Abstract
We present an algorithm for constructing a depth-first search tree in planar digraphs; the algorithm can be implemented in the complexity class \(\text{ AC}^1(\text{ UL }\cap \text{ co-UL})\), which is contained in \(\text{ AC}^2\). Prior to this (for more than a quarter-century), the fastest uniform deterministic parallel algorithm for this problem had a runtime of \(O(\log ^{10}n)\) (corresponding to the complexity class \(\text{ AC}^{10}\subseteq \text{ NC}^{11}\)). We also consider the problem of computing depth-first search trees in other classes of graphs and obtain additional new upper bounds.
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Notes
An earlier version of this work claimed a stronger upper bound, but there was an error in one of the lemmas in that version [3].
Let us give additional motivation for having a dynamically computed ordering on the neighbors of v. We will be considering a DAG whose vertices consist of strongly connected components (SCCs) of the original graph G. We will have already pre-computed several DFS trees of each SCC: one rooted at each node in the SCC. Our final DFS tree will consist of (a) one DFS tree for each SCC (where the root of the DFS tree for SCC C is some node \(r_C \in C\)) along with (b) a selected edge \((v_D,r_C)\) connecting any two SCCs D and C that are adjacent in the DFS tree of the DAG. But of course, to fully specify the DFS tree, we also need to have an ordering on the neighbors of each vertex. In practice, we will be using the (precomputed) DFS tree of D (rooted at \(r_D\)) to determine the order of neighbors of vertex D in the DAG (whose vertices are SCCs). The “lexicographically least” property of our DFS tree of the DAG depends only on the ordering of the neighbors (and not on the selection of the specific edge between vertices in the directed acyclic multigraph).
In case a more detailed definition is necessary, here is what is meant by “the lexicographically least path from s to v.” Let p and \(p'\) be two paths from s to v. If p is shorter than \(p'\), then p precedes \(p'\) in the lexicographic ordering. If p and \(p'\) have the same length and are not equal, then they each start with s and agree up through some vertex x, and first differ at the next vertex. Let us say that p has the edge (x, w) and \(p'\) has the edge \((x,w')\) The vertex x is entered via some edge e (where if \(x=s\), then e is the null edge). The neighbors of x are ordered according to f(x, e). If w precedes \(w'\) in the ordering f(x, e), then p precedes \(p'\) in the lexicographic ordering.
This may seem counterintuitive. If \(C_v\) is not entirely red, then v participates in some red cycle containing edges not in \(C_v\). Whereas if \(C_v\) is all red, then v is not connected to other red parts of G, and thus we color it white.
The interior of a cycle is the subgraph of G induced on the vertices that are embedded inside C, but not on C.
Notice that here we explicitly allow \(k = 1\) so that \(v_1 = v_k\).
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Acknowledgements
We thank the anonymous referees for their helpful and insightful suggestions, which improved the presentation.
Funding
This study was supported in part by NSF Grants CCF-1909216 and CCF-1909683, partially supported by a grant from Infosys foundation and TCS PhD fellowship and partially supported by a grant from Infosys foundation and SERB-MATRICS Grant MTR/2017/000480.
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Allender, E., Chauhan, A. & Datta, S. Depth-first search in directed planar graphs, revisited. Acta Informatica 59, 289–319 (2022). https://doi.org/10.1007/s00236-022-00425-1
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DOI: https://doi.org/10.1007/s00236-022-00425-1