Prefixbound

Prefixbound is a data structure concept that combines binary search trees and prefix sums to achieve fast lookup and range queries. It is primarily used in competitive programming to quickly determine the sum of elements in a specific range of a sorted array.

The prefixbound data structure consists of two components: a binary search tree (BST) and a prefix sum array. The BST stores the indices of the elements in the original array, while the prefix sum array stores the cumulative sum of the elements up to each index. When a query is asked, the BST can quickly determine the range of indices that satisfy the query, and the prefix sum array can then be used to calculate the sum of the elements in that range.

Prefixbound has several benefits that make it a valuable data structure in competitive programming. It allows for fast range queries, which is crucial in problems that require counting or summing elements in a specific range. It also supports point queries, where the sum of a specific element is requested. Furthermore, prefixbound can be adapted to other data structures, such as SAX trees or sparse tables, to support more complex queries.

The time complexity of prefixbound is O(log n) for search and O(n) for construction, where n is the size of the input array. This makes it a competitive data structure for a wide range of problems. However, it requires significant memory to store the prefix sum array, which can be a limitation for large inputs. Nonetheless, prefixbound remains a popular choice for many competitive programming problems, and its versatility has made it a widely used technique in algorithmic thinking.