Sum of maximum of all subarrays | Divide and Conquer

Given an array arr[] of length N, the task is to find the sum of the maximum elements of every possible sub-array of the array.

Examples:

Input : arr[] = {1, 3, 1, 7}
Output : 42
Max of all sub-arrays:
{1} - 1
{1, 3} - 3
{1, 3, 1} - 3
{1, 3, 1, 7} - 7
{3} - 3
{3, 1} - 3
{3, 1, 7} - 7 
{1} - 1
{1, 7} - 7
{7} - 7
1 + 3 + 3 + 7 + 3 + 3 + 7 + 1 + 7 + 7 = 42

Input : arr[] = {1, 1, 1, 1, 1}
Output : 15


We have already discussed an O(N) approach using stack for this problem in this article.

Approach :
In this article, we will learn how to solve this problem using divide and conquer.
Let’s assume that element at ith index is largest of all. For any sub-array that contains index ‘i’, the element at ‘i’ will always be maximum in the sub-array.
If element at ith index is largest, we can safely say, that element ith index will be largest in (i+1)*(N-i) subarrays. So, its total contribution will be arr[i]*(i+1)*(N-i). Now, we will divide the array in two parts, (0, i-1) and (i+1, N-1) and apply the same algorithms to both of them separately.

So our general recurrence relation will be:

maxSumSubarray(arr, l, r) = arr[i]*(r-i+1)*(i-l+1) 
                            + maxSumSubarray(arr, l, i-1)
                            + maxSumSubarray(arr, i+1, r)
where i is index of maximum element in range [l, r].

Now, we need a way to efficiently answer rangeMax() queries. Segment tree will be an efficient way to answer this query. We will need to answer this query at most N times. Thus, the time complexity of our divide and conquer algorithm will O(Nlog(N)).
If we have to answer the problem “Sum of minimum of all subarrays” then we will use the segment tree to answer rangeMin() queries. For this, you can go through the article segment tree range minimim.

Below is the implementation code:

filter_none

edit
close

play_arrow

link
brightness_4
code

// C++ implementation of the above approach
  
#include <bits/stdc++.h>
#define seg_max 51
using namespace std;
  
// Array to store segment tree.
// In first we will store the maximum
// of a range
// In second, we will store index of
// that range
pair<int, int> seg_tree[seg_max];
  
// Size of array declared global
// to maintain simplicity in code
int n;
  
// Function to build segment tree
pair<int, int> buildMaxTree(int l, int r, int i, int arr[])
{
    // Base case
    if (l == r) {
        seg_tree[i] = { arr[l], l };
        return seg_tree[i];
    }
  
    // Finding the maximum among left and right child
    seg_tree[i] = max(buildMaxTree(l, (l + r) / 2, 2 * i + 1, arr),
                      buildMaxTree((l + r) / 2 + 1, r, 2 * i + 2, arr));
  
    // Returning the maximum to parent
    return seg_tree[i];
}
  
// Function to perform range-max query in segment tree
pair<int, int> rangeMax(int l, int r, int arr[],
                        int i = 0, int sl = 0, int sr = n - 1)
{
    // Base cases
    if (sr < l || sl > r)
        return { INT_MIN, -1 };
    if (sl >= l and sr <= r)
        return seg_tree[i];
  
    // Finding the maximum among left and right child
    return max(rangeMax(l, r, arr, 2 * i + 1, sl, (sl + sr) / 2),
               rangeMax(l, r, arr, 2 * i + 2, (sl + sr) / 2 + 1, sr));
}
  
// Function to find maximum sum subarray
int maxSumSubarray(int arr[], int l = 0, int r = n - 1)
{
    // base case
    if (l > r)
        return 0;
  
    // range-max query to determine
    // largest in the range.
    pair<int, int> a = rangeMax(l, r, arr);
  
    // divide the array in two parts
    return a.first * (r - a.second + 1) * (a.second - l + 1)
           + maxSumSubarray(arr, l, a.second - 1)
           + maxSumSubarray(arr, a.second + 1, r);
}
  
// Driver Code
int main()
{
    // Input array
    int arr[] = { 1, 3, 1, 7 };
  
    // Size of array
    n = sizeof(arr) / sizeof(int);
  
    // Builind the segment-tree
    buildMaxTree(0, n - 1, 0, arr);
  
    cout << maxSumSubarray(arr);
  
    return 0;
}

chevron_right


Output:

42

Time complexity : O(Nlog(N))



My Personal Notes arrow_drop_up

Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.