1,550 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

    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 GeG\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 GeG\setminus e, for all edges ee in GG. (ii)(ii) The size of the largest and of the smallest strongly connected components in GeG\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 GeG\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 GeG\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

    Approximating the Smallest Spanning Subgraph for 2-Edge-Connectivity in Directed Graphs

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    Let GG be a strongly connected directed graph. We consider the following three problems, where we wish to compute the smallest strongly connected spanning subgraph of GG that maintains respectively: the 22-edge-connected blocks of GG (\textsf{2EC-B}); the 22-edge-connected components of GG (\textsf{2EC-C}); both the 22-edge-connected blocks and the 22-edge-connected components of GG (\textsf{2EC-B-C}). All three problems are NP-hard, and thus we are interested in efficient approximation algorithms. For \textsf{2EC-C} we can obtain a 3/23/2-approximation by combining previously known results. For \textsf{2EC-B} and \textsf{2EC-B-C}, we present new 44-approximation algorithms that run in linear time. We also propose various heuristics to improve the size of the computed subgraphs in practice, and conduct a thorough experimental study to assess their merits in practical scenarios

    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(nlog2nloglogn)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(log2nloglogn)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 Algorithms for the Massively Parallel Computation Model

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    The Massive Parallel Computing (MPC) model gained popularity during the last decade and it is now seen as the standard model for processing large scale data. One significant shortcoming of the model is that it assumes to work on static datasets while, in practice, real-world datasets evolve continuously. To overcome this issue, in this paper we initiate the study of dynamic algorithms in the MPC model. We first discuss the main requirements for a dynamic parallel model and we show how to adapt the classic MPC model to capture them. Then we analyze the connection between classic dynamic algorithms and dynamic algorithms in the MPC model. Finally, we provide new efficient dynamic MPC algorithms for a variety of fundamental graph problems, including connectivity, minimum spanning tree and matching.Comment: Accepted to the 31st ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2019

    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
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