How to Compute a Moving Sum

TL;DR

This paper explores the computational methods for windowed recurrences, introducing new sequential and parallel algorithms, and providing a comprehensive algebraic framework for efficient calculation of moving sums across various scientific fields.

Contribution

It presents novel algorithms for sequential and parallel computation of windowed recurrences, grounded in algebraic theory, with detailed analysis and practical pseudo-code implementations.

Findings

Developed a low-latency sequential algorithm.

Derived new parallel and vector algorithms using algebraic structures.

Provided extensive examples and pseudo-code for practical implementation.

Abstract

Windowed recurrences are sliding window calculations where a function is applied iteratively across the window of data, and are ubiquitous throughout the natural, social, and computational sciences. In this monograph we explore the computational aspects of these calculations, including sequential and parallel computation, and develop the theory underlying the algorithms and their applicability. We introduce an efficient new sequential algorithm with low latency, and develop new techniques to derive and analyze the complexity and domain of validity of existing sequential algorithms. For parallel computation we derive new parallel and vector algorithms by relating windowed recurrences to the algebraic construction of semidirect products, and to algorithms for exponentiation in semigroups. In the middle chapters of the monograph we further develop the theory of semi-associativity and the algebraic conditions for representing function composition and function application by data. This systematizes the techniques used by practitioners to parallelize recurrence calculations. We end the monograph with an extensive gallery of examples of interest to specialists in many fields. Throughout the monograph new algorithms are described with pseudo-code transcribed from functioning source code.