Space-Efficient DFS and Applications to Connectivity Problems: Simpler, Leaner, Faster

Abstract

The problem of space-efficient depth-first search (DFS) is reconsidered. A particularly simple and fast algorithm is presented that, on a directed or undirected input graph G = (V, E) with n vertices and m edges, carries out a DFS in O(n + m) timeA slightly more complicated variant of the algorithm works in the same time with at most n + (4/5)m + O(log n) bits. It is also shown that a DFS can be carried out in a graph with n vertices and m edges in O(n + m log * n) time with O(n) bits or in O(n + m) time with either O(n log log(4 + m/n)) bits or, for arbitrary integer k ≥ 1, O(n log (k) n) bits. These results among them subsume or improve most earlier results on space-efficient DFS. Some of the new time and space bounds are shown to extend to applications of DFS such as the computation of cut vertices, bridges, biconnected components and 2-edge-connected components in undirected graphs.

Finding Strong Components Using Depth-First Search

“…There has been quite a bit of additional work on reducing the extra space needed to run depth-first search. See Hagerup [Hag20] and the references therein. Most of these results make more assumptions about the way the input graph is represented.…”

Nearly Optimal Space Efficient Algorithm for Depth First Search

“…Recently, Hagerup [20] claimed an algorithm that finds Dfs in O(m log * n + n) time using O(n) bits of space. We improve upon this algorithm giving a near optimal running time for Dfs -it is almost linear in m + n. Our result can be succinctly stated as follows:…”

Succinct Planar Encoding with Minor Operations

“…Many graphs that arise in real-world application are very large. This has given rise to an area of research with the aim of reducing the required space [1,7,8,12,14,19]. Practical examples include large road-networks [26] or social-network graphs [10].…”