Southwest Tree: A Low-Memory Data Structure for Partial Accumulations by Non-Commutative Invertible Operations

TL;DR

The paper introduces the Southwest Tree, a low-memory data structure that efficiently performs partial accumulations with non-commutative invertible operations in logarithmic time, overcoming limitations of existing structures.

Contribution

It presents the Southwest Tree, a novel data structure that handles non-commutative, invertible operations efficiently with minimal memory usage.

Findings

Performs partial accumulations in O(log N) time

Uses no extra memory beyond the input array

Handles non-commutative invertible operations effectively

Abstract

The task of accumulating a portion of a list of values, whose values may be updated at any time, is widely used throughout various applications in computer science. While it is trivial to accomplish this task without any constraints, trivial solutions often sacrifice time complexity in either accumulating or updating the values, one being constant time and the other being linear. To even out the complexity, two well-known data structures have been used to accomplish this task, namely the Segment Tree and the Binary Indexed Tree, which are able to carry out both tasks in O(log_2 N) time for a list of N elements. However, the Segment Tree suffers from requiring auxiliary memory to contain additional values, while the Binary Indexed Tree is unable to handle non-commutative accumulation operations. Here, we present a data structure, called the Southwest Tree, that accomplishes these tasks for non-commutative, invertible accumulation operations in O(log_2 N) time and uses no additional memory to store the structure apart from the initial input array.