Unifying Matrix Data Structures: Simplifying and Speeding up Iterative Algorithms

TL;DR

This paper introduces a unifying framework for matrix data structures that simplifies proofs and enhances the speed of iterative algorithms like the simplex method by reducing complex properties to dynamic matrix inverse maintenance.

Contribution

The paper presents a general framework that captures various matrix data structures, simplifying their proofs and enabling faster iterative algorithms through reduction to dynamic matrix inverse maintenance.

Findings

Short proofs for many existing data structures.

Speed improvements in iterative algorithms such as the simplex method.

Reduction of complex matrix property maintenance to dynamic matrix inverse problems.

Abstract

Many algorithms use data structures that maintain properties of matrices undergoing some changes. The applications are wide-ranging and include for example matchings, shortest paths, linear programming, semi-definite programming, convex hull and volume computation. Given the wide range of applications, the exact property these data structures must maintain varies from one application to another, forcing algorithm designers to invent them from scratch or modify existing ones. Thus it is not surprising that these data structures and their proofs are usually tailor-made for their specific application and that maintaining more complicated properties results in more complicated proofs. In this paper we present a unifying framework that captures a wide range of these data structures. The simplicity of this framework allows us to give short proofs for many existing data structures regardless of how complicated the to be maintained property is. We also show how the framework can be used to speed up existing iterative algorithms, such as the simplex algorithm. More formally, consider any rational function $f(A_1,...,A_d)$ with input matrices $A_1,...,A_d$. We show that the task of maintaining $f(A_1,...,A_d)$ under updates to $A_1,...,A_d$ can be reduced to the much simpler problem of maintaining some matrix inverse $M^{-1}$ under updates to $M$. The latter is a well studied problem called dynamic matrix inverse. By applying our reduction and using known algorithms for dynamic matrix inverse we can obtain fast data structures and iterative algorithms for much more general problems.