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Sensitivity Analysis of Minimum Spanning Trees in SubInverseAckermann Time
, 2015
"... We present a deterministic algorithm for computing the sensitivity of a minimum spanning tree (MST) or shortest path tree in O(m logα(m,n)) time, where α is the inverseAckermann function. This improves upon a long standing bound of O(mα(m,n)) established by Tarjan. Our algorithms are based on an e ..."
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We present a deterministic algorithm for computing the sensitivity of a minimum spanning tree (MST) or shortest path tree in O(m logα(m,n)) time, where α is the inverseAckermann function. This improves upon a long standing bound of O(mα(m,n)) established by Tarjan. Our algorithms are based on an efficient splitfindmin data structure, which maintains a collection of sequences of weighted elements that may be split into smaller subsequences. As far as we are aware, our splitfindmin algorithm is the first with superlinear but subinverseAckermann complexity. We also give a reduction from MST sensitivity to the MST problem itself. Together with the randomized linear time MST algorithm of Karger, Klein, and Tarjan, this gives another randomized linear time MST sensitivity algorithm.
Fast Random Graph Generation ∗
"... Today, several database applications call for the generation of random graphs. A fundamental, versatile random graph model adopted for that purpose is the ErdősRényi Γv,p model. This model can be used for directed, undirected, and multipartite graphs, with and without selfloops; it induces algorit ..."
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Today, several database applications call for the generation of random graphs. A fundamental, versatile random graph model adopted for that purpose is the ErdősRényi Γv,p model. This model can be used for directed, undirected, and multipartite graphs, with and without selfloops; it induces algorithms for both graph generation and sampling, hence is useful not only in applications necessitating the generation of random structures but also for simulation, sampling and in randomized algorithms. However, the commonly advocated algorithm for random graph generation under this model performs poorly when generating large graphs, and fails to make use of the parallel processing capabilities of modern hardware. In this paper, we propose PPreZER, an alternative, data parallel algorithm for random graph generation under the ErdősRényi model, designed and implemented in a graphics processing unit (GPU). We are led to thischiefcontributionofoursviaasuccessionofsevenintermediary algorithms, both sequential and parallel. Our extensive experimental study shows an average speedup of 19 for PPreZER with respect to the baseline algorithm.
A Simpler Implementation and Analysis of Chazelle’s Soft Heaps
 In Proc. of the 19th ACMSIAM Symposium on Discrete Algorithms
, 2009
"... Chazelle (JACM 47(6), 2000) devised an approximate meldable priority queue data structure, called Soft Heaps, and used it to obtain the fastest known deterministic comparisonbased algorithm for computing minimum spanning trees, as well as some new algorithms for selection and approximate sorting pr ..."
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Chazelle (JACM 47(6), 2000) devised an approximate meldable priority queue data structure, called Soft Heaps, and used it to obtain the fastest known deterministic comparisonbased algorithm for computing minimum spanning trees, as well as some new algorithms for selection and approximate sorting problems. If n elements are inserted into a collection of soft heaps, then up to εn of the elements still contained in these heaps, for a given error parameter ε, maybecorrupted, i.e., have their keys artificially increased. In exchange for allowing these corruptions, each soft heap operation is performed in O(log 1 ε) amortized time. Chazelle’s soft heaps are derived from the binomial heaps data structure in which each priority queue is composed of a collection of binomial trees. We describe a simpler and more direct implementation of soft heaps in which each priority queue is composed of a collection of standard binary trees. Our implementation has the advantage that no cleanup operations similar to the ones used in Chazelle’s implementation are required. We also present a concise and unified potentialbased amortized analysis of the new implementation. 1
A QuasiPolynomial Time Partition Oracle for Graphs with an Excluded Minor
, 2013
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