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A binary indexed tree, also known as a Fenwick tree, is a data structure that efficiently supports updating and querying prefix sums of an array. It is particularly useful when there is a need to perform frequent updates on elements of an array and calculate prefix sums of subarrays.
The binary indexed tree is represented as an array of nodes, where each node stores the cumulative sum of a range of elements. The index of each node in the array corresponds to the binary representation of the index in the original array. The tree structure allows for efficient updates and queries by utilizing the binary representation of the indices.
To construct a binary indexed tree, the initial array is used to update the tree. Each element of the array is added to the corresponding nodes in the tree, based on their binary representation. This process is repeated until all elements have been processed.
Updating an element in the binary indexed tree involves updating the corresponding nodes in the tree. This is done by adding the difference between the new value and the old value to the nodes that cover the index of the updated element. The process continues by updating the nodes until reaching the root of the tree.
Querying the prefix sum of a subarray is also efficiently performed using the binary indexed tree. By traversing the tree from the given index to the root, the cumulative sum of the subarray can be calculated by summing the values of the nodes encountered during the traversal.
The binary indexed tree provides a balance between space complexity and time complexity. It requires O(n) space to store the tree, where n is the number of elements in the original array. The time complexity for updating an element or querying a prefix sum is O(log n), making it an efficient data structure for certain types of problems, such as range sum queries.