Range sum queries without updates

Given an array arr of integers of size n. We need to compute the sum of elements from index i to index j. The queries consisting of i and j index values will be executed multiple times.

Examples: 

Input : arr[] = {1, 2, 3, 4, 5}
i = 1, j = 3
i = 2, j = 4
Output : 9
12
Input : arr[] = {1, 2, 3, 4, 5}
i = 0, j = 4
i = 1, j = 2
Output : 15
5

[Naive Solution] - Simple Sum - O(n) for Every Query and O(1) Space

A Simple Solution is to compute the sum for every query.

Output

9
12

[Expected Solution] - Prefix Sum - O(1) for Every Query and O(n) Space

Follow the given steps to solve the problem:

• Create the prefix sum array of the given input array
• Now for every query (1-based indexing)
  • If L is greater than 1, then print prefixSum[R] - prefixSum[L-1]
  • else print prefixSum[R]

9
12

Here time complexity of every range sum query is O(1) and the overall time complexity is O(n).

Auxiliary Space required = O(n), where n is the size of the given array.

The question becomes complicated when updates are also allowed. In such situations when using advanced data structures like Segment Tree or Binary Indexed Tree.