prefixsummor

The prefixsummor is a term used to describe a class of data structures and algorithms designed to maintain the sums of prefixes of a numeric sequence under dynamic updates. It focuses on efficient access to cumulative totals that begin at the start of the sequence.

It supports two primary operations: update(i, delta), which adjusts the value at position i by delta, and prefix_sum(k), which returns the sum of the first k elements. Initialization takes O(n) time to build from an initial sequence, after which the data structure is ready for subsequent updates and queries.

In its simplest form, a prefixsummor can be implemented by a plain array, recomputing sums after each update, which yields O(n) time per operation. More sophisticated implementations use tree-based structures such as a Fenwick tree (Binary Indexed Tree) or a segment tree to achieve faster performance. These structures provide efficient mechanisms to maintain and query prefix sums as the underlying sequence changes.

A Fenwick tree offers logarithmic time for both updates and prefix queries, while a segment tree provides similar capabilities and can be extended to handle more general range sums and related queries. The choice of structure depends on the expected combination of updates and queries, as well as memory constraints and implementation details.

Applications include real-time analytics, financial time-series processing, streaming data dashboards, and certain dynamic programming problems where frequent prefix calculations are needed. The concept is closely related to static prefix sums and cumulative sums but emphasizes dynamic maintenance under updates.

Overall, the prefixsummor serves as a practical framework for efficiently maintaining and querying the sums of prefixes in mutable sequences, linking concepts from prefix sums to dynamic data structures.