Improved Algorithms for Effective Resistance Computation on Graphs

TL;DR

This paper introduces improved algorithms for approximating effective resistance in graphs, reducing computational complexity and establishing lower bounds, with extensions to index-based methods for fast queries.

Contribution

It presents new local and index-based algorithms for effective resistance approximation with better time and space complexity, and proves lower bounds on local algorithm performance.

Findings

Achieved $ ilde{O}(rac{ ext{sqrt}(d)}{ ext{epsilon}})$ time for local ER approximation on expanders.

Established a lower bound of $ ilde{ ext{Omega}}(1/ ext{epsilon})$ for local algorithms on expanders.

Proposed an index-based ER approximation algorithm with $ ilde{O}( ext{min}igrace m+rac{n}{ ext{epsilon}^{1.5}}, rac{ ext{sqrt}(nm)}{ ext{epsilon}}ig)$ processing time.

Abstract

Effective Resistance (ER) is a fundamental tool in various graph learning tasks. In this paper, we address the problem of efficiently approximating ER on a graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ with $n$ vertices and $m$ edges. First, we focus on local online-computation algorithms for ER approximation, aiming to improve the dependency on the approximation error parameter $\epsilon$. Specifically, for a given vertex pair $(s,t)$, we propose a local algorithm with a time complexity of $\tilde{O}(\sqrt{d}/\epsilon)$ to compute an $\epsilon$-approximation of the $s,t$-ER value for expander graphs, where $d=\min \{d_s,d_t\}$. This improves upon the previous state-of-the-art, including an $\tilde{O}(1/\epsilon^2)$ time algorithm based on random walk sampling by Andoni et al. (ITCS'19) and Peng et al. (KDD'21). Our method achieves this improvement by combining deterministic search with random walk sampling to reduce variance. Second, we establish a lower bound for ER approximation on expander graphs. We prove that for any $\epsilon\in (0,1)$, there exist an expander graph and a vertex pair $(s,t)$ such that any local algorithm requires at least $\Omega(1/\epsilon)$ time to compute the $\epsilon$-approximation of the $s,t$-ER value. Finally, we extend our techniques to index-based algorithms for ER computation. We propose an algorithm with $\tilde{O}(\min \{m+n/\epsilon^{1.5},\sqrt{nm}/\epsilon\})$ processing time, $\tilde{O}(n/\epsilon)$ space complexity and $O(1)$ query complexity, which returns an $\epsilon$-approximation of the $s,t$-ER value for any $s,t\in \mathcal{V}$ for expander graphs. Our approach improves upon the state-of-the-art $\tilde{O}(m/\epsilon)$ processing time by Dwaraknath et al. (NeurIPS'24) and the $\tilde{O}(m+n/\epsilon^2)$ processing time by Li and Sachdeva (SODA'23).