Fully Dynamic Effective Resistances

TL;DR

This paper introduces a fully dynamic data structure for maintaining approximate all-pairs effective resistances in graphs, enabling efficient updates and queries in complex network scenarios.

Contribution

It presents the first fully dynamic algorithm with sublinear update time for maintaining approximate effective resistances using a novel vertex resistance sparsifier approach.

Findings

Achieves ( m^{4/5} psilon^{-4} ) expected amortized update and query time.

Maintains a dynamic Schur complement via random walks and vertex subset sampling.

Provides a new local representation of vertex sparsifiers that may be of independent interest.

Abstract

In this paper we consider the \emph{fully-dynamic} All-Pairs Effective Resistance problem, where the goal is to maintain effective resistances on a graph $G$ among any pair of query vertices under an intermixed sequence of edge insertions and deletions in $G$. The effective resistance between a pair of vertices is a physics-motivated quantity that encapsulates both the congestion and the dilation of a flow. It is directly related to random walks, and it has been instrumental in the recent works for designing fast algorithms for combinatorial optimization problems, graph sparsification, and network science. We give a data-structure that maintains $(1+\epsilon)$-approximations to all-pair effective resistances of a fully-dynamic unweighted, undirected multi-graph $G$ with $\tilde{O}(m^{4/5}\epsilon^{-4})$ expected amortized update and query time, against an oblivious adversary. Key to our result is the maintenance of a dynamic \emph{Schur complement}~(also known as vertex resistance sparsifier) onto a set of terminal vertices of our choice. This maintenance is obtained (1) by interpreting the Schur complement as a sum of random walks and (2) by randomly picking the vertex subset into which the sparsifier is constructed. We can then show that each update in the graph affects a small number of such walks, which in turn leads to our sub-linear update time. We believe that this local representation of vertex sparsifiers may be of independent interest.