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

• Published in: Algorithmica Vol. 82; no. 4; pp. 1033 - 1056
• Main Author: Hagerup, Torben
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2020; Springer Nature B.V
• Subjects: Algorithm Analysis and Problem Complexity; Algorithms; Apexes; Computer Science; Computer Systems Organization and Communication Networks; Data Structures and Information Theory; Graph theory; Mathematics of Computing; Theory of Computation; Connectivity; Graph algorithms; Space efficiency; Depth-first search
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-019-00629-x

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 ) time with n + ∑ v ∈ V ≥ 3 ⌈ log 2 ( d v - 1 ) ⌉ + O ( log n ) ≤ n + m + O ( log n ) bits of working memory, where d v is the (total) degree of v , for each v ∈ V , and V ≥ 3 = { v ∈ V ∣ d v ≥ 3 } . A 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 + min { 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.