$i], 1);
        }

        /* If this element is 0, then the
        maximum product cannot end here,
        make both max_ending_here and 
        min_ending_here 0
        Assumption: Output is alway 
        greater than or equal to 1. */
        else if ($arr[$i] == 0)
        {
            $max_ending_here = 1;
            $min_ending_here = 1;
        }

        /* If element is negative. This
        is tricky max_ending_here can
        either be 1 or positive. 
        min_ending_here can either be 1 or
        negative. next min_ending_here will
        always be prev. max_ending_here * 
        arr[i] next max_ending_here will be
        1 if prev min_ending_here is 1,
        otherwise next max_ending_here will
        be prev min_ending_here * arr[i] */
        else
        {
            $temp = $max_ending_here;
            $max_ending_here =
                   max ($min_ending_here 
                           * $arr[$i], 1);
                           
            $min_ending_here =
                          $temp * $arr[$i];
        }

        // update max_so_far, if needed
        if ($max_so_far < $max_ending_here)
            $max_so_far = $max_ending_here;
    }

    return $max_so_far;
}

// Driver Program to test above function
    $arr = array(1, -2, -3, 0, 7, -8, -2);
    $n = sizeof($arr) / sizeof($arr[0]);
    echo("Maximum Sub array product is ");
    echo (maxSubarrayProduct($arr, $n));

// This code is contributed by nitin mittal 
?>

Output:

Maximum Sub array product is 112

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

This article is compiled by Dheeraj Jain and reviewed by GeeksforGeeks team. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above



My Personal Notes arrow_drop_up

Improved By : nitin mittal


 

Recommended Posts:



3.5 Average Difficulty : 3.5/5.0
Based on 243 vote(s)






User Actions