496 research outputs found

    Incremental 22-Edge-Connectivity in Directed Graphs

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    In this paper, we initiate the study of the dynamic maintenance of 22-edge-connectivity relationships in directed graphs. We present an algorithm that can update the 22-edge-connected blocks of a directed graph with nn vertices through a sequence of mm edge insertions in a total of O(mn)O(mn) time. After each insertion, we can answer the following queries in asymptotically optimal time: (i) Test in constant time if two query vertices vv and ww are 22-edge-connected. Moreover, if vv and ww are not 22-edge-connected, we can produce in constant time a "witness" of this property, by exhibiting an edge that is contained in all paths from vv to ww or in all paths from ww to vv. (ii) Report in O(n)O(n) time all the 22-edge-connected blocks of GG. To the best of our knowledge, this is the first dynamic algorithm for 22-connectivity problems on directed graphs, and it matches the best known bounds for simpler problems, such as incremental transitive closure.Comment: Full version of paper presented at ICALP 201

    Decremental Single-Source Reachability in Planar Digraphs

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    In this paper we show a new algorithm for the decremental single-source reachability problem in directed planar graphs. It processes any sequence of edge deletions in O(nlog⁥2nlog⁥log⁥n)O(n\log^2{n}\log\log{n}) total time and explicitly maintains the set of vertices reachable from a fixed source vertex. Hence, if all edges are eventually deleted, the amortized time of processing each edge deletion is only O(log⁥2nlog⁥log⁥n)O(\log^2 n \log \log n), which improves upon a previously known O(n)O(\sqrt{n}) solution. We also show an algorithm for decremental maintenance of strongly connected components in directed planar graphs with the same total update time. These results constitute the first almost optimal (up to polylogarithmic factors) algorithms for both problems. To the best of our knowledge, these are the first dynamic algorithms with polylogarithmic update times on general directed planar graphs for non-trivial reachability-type problems, for which only polynomial bounds are known in general graphs

    Dynamic Dominators and Low-High Orders in DAGs

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    We consider practical algorithms for maintaining the dominator tree and a low-high order in directed acyclic graphs (DAGs) subject to dynamic operations. Let G be a directed graph with a distinguished start vertex s. The dominator tree D of G is a tree rooted at s, such that a vertex v is an ancestor of a vertex w if and only if all paths from s to w in G include v. The dominator tree is a central tool in program optimization and code generation, and has many applications in other diverse areas including constraint programming, circuit testing, biology, and in algorithms for graph connectivity problems. A low-high order of G is a preorder of D that certifies the correctness of D, and has further applications in connectivity and path-determination problems. We first provide a practical and carefully engineered version of a recent algorithm [ICALP 2017] for maintaining the dominator tree of a DAG through a sequence of edge deletions. The algorithm runs in O(mn) total time and O(m) space, where n is the number of vertices and m is the number of edges before any deletion. In addition, we present a new algorithm that maintains a low-high order of a DAG under edge deletions within the same bounds. Both results extend to the case of reducible graphs (a class that includes DAGs). Furthermore, we present a fully dynamic algorithm for maintaining the dominator tree of a DAG under an intermixed sequence of edge insertions and deletions. Although it does not maintain the O(mn) worst-case bound of the decremental algorithm, our experiments highlight that the fully dynamic algorithm performs very well in practice. Finally, we study the practical efficiency of all our algorithms by conducting an extensive experimental study on real-world and synthetic graphs

    Controlling edge dynamics in complex networks

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    The interaction of distinct units in physical, social, biological and technological systems naturally gives rise to complex network structures. Networks have constantly been in the focus of research for the last decade, with considerable advances in the description of their structural and dynamical properties. However, much less effort has been devoted to studying the controllability of the dynamics taking place on them. Here we introduce and evaluate a dynamical process defined on the edges of a network, and demonstrate that the controllability properties of this process significantly differ from simple nodal dynamics. Evaluation of real-world networks indicates that most of them are more controllable than their randomized counterparts. We also find that transcriptional regulatory networks are particularly easy to control. Analytic calculations show that networks with scale-free degree distributions have better controllability properties than uncorrelated networks, and positively correlated in- and out-degrees enhance the controllability of the proposed dynamics.Comment: Preprint. 24 pages, 4 figures, 2 tables. Source code available at http://github.com/ntamas/netctr

    Strong Connectivity in Directed Graphs under Failures, with Application

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    In this paper, we investigate some basic connectivity problems in directed graphs (digraphs). Let GG be a digraph with mm edges and nn vertices, and let G∖eG\setminus e be the digraph obtained after deleting edge ee from GG. As a first result, we show how to compute in O(m+n)O(m+n) worst-case time: (i)(i) The total number of strongly connected components in G∖eG\setminus e, for all edges ee in GG. (ii)(ii) The size of the largest and of the smallest strongly connected components in G∖eG\setminus e, for all edges ee in GG. Let GG be strongly connected. We say that edge ee separates two vertices xx and yy, if xx and yy are no longer strongly connected in G∖eG\setminus e. As a second set of results, we show how to build in O(m+n)O(m+n) time O(n)O(n)-space data structures that can answer in optimal time the following basic connectivity queries on digraphs: (i)(i) Report in O(n)O(n) worst-case time all the strongly connected components of G∖eG\setminus e, for a query edge ee. (ii)(ii) Test whether an edge separates two query vertices in O(1)O(1) worst-case time. (iii)(iii) Report all edges that separate two query vertices in optimal worst-case time, i.e., in time O(k)O(k), where kk is the number of separating edges. (For k=0k=0, the time is O(1)O(1)). All of the above results extend to vertex failures. All our bounds are tight and are obtained with a common algorithmic framework, based on a novel compact representation of the decompositions induced by the 11-connectivity (i.e., 11-edge and 11-vertex) cuts in digraphs, which might be of independent interest. With the help of our data structures we can design efficient algorithms for several other connectivity problems on digraphs and we can also obtain in linear time a strongly connected spanning subgraph of GG with O(n)O(n) edges that maintains the 11-connectivity cuts of GG and the decompositions induced by those cuts.Comment: An extended abstract of this work appeared in the SODA 201

    Planar Reachability Under Single Vertex or Edge Failures

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    International audienceIn this paper we present an efficient reachability oracle under single-edge or single-vertex failures for planar directed graphs. Specifically, we show that a planar digraph G can be preprocessed in O(n log 2 n/log log n) time, producing an O(n log n)-space data structure that can answer in O(log n) time whether u can reach v in G if the vertex x (the edge f) is removed from G, for any query vertices u, v and failed vertex x (failed edge f). To the best of our knowledge, this is the first data structure for planar directed graphs with nearly optimal preprocessing time that answers all-pairs queries under any kind of failures in polylogarithmic time. We also consider 2-reachability problems, where we are given a planar digraph G and we wish to determine if there are two vertex-disjoint (edge-disjoint) paths from u to v, for query vertices u, v. In this setting we provide a nearly optimal 2-reachability oracle, which is the existential variant of the reachability oracle under single failures, with the following bounds. We can construct in O(n polylog n) time an O(n log 3+o(1) n)-space data structure that can check in O(log 2+o(1) n) time for any query vertices u, v whether v is 2-reachable from u, or otherwise find some separating vertex (edge) x lying on all paths from u to v in G. To obtain our results, we follow the general recursive approach of Thorup for reachability in planar graphs [J. ACM '04] and we present new data structures which generalize dominator trees and previous data structures for strong-connectivity under failures [Georgiadis et al., SODA '17]. Our new data structures work also for general digraphs and may be of independent interest

    Dominators in Directed Graphs: A Survey of Recent Results, Applications, and Open Problems

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    The computation of dominators is a central tool in program optimization and code generation, and it has applications in other diverse areas includingconstraint programming, circuit testing, and biology. In this paper we survey recent results, applications, and open problems related to the notion of dominators in directed graphs,including dominator verification and certification, computing independent spanning trees, and connectivity and path-determination problems in directed graphs
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