O(n) time and the update operation taking O(1) time.

Alternatively, another array can be constructed to store the cumulative sum up to the ith index. This allows range sums to be calculated in O(1) time but makes update operations take O(n) time. This approach is suitable when query operations are frequent but updates are rare.

Method 2 - Segment Tree

To solve the problem efficiently, we can use a Segment Tree data structure to handle two types of operations:

Update queries to change values at specific indices in the array.
Range sum queries to compute the sum of elements within a given range.

The Segment Tree allows us to perform both update and range sum operations efficiently in O(log n) time.

Representation of Segment Tree

The leaf nodes in a segment tree correspond to the elements of the input array.
Each internal node represents the result of merging its child nodes, with the type of merging depending on the problem. For this problem, the merging is the sum of the elements under a node. Segment trees are often represented using arrays: the left child of a node at index i is at 2*i+1, the right child at 2*i+2, and the parent node at (i-1)/2.

Construction of Segment Tree

To construct a segment tree, start with the segment arr[0...n-1]. Recursively divide the segment into two halves (unless it is already of length 1) and repeat the process for each half. At every segment, store the sum in the corresponding tree node.

The segment tree is a [[Full Binary Tree]] because segments are always split into two halves. All tree levels, except possibly the

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