Linear-Time Algorithms for Dominators and Other Path-Evaluation Problems

Buchsbaum, Adam L.; Georgiadis, Loukas; Kaplan, Haim; Rogers, Anne; Tarjan, Robert E.; Westbrook, Jeffery

doi:10.1137/070693217

Cited by 66 publications

(76 citation statements)

References 45 publications

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“…Step 1 takes O(m) time by [3], and Step 2 takes O(m) time by Lemma 3.10. From Lemma 3.7 we have that the total number of vertices and the total number of edges in all auxiliary graphs H of G are O(n) and O(m) respectively.…”

Section: A Linear-time Algorithm Although Algorithmsmentioning

confidence: 99%

“…Each recursive call refines the current partition of V , thus we have at most n − 1 recursive calls. By [3,25] and Lemma 3.10, the total work per recursive call is O(m). 2…”

mentioning

confidence: 99%

“…Subsequently, several linear-time algorithms were discovered [1,3,4,8,9,12]. Tarjan [25] showed that the bridges of flow graph G(s) can be computed in O(m) time given D(s).…”

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confidence: 99%

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2-Edge Connectivity in Directed Graphs

Georgiadis

Italiano

Laura

et al. 2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

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Edge and vertex connectivity are fundamental concepts in graph theory. While they have been thoroughly studied in the case of undirected graphs, surprisingly not much has been investigated for directed graphs. In this paper we study 2-edge connectivity problems in directed graphs and, in particular, we consider the computation of the following natural relation: We say that two vertices v and w are 2-edge-connected if there are two edge-disjoint paths from v to w and two edgedisjoint paths from w to v. This relation partitions the vertices into blocks such that all vertices in the same block are 2-edge-connected. Differently from the undirected case, those blocks do not correspond to the 2-edge-connected components of the graph. The main result of this paper is an algorithm for computing the 2-edge-connected blocks of a directed graph in linear time. Besides being asymptotically optimal, our algorithm improves significantly over previous bounds. Once the 2-edge-connected blocks are available, we can test in constant time if two vertices are 2-edge-connected. Additionally, we also show how to compute in linear time a sparse certificate for this relation, i.e., a subgraph of the input graph that has O(n) edges and maintains the same 2-edge-connected blocks as the input graph, where n is the number of vertices.

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Section: A Linear-time Algorithm Although Algorithmsmentioning

confidence: 99%

“…Each recursive call refines the current partition of V , thus we have at most n − 1 recursive calls. By [3,25] and Lemma 3.10, the total work per recursive call is O(m). 2…”

mentioning

confidence: 99%

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2-Edge Connectivity in Directed Graphs

Georgiadis

Italiano

Laura

et al. 2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…The properties of dfs imply that every cycle of G s is contained in a loop. A loop nesting forest can be computed in linear time [4,25]. Since here we deal with strongly connected digraphs, each vertex is contained in a loop, so H is a tree rooted at s. Therefore, we will refer to H as the loop nesting tree of G s .…”

Section: Algorithmsmentioning

confidence: 99%

“…Throughout the paper we let s be a fixed but arbitrary start vertex of G. Since G is strongly connected, all vertices are reachable from s and reach s, so we can view both G and G R as flow graphs with start vertex s. We will denote those flow graphs respectively by G s and G . We let C(v) (resp., C R (v)) denote the set of children in D (resp., D R ) of a vertex v. The dominator tree of a flow graph can be computed in linear time [1,4]. A vertex v = s is a strong articulation point of G if and only if it is not a leaf in D or in D R [13].…”

Section: Algorithmsmentioning

confidence: 99%

Computing 2-Connected Components and Maximal 2-Connected Subgraphs in Directed Graphs: An Experimental Study

Georgiadis¹,

Italiano²,

Karanasiou³

et al. 2018

2018 Proceedings of the Twentieth Workshop on Algorithm Engineering and Experiments (ALENEX)

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Motivated by very recent work on 2-connectivity in directed graphs, we revisit the problem of computing the 2-edge-and 2-vertex-connected components, and the maximal 2-edge-and 2-vertex-connected subgraphs of a directed graph G. We explore the design space for efficient algorithms in practice, based on recently proposed techniques, and conduct a thorough empirical study to highlight the merits and weaknesses of each technique.

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