Computing Edge-Connectivity in Multigraphs and Capacitated Graphs

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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“…Lemma 1 [38], [39]. The graph k is l-edge connected if and only if q is l-edge connected for any integer l with I l k.…”

Section: The Sparse K-edge (K-vertex) Certificatementioning

confidence: 99%

Fully dynamic maintenance of k-connectivity in parallel

Liang

Brent

Shen

2001

IEEE Trans. Parallel Distrib. Syst.

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“…In Step 1, we first add k edges between s and V so that H satisfies (4.1) and we attain 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Step 2, for each pair {u, v} ⊆ V , after splitting min{d By using further analysis, we can prove that it can be implemented to run in O(n(n 2 +k 3 )(p+ m + n log n) log k + pmn 2 log (n 2 /m)), whose proof is omitted. As a result, this total complexity can be reduced to O(m + n(k 3 + n 2 )(p + kn + n log n) log k + pkn 3 log (n/k)) by applying the procedure to a sparse spanning subgraph of G with O(kn) edges, where such sparsification takes O(m + n log n) time [16,17].…”

Section: Outputmentioning

confidence: 99%

Minimum Augmentation of Edge-Connectivity between Vertices and Sets of Vertices in Undirected Graphs

Ishii

Akiyama²,

Nagamochi

2008

Algorithmica

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Given an undirected multigraph G = (V, E), a family W of areas W ⊆ V , and a target connectivity k ≥ 1, we consider the problem of augmenting G by the smallest number of new edges so that the resulting graph has at least k edge-disjoint paths between v and W for every pair of a vertex v ∈ V and an area W ∈ W. So far this problem was shown to be NP-complete in the case of k = 1 and polynomially solvable in the case of k = 2. In this paper, we show that the problem for k ≥ 3 can be solved in O(m+n(k 3 +n 2 )(p+kn+n log n) log k+pkn 3 log (n/k)) time, where n = |V |, m = |{{u, v}|(u, v) ∈ E}|, and p = |W|. * An extended abstract of this paper was presented at 9th

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“…Nagamochi and Ibaraki [19] published the first deterministic min-cut algorithm that is not based on a flow algorithm, has the fastest running time of O(nm), but is rather complicated. Stoer and Wagner [20] published a min-cut algorithm with the same running time as Nagamochi and Ibaraki's, but is very simple.…”

Section: Man-c(h); Man-c(h′); Else Return G; End If Endmentioning

confidence: 99%

A Graph Clustering Algorithm Based on Minimum and Normalized Cut

Wang

Peng

et al. 2007

Computational Science – ICCS 2007

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Abstract. Clustering is the unsupervised classification of patterns into groups. In this paper, a clustering algorithm for weighted similarity graph is proposed based on minimum and normalized cut. The minimum cut is used as the stopping condition of the recursive algorithm, and the normalized cut is used to partition a graph into two subgraphs. The algorithm has the advantage of many existing algorithms: nonparametric clustering method, low polynomial complexity, and the provable properties. The algorithm is applied to image segmentation; the provable properties together with experimental results demonstrate that the algorithm performs well.

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Computing Edge-Connectivity in Multigraphs and Capacitated Graphs

Cited by 342 publications

References 5 publications

Fully dynamic maintenance of k-connectivity in parallel

Fully dynamic maintenance of k-connectivity in parallel

Minimum Augmentation of Edge-Connectivity between Vertices and Sets of Vertices in Undirected Graphs

A Graph Clustering Algorithm Based on Minimum and Normalized Cut

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