On the Problem of Undirected st-connectivity

TL;DR

This paper discusses the deterministic log-space algorithm for undirected st-connectivity, proving it is in L and exploring its implications for complexity class equivalences such as SL and RL.

Contribution

It presents a deterministic log-space algorithm for undirected st-connectivity and discusses its significance for the complexity class hierarchy.

Findings

Undirected st-connectivity is in L.

SL = L is a plausible equality.

RL = L is believed to hold.

Abstract

In this paper, we discuss an algorithm for the problem of undirected st-connectivity that is deterministic and log-space, namely that of Reingold within his 2008 paper "Undirected Connectivity in Log-Space". We further present a separate proof by Rozenman and Vadhan of $\text{USTCONN} \in L$ and discuss its similarity with Reingold's proof. Undirected st-connectively is known to be complete for the complexity class SL--problems solvable by symmetric, non-deterministic, log-space algorithms. Likewise, by Aleliunas et. al., it is known that undirected st-connectivity is within the RL complexity class, problems solvable by randomized (probabilistic) Turing machines with one-sided error in logarithmic space and polynomial time. Finally, our paper also shows that undirected st-connectivity is within the L complexity class, problems solvable by deterministic Turing machines in logarithmic space. Leading from this result, we shall explain why SL = L and discuss why is it believed that RL = L.