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

2025-02-03

AI Summary

This paper addresses the problem of supporting efficient range queries and point updates for non-commutative, invertible cumulative operations—such as matrix chain multiplication and function composition—on dynamic arrays. Conventional solutions suffer from fundamental limitations: segment trees require O(N) auxiliary space, while Fenwick trees (binary indexed trees) mandate commutativity. We propose the first data structure achieving both asymptotic time and space optimality: it uses zero extra space (O(1) space complexity) while supporting both operations in O(log₂N) time. Our method constructs an implicit tree structure via coordinate transformation and in-place index mapping, augmented by inverse-element computation and a southwest-directed traversal strategy. Experimental results demonstrate approximately 40% memory reduction over segment trees and establish, for the first time, simultaneous time–space optimality for non-commutative, invertible settings.

Technology Category

• Search and Optimization: Distributed Search
• Machine Learning: Matrix & Tensor Methods
• Data Mining & Knowledge Management: Data Compression

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Data transparency and provenance
• Web Mining and Content Analysis: Bridging structured and unstructured data

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.

Problem

• Efficient Computation
• Data Structure
• Complex Operations

Innovation

• Southwest Tree
• O(log_2 N) Efficiency
• Space Optimization