Range Sum Query - Mutable

Problem Description

Given an integer array nums, handle multiple queries of the following types:

1. Update the value of an element in nums.
2. Calculate the sum of the elements of nums between indices left and right inclusive.

Implement the NumArray class:

• NumArray(int[] nums) Initializes the object with the integer array nums.
• void update(int index, int val) Updates the value of nums[index] to be val.
• int sumRange(int left, int right) Returns the sum of the elements of nums between indices left and right inclusive.

Key Insights

• Efficiently handling multiple updates and range sum queries is crucial.
• A naive approach using simple iteration for each query would lead to O(n) time complexity per query, which is inefficient for large input sizes.
• Data structures like Segment Trees or Binary Indexed Trees (Fenwick Trees) can optimize updates and queries to O(log n) time complexity.

Space and Time Complexity

Time Complexity:

• O(log n) for both update and sumRange operations using a Binary Indexed Tree or Segment Tree.

Space Complexity:

• O(n) for storing the original array and any auxiliary structures.

Solution

To solve this problem, we can use a Binary Indexed Tree (Fenwick Tree) or a Segment Tree. Both data structures allow us to efficiently update elements and calculate prefix sums, which we can leverage to compute range sums.

1. Binary Indexed Tree (Fenwick Tree):

   • This tree allows us to update an element and calculate prefix sums in logarithmic time.
   • We maintain an array that represents cumulative frequencies, enabling us to compute the sum from index left to right by calculating prefix sums.

2. Segment Tree:

   • A segment tree can also efficiently handle the updates and range queries.
   • Each node in the segment tree represents the sum of a segment of the array, allowing for quick updates and queries.

For this solution, we will implement the Binary Indexed Tree due to its simplicity and efficiency.

Code Solutions