Depth-First Search Using $O(n)$ Bits

Abstract

We provide algorithms performing Depth-First Search (DFS) on a directed or undirected graph with $n$ vertices and $m$ edges using only $O(n)$ bits. One algorithm uses $O(n)$ bits and runs in $O(m \log n)$ time. Another algorithm uses $n+o(n)$ bits and runs in polynomial time. Furthermore, we show that DFS on a directed acyclic graph can be done in space $n/2^{\varOmega (\sqrt{\log n})}$ and in polynomial time, and we also give a simple linear-time $O(\log n)$-space algorithm for the depth-first traversal of an undirected tree. Finally, we also show that for a graph having an $O(1)$-size feedback set, DFS can be done in $O(\log n)$ space. Our algorithms are based on the analysis of properties of DFS and applications of the $s$-$t$ connectivity algorithms due to Reingold and Barnes et al., both of which run in sublinear space.