Maximum Subarray Sum in a given Range

Given an array of n numbers, the task is to answer the following queries: 

maximumSubarraySum(start, end) : Find the maximum 
subarray sum in the range from array index 'start' 
to 'end'.

Also see : Range Query With Update Required

Examples: 

Input : arr[] = {1, 3, -4, 5, -2}
 Query 1: start = 0, end = 4
 Query 2: start = 0, end = 2
Output : 5
 4
Explanation:
For Query 1, [1, 3, -4, 5] or ( [5] ) 
represent the maximum sum sub arrays 
with sum = 5.

For Query 2, [1, 3] represents the 
maximum sum subarray in the query range
with sum = 4

Segment Trees can be used to solve this problem. Here, we need to keep information regarding various cumulative sums. At every Node we store the following: 

1. Maximum Prefix Sum, 
2. Maximum Suffix Sum, 
3. Total Sum, 
4. Maximum Subarray Sum

A classical Segment Tree with each Node storing the above information should be enough to answer each query. The only focus here is on how the left and the right Nodes of the tree are merged together. Now, we will discuss how each of the information is constructed in each of the segment tree Nodes using the information of its left and right child. 

Constructing the Maximum Prefix Sum using Left and Right child

There can be two cases for maximum prefix sum of a Node: 

• The maximum prefix sum occurs in the left child, 

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In this Case,
Maximum Prefix Sum = Maximum Prefix Sum of Left Child

• The maximum prefix sum contains every array element of the left child and the elements contributing to the maximum prefix sum of the right child, 
   

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In this Case,
Maximum Prefix Sum = Total Sum of Left Child + 
 Maximum Prefix Sum of Right Child

Constructing the Maximum Suffix Sum using Left and Right child

There can be two cases for maximum suffix sum of a Node: 

• The maximum suffix sum occurs in the right child, 

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In this Case,
Maximum Suffix Sum = Maximum Suffix Sum of Right Child

• The maximum suffix sum contains every array element of the Right child and the elements contributing to the maximum suffix sum of the left child, 
   

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In this Case,
Maximum Suffix Sum = Total Sum of Right Child + 
 Maximum Suffix Sum of Left Child

Constructing the Maximum Subarray Sum using Left and Right child

There can be three cases for the maximum sub-array sum of a Node: 
 

• The maximum sub-array sum occurs in the left child, 
   

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In this Case,
Maximum Sub-array Sum = Maximum Subarray Sum of Left Child

• The maximum sub-array sum occurs in the right child, 

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In this Case,
Maximum Sub-array Sum = Maximum Subarray Sum of Right Child

• The maximum subarray sum, contains array elements of the right child contributing to the maximum prefix sum of the right child, and the array elements of the Left child contributing to the maximum suffix sum of the left child, 
   

👁 Image

 

In this Case,
Maximum Subarray Sum = Maximum Prefix Sum of Right Child 
 + 
 Maximum Suffix Sum of Left Child

Implementation:

Maximum Sub-Array Sum between 0 and 4 = 5
Maximum Sub-Array Sum between 0 and 2 = 4

Time Complexity: O(logn) for each query.