On-line maintenance of the four-connected components of a graph

Kanevsky, Arkady; Tamassia, Roberto; Battista, Giuseppe Di; Chen, J.

doi:10.1109/sfcs.1991.185451

Cited by 37 publications

(29 citation statements)

References 23 publications

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“…As for k b Q, there are only a few results available. Kanevsky et al [30] presented a partially dynamic algorithm for the maintenance of 4-vertex connected components which requires yqY n amortized time per update and per query provided that the graph is triconnected, where q is the number of operations performed. Dinitz [9] presented a partially dynamic algorithm for the maintenance of 4-edge connected components.…”

Section: Previous Related Resultsmentioning

confidence: 99%

Fully dynamic maintenance of k-connectivity in parallel

Liang

Brent

Shen

2001

IEEE Trans. Parallel Distrib. Syst.

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AbstractÐGiven a graph q Y i with n vertices and m edges, the k-connectivity of q denotes either the k-edge connectivity or the k-vertex connectivity of q. In this paper, we deal with the fully dynamic maintenance of k-connectivity of q in the parallel setting for k PY Q. We study the problem of maintaining k-edge/vertex connected components of a graph undergoing repeatedly dynamic updates, such as edge insertions and deletions, and answering the query of whether two vertices are included in the same k-edge/ vertex connected component. Our major results are the following: 1) An NC algorithm for the 2-edge connectivity problem is proposed, which runs in ylog n logman time using yn QaR processors per update and query.2) It is shown that the biconnectivity problem can be solved in ylog P n time using ynPnY na log n processors per update and yI time with a single processor per query or in ylog n log m n time using ynPnY na log n processors per update and ylog n time using ynPnY na log n processors per query, where XY X is the inverse of Ackermann's function. 3) An NC algorithm for the triconnectivity problem is also derived, which takes ylog n log m n log n log log naQnY n time using ynQnY na log n processors per update and yI time with a single processor per query. 4) An NC algorithm for the 3-edge connectivity problem is obtained, which has the same time and processor complexities as the algorithm for the triconnectivity problem. To the best of our knowledge, the proposed algorithms are the first NC algorithms for the problems using yn processors in contrast to m processors for solving them from scratch. In particular, the proposed NC algorithm for the 2-edge connectivity problem uses only yn QaR processors. All the proposed algorithms run on a CRCW PRAM.Index TermsÐNC algorithms, 2-edge/vertex connectivity, 3-edge/vertex connectivity, dynamic data structures, parallel algorithm design and analysis, graph problems.

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Section: Previous Related Resultsmentioning

confidence: 99%

Fully dynamic maintenance of k-connectivity in parallel

Liang

Brent

Shen

2001

IEEE Trans. Parallel Distrib. Syst.

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“…The 4-block tree of a 3-connected planar graph can be computed in O(nα(n)) time [KTBC92]. The 4-block tree has at most (2n − 4)/3 leaves [BDD + 04].…”

Section: Cutting Leavesmentioning

confidence: 99%

“…We first compute its 4-block tree T 4 in O(nα(n)) time [KTBC92], choose an arbitrary node of T 4 , and direct all edges towards it. We call a node u of T 4 a predecessor of v if there is a directed path from u to v in T 4 .…”

Section: -Connected Planar Graphsmentioning

confidence: 99%

Computing large matchings fast

Rutter

Wolff²

2010

ACM Trans. Algorithms

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In this paper we present algorithms for computing large matchings in 3-regular graphs, graphs with maximum degree 3, and 3-connected planar graphs. The algorithms give a guarantee on the size of the computed matching and take linear or slightly superlinear time. Thus they are faster than the best-known algorithm for computing maximum matchings in general graphs, which runs in O( √ nm) time, where n denotes the number of vertices and m the number of edges of the given graph. For the classes of 3-regular graphs and graphs with maximum degree 3 the bounds we achieve are known to be best possible.We also investigate graphs with block trees of bounded degree, where the d-block tree is the adjacency graph of the d-connected components of the given graph. In 3-regular graphs and 3-connected planar graphs with bounded-degree 2-and 4-block trees, respectively, we show how to compute maximum matchings in slightly superlinear time.

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“…During each update, all edges inserted into G x,I are also inserted into H and H * , but no edge will be deleted from H and H * . We also construct the data structure of Kannevsky et al [6] described in the previous section in order to test the fourconnectivity between two vertices in H * , and we call it D * . Finally, we maintain a list L of all v ∈ V G with deg G (v) > 3.…”

Section: An O(m + Nα(n N))-time Algorithm For the 2-vdppmentioning

confidence: 99%

“…al. [6]. This data structure being initialized with a triconnected graph supports the insertion of additional edges, can answer queries whether two vertices v and w in the current graph are 4-connected, and finally, if they are neither adjacent nor 4-connected it can output a 3-separator separating v and w. One can use this data structure to identify a vertex v in G x,I not 4-connected to x (and hence non-adjacent to x since one can show that the vertices s 1 , s 2 , t 1 , and t 2 adjacent to x are 4-connected to x) and a 3-separator S separating v and x, and then run a ∆-replacement by S deleting v. However, since D does not support edge deletions, we use data structure D instead for G x,I for a graph H that only before the first ∆-replacement should be equal to G x,I .…”

Section: Introductionmentioning

confidence: 99%

Improved Algorithms for the 2-Vertex Disjoint Paths Problem

Tholey

2009

Lecture Notes in Computer Science

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Abstract. Given distinct vertices s1, s2, t1, and t2 the 2-vertex-disjoint paths problem consists in determining two vertex-disjoint paths p1, from s1 to t1, and p2, from s2 to t2, if such paths exist. As a first result we show that by using some kind of sparsification technique the previously best known time bound of O(n + mα(m, n)) can be reduced to O(m + nα(n, n)), where α denotes the inverse of the Ackermann function. Moreover, we extend the very practical and simple algorithm of Hagerup for solving the 2-vertex-disjoint-paths problem on 3-connected planar graphs to a practical linear time algorithm for the 2-VDPP on general planar graphs thereby avoiding the computation of planar embeddings or triconnected components.

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On-line maintenance of the four-connected components of a graph

Cited by 37 publications

References 23 publications

Fully dynamic maintenance of k-connectivity in parallel

Fully dynamic maintenance of k-connectivity in parallel

Computing large matchings fast

Improved Algorithms for the 2-Vertex Disjoint Paths Problem

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