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On Clusterings: Good, Bad and Spectral
, 2000
"... We motivate and develop a natural bicriteria measure for assessing the quality of a clustering which avoids the drawbacks of existing measures. A simple recursive heuristic has polylogarithmic worstcase guarantees under the new measure. The main result of the paper is the analysis of a popular spe ..."
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Cited by 254 (12 self)
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We motivate and develop a natural bicriteria measure for assessing the quality of a clustering which avoids the drawbacks of existing measures. A simple recursive heuristic has polylogarithmic worstcase guarantees under the new measure. The main result of the paper is the analysis of a popular spectral algorithm. One variant of spectral clustering turns out to have effective worstcase guarantees
Better Random Sampling Algorithms for Flows in Undirected Graphs
, 1997
"... We present better random sampling algorithms for maximum flows in undirected graphs. Our algorithms apply to capacitated or uncapacitated graphs, and find a maximum flow of value v in ~ O( p mnv) time. This improves on a previous bound of ~ O(m 2=3 n 1=3 v) given by the author recently, which ..."
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Cited by 12 (4 self)
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We present better random sampling algorithms for maximum flows in undirected graphs. Our algorithms apply to capacitated or uncapacitated graphs, and find a maximum flow of value v in ~ O( p mnv) time. This improves on a previous bound of ~ O(m 2=3 n 1=3 v) given by the author recently, which in turn improved on the O(mv) time bound for a typical augmenting path algorithm. In uncapacitated graphs without parallel edges, the bound is no worse than ~ O(n 5=2 ). We give another algorithm that finds a (1 \Gamma ffl) times maximum flow in time ~ O(m p n=ffl), regardless of v. 1 Introduction. Random sampling has been a useful tool for solving cut problems in undirected graphs. In previous work [Kar97a], this author showed that randomly choosing edges from a graph yields a sampled graph in which every cut is close to its expected value with high probability. This led to algorithms for approximating [Kar97a] and exactly finding [Kar96] global mincuts in nearlinear time with hig...
Randomized Approximation Schemes for Cuts and Flows in Capacitated Graphs
, 2011
"... We describe random sampling techniques for approximately solving problems that involve cuts and flows in graphs. We give a nearlineartime randomized combinatorial construction that transforms any graph on n vertices into an O(n log n)edge graph on the same vertices whose cuts have approximately t ..."
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Cited by 9 (0 self)
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We describe random sampling techniques for approximately solving problems that involve cuts and flows in graphs. We give a nearlineartime randomized combinatorial construction that transforms any graph on n vertices into an O(n log n)edge graph on the same vertices whose cuts have approximately the same value as the original graph’s. In this new graph, for example, we can run the Õ(m3/2)time maximum flow algorithm of Goldberg and Rao to find an s– t minimum cut in Õ(n3/2) time. This corresponds to a (1 + ɛ)times minimum s–t cut in the original graph. A related approach leads to a randomized divide and conquer algorithm producing an approximately maximum flow in Õ(m √ n) time. Our algorithm is also used to improve the running time of sparsest cut algorithms from Õ(mn) to Õ(n²). Our approach also accelerates several other recent cut and flow algorithms. Our algorithms are based on a general theorem analyzing the concentration of cut values near their expectation in random graphs.