We provide data structures that maintain a graph as edges are inserted and deleted, and keep track of the following properties with the following times: minimum spanning forests, graph connectivity, graph 2-edge conne...
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We provide data structures that maintain a graph as edges are inserted and deleted, and keep track of the following properties with the following times: minimum spanning forests, graph connectivity, graph 2-edge connectivity, and bipartiteness in time O(n(1/2)) per change;3-edge connectivity, in time O(n(2/3)) per change;4-edge connectivity, in time O(ncr(n)) per change;k-edge connectivity for constant k, in time O(nlogn) per change;2-vertex connectivity, and 3-vertex connectivity, in time O(n) per change;and 4-vertex connectivity, in time O(n alpha(n)) per change. Further results speed up the insertion times to match the bounds of known partially dynamicalgorithms. All our algorithms are based on a new technique that transforms an algorithm for sparse graphs into one that will work on any graph, which we call sparsification.
This paper introduces compressed certificates for planarity, biconnectivity and triconnectivity in planar graphs, and prove many structural properties of certificates in planar graphs. As an application of our compres...
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This paper introduces compressed certificates for planarity, biconnectivity and triconnectivity in planar graphs, and prove many structural properties of certificates in planar graphs. As an application of our compressed certificates, we develop efficient dynamic planar algorithms. In particular, we consider the following three operations on a planar graph G: (i) insert an edge if the resultant graph remains planar;(ii) delete an edge;and (iii) test whether an edge could be added to the graph without violating planarity. We show how to support each of the above operations in O(n(2/3)) time, where n is the number of vertices in the graph. The bound for tests and deletions is worst-case, while the bound for insertions is amortized. This is the first algorithm for this problem with sub-linear running time, and it affirmatively answers a question posed in Eppstein ct al. [1992]. We use our compressed certificates for biconnectivity and triconnectivity to maintain the biconnected and triconnected components of a dynamic planar graph. The time bounds are the same: O(n(2/3)) worst-case time per edge deletion, O(n(2/3)) amortized time per edge insertion, and O(n(2/3)) worst-case time to check whether two vertices are either biconnected or triconnected.
We give algorithms and data structures that maintain the 2-edge and 2-vertex-connected components of a graph under insertions and deletions of edges and vertices, where deletions occur in a backtracking fashion (i.e.,...
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We give algorithms and data structures that maintain the 2-edge and 2-vertex-connected components of a graph under insertions and deletions of edges and vertices, where deletions occur in a backtracking fashion (i.e., deletions undo the insertions in the reverse order). Our algorithms run in Theta(log n) worst-case time per operation and use Theta(n) space, where n is the number of vertices. Using our data structure we can answer queries, which ask whether vertices u and v belong to the same 2-connected component, in Theta(log n) worst-case time.
In this paper we propose a uniform approach to deal with incremental problems on digraphs and with decremental problems on dags generalizing a technique used by La Poutre and van Leeuwen in [17] for updating the trans...
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In this paper we propose a uniform approach to deal with incremental problems on digraphs and with decremental problems on dags generalizing a technique used by La Poutre and van Leeuwen in [17] for updating the transitive closure and the transitive reduction of a dag. We define a propagation property on a binary relationship over the vertices of a digraph as a simple sufficient condition to apply this approach. The proposed technique is suitable for a very simple implementation which does not depend on the particular problem;in other words, the same procedures can be used to deal with different problems by simply setting appropriate boundary conditions. In particular, we provide semi-dynamicalgorithms and data structures for maintaining a binary relationship over the vertices of a digraph (dag) with n vertices and m edges, requiring O(nmax{q,m}) total time for any sequence of q edge insertions (deletions). This gives O(n) amortized time per operation over a sequence of Omega(m) edge insertions (deletions), Queries can be answered in constant time. The space required is O(n(2)). We apply the proposed technique to various problems about dominance, providing a solution to the problems of maintaining the dominance relationship, the dominator tree, and the nearest common dominator of a digraph in the incremental case, and of a dag in the decremental case;no dynamic solution was previously known for some of these problems. Finally we mention that the algorithms indeed work correctly also for interleaved sequences of insertion and deletion of edges in a dag, although the complexity bound holds for monotone sequence of updates only. (C) 1998-Elsevier Science B.V. All rights reserved.
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