Faster Walks in Graphs: A $\tilde O(n^2)$ Time-Space Trade-off for Undirected s-t Connectivity

TL;DR

This paper introduces a new randomized algorithm using Metropolis-type walks that significantly improves the time-space trade-off for undirected s-t connectivity in graphs, achieving near-linear time with reduced space compared to previous methods.

Contribution

The paper presents a novel family of algorithms for USTCON with a time-space product of  O(n^2), improving upon the previous  O(n m) trade-off, and demonstrates how to optimize walk parameters for better performance.

Findings

Achieves  O(n+m) running time with improved space efficiency.

Provides a method to tune walks to match random walk performance.

Maintains  O(n^2) worst-case cover time.

Abstract

In this paper, we make use of the Metropolis-type walks due to Nonaka et al. (2010) to provide a faster solution to the $S$-$T$-connectivity problem in undirected graphs (USTCON). As our main result, we propose a family of randomized algorithms for USTCON which achieves a time-space product of $S\cdot T = \tilde O(n^2)$ in graphs with $n$ nodes and $m$ edges (where the $\tilde O$-notation disregards poly-logarithmic terms). This improves the previously best trade-off of $\tilde O(n m)$, due to Feige (1995). Our algorithm consists in deploying several short Metropolis-type walks, starting from landmark nodes distributed using the scheme of Broder et al. (1994) on a modified input graph. In particular, we obtain an algorithm running in time $\tilde O(n+m)$ which is, in general, more space-efficient than both BFS and DFS. We close the paper by showing how to fine-tune the Metropolis-type walk so as to match the performance parameters (e.g., average hitting time) of the unbiased random walk for any graph, while preserving a worst-case bound of $\tilde O(n^2)$ on cover time.