Trans-dichotomous algorithms for minimum spanning trees and shortest paths
Journal of Computer and System Sciences - Special issue: 31st IEEE conference on foundations of computer science, Oct. 22–24, 1990
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Journal of the ACM (JACM)
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STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
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Journal of the ACM (JACM)
The discrepancy method: randomness and complexity
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SIAM Journal on Computing
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ICALP '00 Proceedings of the 27th International Colloquium on Automata, Languages and Programming
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FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
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FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
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IEEE Annals of the History of Computing
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APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Introduction to testing graph properties
Property testing
Introduction to testing graph properties
Property testing
Proceedings of the forty-third annual ACM symposium on Theory of computing
Introduction to testing graph properties
Studies in complexity and cryptography
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Studies in complexity and cryptography
Facility location in sublinear time
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
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We present a probabilistic algorithm that, given a connected graph G (represented by adjacency lists) of maximum degree d, with edge weights in the set {1; ... ; ω}, and given a parameter 0 O(dωƐċ2 log ω/Ɛ) the weight of the minimum spanning tree of G with a relative error of at most Ɛ. Note that the running time does not depend on the number of vertices in G. We also prove a nearly matching lower bound of Ω(dωƐċ2) on the probe and time complexity of any approximation algorithm for MST weight. The essential component of our algorithm is a procedure for estimating in time O(dƐċ2 logƐċ1) the number of connected components of an unweighted graph to within an additive error of ∈n. The time bound is shown to be tight up to within the log Ɛċ1 factor. Our connected-components algorithm picks O(1/∈2) vertices in the graph and then grows "local spanning trees" whose sizes are specified by a stochastic process. From the local information collected in this way, the algorithm is able to infer, with high confidence, an estimate of the number of connected components. We then show how estimates on the number of components in various subgraphs of G can be used to estimate the weight of its MST.