Augmenting Undirected Edge Connectivity in Õ(n2) Time

Benczúr, András A.; Karger, David R.

doi:10.1006/jagm.2000.1093

Cited by 23 publications

(19 citation statements)

References 17 publications

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“…Hence we only have to consider the case where f is symmetric and X and Y do not cross each other (i.e., X and Y intersect each other). In this case, by symmetry of f , the intersecting subsets X and Y satisfy f ( 3 )}, and d + : 2 V → + be the cut function such that d + (X) denotes the number of arcs outgoing from X to V − X. The set function d + is known to be fully submodular.…”

Section: Preliminariesmentioning

confidence: 98%

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Minimum Degree Orderings

Nagamochi

2008

Algorithmica

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It is known that, given an edge-weighted graph, a maximum adjacency ordering (MA ordering) of vertices can find a special pair of vertices, called a pendent pair, and that a minimum cut in a graph can be found by repeatedly contracting a pendent pair, yielding one of the fastest and simplest minimum cut algorithms. In this paper, we provide another ordering of vertices, called a minimum degree ordering (MD ordering) as a new fundamental tool to analyze the structure of graphs. We prove that an MD ordering finds a different type of special pair of vertices, called a flat pair, which actually can be obtained as the last two vertices after repeatedly removing a vertex with the minimum degree. By contracting flat pairs, we can find not only a minimum cut but also all extreme subsets of a given graph. These results can be extended to the problem of finding extreme subsets in symmetric submodular set functions.

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Section: Preliminariesmentioning

confidence: 98%

“…Currently it is known [16] that all extreme sets in an edge-weighted graph can be computed in O(mn + n 2 log n) time. Benczúr and Karger [3] gave an O(n 2 log n) time randomized algorithm for computing extreme sets with high probability.…”

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confidence: 99%

Minimum Degree Orderings

Nagamochi

2008

Algorithmica

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“…There are several existing algorithms for this task (e.g., [14,5,7]) in undirected and directed graphs, but most algorithms only preserve global edge connectivity (i.e., the value of the global min-cut). Our results in this section apply to the general setting where all pairs edge connectivities are preserved.…”

Section: Edge Splitting-offmentioning

confidence: 99%

Graph Connectivities, Network Coding, and Expander Graphs

Cheung

Lau

Leung

2011

2011 IEEE 52nd Annual Symposium on Foundations of Computer Science

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Abstract. We present a new algebraic formulation for computing edge connectivities in a directed graph, using ideas developed in network coding. This reduces the problem of computing edge connectivities to solving systems of linear equations, thus allowing us to use tools in linear algebra to design new algorithms. Using the algebraic formulation, we obtain faster algorithms for computing single source edge connectivities and all pairs edge connectivities. In some settings, the amortized time to compute the edge connectivity for one pair is sublinear. Through this connection, we have also found an interesting use of expanders and superconcentrators to design fast algorithms for some graph connectivity problems.Key words. graph connectivities, network coding, expander graphs, linear equations, randomized algorithms AMS subject classifications. 05C40, 05C50, 05C85, 68Q25, 68W20 DOI. 10.1137/1108449701. Introduction. Graph connectivity is a basic concept that measures the reliability and efficiency of a graph. The edge connectivity from vertex s to vertex t is defined as the size of a minimum s-t cut or, equivalently, the maximum number of edge disjoint paths from s to t. Computing edge connectivities is a classical and wellstudied problem in combinatorial optimization. Most known algorithms for solving this problem are based on network flow techniques (see, e.g., [34]).Even and Tarjan [12] introduced the fastest algorithm to compute s-t edge connectivity in a simple directed graph and running in O(min{m 1/2 , n 2/3 } · m) time, where m is the number of edges and n is the number of vertices. To compute the edge connectivities for many pairs, however, it is not known how to do it faster than computing edge connectivity for each pair separately, even when the pairs share the source or the sink. For instance, it is not known how to compute all pairs edge connectivities faster than computing s-t edge connectivity for Θ(n 2 ) pairs. This is in contrast to the problem in undirected graphs, where all pairs edge connectivities can be computed inÕ(mn) time by constructing a Gomory-Hu tree [6].Network coding is an innovative method for transmitting information in a computer network. The fundamental result is a max-information-flow min-cut theorem for multicasting [1]: if the edge connectivity from the source vertex s to each sink vertex t i is at least k, then one can transmit k units of information to all sink vertices simultaneously by performing encoding and decoding at the vertices. An elegant algebraic framework has been developed for constructing efficient network coding schemes for multicasting [29,26].In this paper, we use the techniques developed in network coding to obtain a new algebraic formulation for computing edge connectivities. This reduces the problem

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“…between every pair of unconnected vertices there exist at least k vertex-disjoint paths. Edge connectivity augmentation algorithms like [38,8] can be used to compute the minimum set of additional edges, required to make a graph k-connected. Additionally, the union of k edge-disjoint spanning trees, will result in a k-connected graph, since it is a packing of k paths for every vertex pair.…”

Section: Bereitgestellt Von | Universitätsbibliothek Ilmenaumentioning

confidence: 99%