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

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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 \(O(n+m)\) time with \(n+\sum _{v\in V_{\ge 3}}\lceil \log _2(d_v-1)\rceil +O(\log n)\le n+m+O(\log n)\) bits of working memory, where \(d_v\) is the (total) degree of v, for each \(v\in V\), and \(V_{\ge 3}=\{v\in V\mid d_v\ge 3\}\). 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 \(k\ge 1\), \(O(n\log ^{(k)}\! n)\) 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.

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Correspondence to Torben Hagerup.

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Hagerup, T. Space-Efficient DFS and Applications to Connectivity Problems: Simpler, Leaner, Faster. Algorithmica (2019). https://doi.org/10.1007/s00453-019-00629-x

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Keywords

  • Graph algorithms
  • Space efficiency
  • Depth-first search
  • Connectivity