Mantaining Dynamic Matrices for Fully Dynamic Transitive Closure

TL;DR

This paper presents a novel matrix-based framework for fully dynamic transitive closure, achieving improved amortized update times and breaking previous complexity barriers for directed acyclic graphs.

Contribution

It introduces a general polynomial reevaluation technique for dynamic transitive closure, leading to faster algorithms with near-optimal query and update times.

Findings

Deterministic algorithm with O(n^2) amortized update time for general graphs.

Faster updates in O(n) amortized time for deletions only.

Breakthrough in single-operation complexity for directed acyclic graphs, with subquadratic algorithms.

Abstract

In this paper we introduce a general framework for casting fully dynamic transitive closure into the problem of reevaluating polynomials over matrices. With this technique, we improve the best known bounds for fully dynamic transitive closure. In particular, we devise a deterministic algorithm for general directed graphs that achieves $O(n^2)$ amortized time for updates, while preserving unit worst-case cost for queries. In case of deletions only, our algorithm performs updates faster in O(n) amortized time. Our matrix-based approach yields an algorithm for directed acyclic graphs that breaks through the $O(n^2)$ barrier on the single-operation complexity of fully dynamic transitive closure. We can answer queries in $O(n^\epsilon)$ time and perform updates in $O(n^{\omega(1,\epsilon,1)-\epsilon}+n^{1+\epsilon})$ time, for any $\epsilon\in[0,1]$, where $\omega(1,\epsilon,1)$ is the exponent of the multiplication of an $n\times n^{\epsilon}$ matrix by an $n^{\epsilon}\times n$ matrix. The current best bounds on $\omega(1,\epsilon,1)$ imply an $O(n^{0.58})$ query time and an $O(n^{1.58})$ update time. Our subquadratic algorithm is randomized, and has one-side error.