Maximum Subarray Sum using Divide and Conquer algorithm

Given an array arr[], the task is to find the subarray that has the maximum sum and return its sum.

Examples:

Input: arr[] = [2, 3, -8, 7, -1, 2, 3]
Output: 11
Explanation: The subarray [7, -1, 2, 3] has the largest sum 11.

Input: arr[] = [-2, -4]
Output: –2
Explanation: The subarray [-2] has the largest sum -2.

Input: arr[] = [5, 4, 1, 7, 8]
Output: 25
Explanation: The subarray [5, 4, 1, 7, 8] has the largest sum 25.

[Naive Approach] By iterating over all subarrays - O(n^2) Time and O(1) Space

The idea is to run two nested loops to iterate over all possible subarrays and find the maximum sum. The outer loop will mark the starting point of a subarray and inner loop will mark the ending point of the subarray. Please refer to Maximum Subarray Sum for implementation.

[Better Approach] Using Divide and Conquer - O(n*logn) time and O(n) space

Divide the given array in two halves and return the maximum of following three:

1. Maximum subarray sum in left half.
2. Maximum subarray sum in right half.
3. Maximum subarray sum such that the subarray crosses the midpoint.

Maximum subarray in left and right half can be found easily by two recursive calls. To find maximum subarray sum such that the subarray crosses the midpoint, find the maximum sum starting from mid point and ending at some point on left of mid, then find the maximum sum starting from mid + 1 and ending with some point on right of mid + 1. Finally, combine the two and return the maximum among left, right and combination of both.

Below is the implementation of the above approach: 

// C++ program for maximum subarray sum
// using Divide and Conquer
#include <bits/stdc++.h>
using namespace std;

// Find the maximum possible sum in arr[] such that arr[m]
// is part of it
int maxCrossingSum(vector<int> &arr, int l, int m, int h) {
  
    // Include elements on left of mid.
    int sum = 0;
    int leftSum = INT_MIN;
    for (int i = m; i >= l; i--) {
        sum = sum + arr[i];
        if (sum > leftSum)
            leftSum = sum;
    }

    // Include elements on right of mid
    sum = 0;
    int rightSum = INT_MIN;
    for (int i = m + 1; i <= h; i++) {
        sum = sum + arr[i];
        if (sum > rightSum)
            rightSum = sum;
    }

    // Return the sum of maximum left, right, and 
  	// cross subarray
    return (leftSum + rightSum);
}

// Returns sum of maximum sum subarray in arr[l..h]
int MaxSum(vector<int> &arr, int l, int h) {
  
    // Invalid Range: low is greater than high
    if (l > h)
        return INT_MIN;

    // Base Case: Only one element
    if (l == h)
        return arr[l];

    // Find middle point
    int m = l + (h - l) / 2;

    // Compute the maximum of the three cases:
    int leftSum = MaxSum(arr, l, m);
    int rightSum = MaxSum(arr, m + 1, h);
    int crossSum = maxCrossingSum(arr, l, m, h);

    // Return the maximum of the three
    if (leftSum >= rightSum && leftSum >= crossSum)
        return leftSum;
    else if (rightSum >= leftSum && rightSum >= crossSum)
        return rightSum;
    else
        return crossSum;
}

int maxSubarraySum(vector<int> &arr) {
    int n = arr.size();
    return MaxSum(arr, 0, n - 1);
}

int main() {
    vector<int> arr = {2, 3, 4, 5, 7};
    cout << maxSubarraySum(arr);
    return 0;
}

// Java program for maximum subarray sum
// using Divide and Conquer
import java.util.*;

class GfG {
  
    static int max(int a, int b) { return (a > b) ? a : b; }
    static int max(int a, int b, int c) {
        return max(max(a, b), c);
    }

    // Find the maximum possible sum in arr[] such that
    // arr[m] is part of it
    static int maxCrossingSum(int[] arr, int l, int m,
                              int h) {

        // Include elements on left of mid
        int sum = 0;
        int leftSum = Integer.MIN_VALUE;
        for (int i = m; i >= l; i--) {
            sum += arr[i];
            if (sum > leftSum)
                leftSum = sum;
        }

        // Include elements on right of mid
        sum = 0;
        int rightSum = Integer.MIN_VALUE;
        for (int i = m; i <= h; i++) {
            sum += arr[i];
            if (sum > rightSum)
                rightSum = sum;
        }

        // Return maximum of left sum, right sum, and their
        // combination
        return max(leftSum + rightSum - arr[m], leftSum,
                   rightSum);
    }

    // Recursive function to calculate maximum subarray sum
    static int MaxSum(int[] arr, int l,
                                       int h) {

        // Invalid range
        if (l > h)
            return Integer.MIN_VALUE;

        // Base case: one element
        if (l == h)
            return arr[l];

        // Find middle point
        int m = l + (h - l) / 2;

        // Return the maximum of three cases:
        // 1. Maximum subarray sum in left half
        // 2. Maximum subarray sum in right half
        // 3. Maximum subarray sum crossing the midpoint
        return max(MaxSum(arr, l, m),
                   MaxSum(arr, m + 1, h),
                   maxCrossingSum(arr, l, m, h));
    }

    static int maxSubarraySum(int[] arr) {
        return MaxSum(arr, 0, arr.length - 1);
    }

    public static void main(String[] args) {
        int[] arr = { 2, 3, 4, 5, 7 };
        System.out.println(maxSubarraySum(arr));
    }
}

# Python program for maximum subarray sum
# using Divide and Conquer

def max(a, b, c=None):
    if c is None:
        return a if a > b else b
    return max(max(a, b), c)


def maxCrossingSum(arr, l, m, h):

    # Include elements on the left of mid
    leftSum = float('-inf')
    sum = 0
    for i in range(m, l - 1, -1):
        sum += arr[i]
        leftSum = max(leftSum, sum)

    # Include elements on the right of mid
    rightSum = float('-inf')
    sum = 0
    for i in range(m, h + 1):
        sum += arr[i]
        rightSum = max(rightSum, sum)

    # Return the maximum of left sum, right sum, and their combination
    return max(leftSum + rightSum - arr[m], leftSum, rightSum)


def MaxSum(arr, l, h):
    if l > h:
        return float('-inf')
    if l == h:

        # Base case: one element
        return arr[l]

    # Find the middle point
    m = l + (h - l) // 2

    # Return the maximum of three cases
    return max(
        MaxSum(arr, l, m),
        MaxSum(arr, m + 1, h),
        maxCrossingSum(arr, l, m, h),
    )


def maxSubarraySum(arr):
    return MaxSum(arr, 0, len(arr) - 1)

arr = [2, 3, 4, 5, 7]
print(maxSubarraySum(arr))

// C# program for maximum subarray sum
// using Divide and Conquer
using System;

class GfG {

    static int max(int a, int b) { return (a > b) ? a : b; }
    static int max(int a, int b, int c) {
        return max(max(a, b), c);
    }

    // Function to calculate the maximum crossing subarray
    // sum
    static int maxCrossingSum(int[] arr, int l, int m,
                              int h) {
        int sum = 0;

        // Calculate maximum sum on the left side of the
        // midpoint
        int leftSum = int.MinValue;
        for (int i = m; i >= l; i--) {
            sum += arr[i];
            leftSum = Math.Max(leftSum, sum);
        }

        // Calculate maximum sum on the right side of the
        // midpoint
        sum = 0;
        int rightSum = int.MinValue;
        for (int i = m; i <= h; i++) {
            sum += arr[i];
            rightSum = Math.Max(rightSum, sum);
        }

        // Return the maximum combined sum, or either left
        // or right sum
        return max(leftSum + rightSum - arr[m], leftSum,
                   rightSum);
    }

    static int MaxSum(int[] arr, int l,
                                       int h) {

        // Base case: invalid range
        if (l > h)
            return int.MinValue;

        // Base case: single element
        if (l == h)
            return arr[l];

        int m = l + (h - l) / 2;

        return max(MaxSum(arr, l, m),
                   MaxSum(arr, m + 1, h),
                   maxCrossingSum(arr, l, m, h));
    }

    static int maxSubarraySum(int[] arr) {
        return MaxSum(arr, 0, arr.Length - 1);
    }

    static void Main(string[] args) {
        int[] arr = { 2, 3, 4, 5, 7 };
        Console.WriteLine(maxSubarraySum(arr));
    }
}

// JavaScript program for maximum subarray sum
// using Divide and Conquer

function max(a, b, c = null) {
    if (c === null)
        return a > b ? a : b;
    return Math.max(a, Math.max(b, c));
}

function maxCrossingSum(arr, l, m, h) {
    let leftSum = -Infinity, sum = 0;

    // Calculate maximum sum on the left side of the
    // midpoint
    for (let i = m; i >= l; i--) {
        sum += arr[i];
        leftSum = Math.max(leftSum, sum);
    }

    let rightSum = -Infinity;
    sum = 0;

    // Calculate maximum sum on the right side of the
    // midpoint
    for (let i = m; i <= h; i++) {
        sum += arr[i];
        rightSum = Math.max(rightSum, sum);
    }

    // Return the maximum combined sum, or either left or
    // right sum
    return max(leftSum + rightSum - arr[m], leftSum,
               rightSum);
}

function MaxSum(arr, l, h) {

    // Base case: invalid range
    if (l > h)
        return -Infinity;

    // Base case: single element
    if (l === h)
        return arr[l];

    const m = l + Math.floor((h - l) / 2);

    return max(MaxSum(arr, l, m),
               MaxSum(arr, m + 1, h),
               maxCrossingSum(arr, l, m, h));
}

function maxSubarraySum(arr) {
    return MaxSum(arr, 0, arr.length - 1);
}

//driver code
const arr = [ 2, 3, 4, 5, 7 ];
console.log(maxSubarraySum(arr));

21

Note: The above recurrence is similar to Merge Sort and can be solved either using Recurrence Tree method or Master method.

[Expected Approach] Using Kadane’s Algorithm - O(n) Time and O(1) Space

The Kadane's Algorithm for this problem takes O(n) time. Therefore the Kadane's algorithm is better than the Divide and Conquer approach, but this problem can be considered as a good example to show power of Divide and Conquer. The above simple approach where we divide the array in two halves, reduces the time complexity from O(n^2) to O(nLogn). Please refre to kadane's algorithm implementation Maximum Subarray Sum.

kartik

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