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7008 results in Algorithmics, Complexity, Computer Algebra, Computational Geometry

Page 191 of 351

  • Randomized Greedy Algorithms for Independent Sets and Matchings in Regular Graphs: Exact Results and Finite Girth Corrections

  • DAVID GAMARNIK, DAVID A. GOLDBERG
  • Journal:
    Combinatorics, Probability and Computing / Volume 19 / Issue 1 / January 2010
    Published online by Cambridge University Press:
    22 June 2009, pp. 61-85

  • We derive new results for the performance of a simple greedy algorithm for finding large independent sets and matchings in constant-degree regular graphs. We show that for r-regular graphs with n nodes and girth at least g, the algorithm finds an independent set of expected cardinalitywhere f(r) is a function which we explicitly compute. A similar result is established for matchings. Our results imply improved bounds for the size of the largest independent set in these graphs, and provide the first results of this type for matchings. As an implication we show that the greedy algorithm returns a nearly perfect matching when both the degree r and girth g are large. Furthermore, we show that the cardinality of independent sets and matchings produced by the greedy algorithm in arbitrary bounded-degree graphs is concentrated around the mean. Finally, we analyse the performance of the greedy algorithm for the case of random i.i.d. weighted independent sets and matchings, and obtain a remarkably simple expression for the limiting expected values produced by the algorithm. In fact, all the other results are obtained as straightforward corollaries from the results for the weighted case.

  • The Linus Sequence

  • PAUL BALISTER, STEVE KALIKOW, AMITES SARKAR
  • Journal:
    Combinatorics, Probability and Computing / Volume 19 / Issue 1 / January 2010
    Published online by Cambridge University Press:
    22 June 2009, pp. 21-46

  • Define the Linus sequence Ln for n ≥ 1 as a 0–1 sequence with L1 = 0, and Ln chosen so as to minimize the length of the longest immediately repeated block Ln−2r+1 ⋅⋅⋅ Ln−r = Ln−r+1 ⋅⋅⋅ Ln. Define the Sally sequence Sn as the length r of the longest repeated block that was avoided by the choice of Ln. We prove several results about these sequences, such as exponential decay of the frequency of highly periodic subwords of the Linus sequence, zero entropy of any stationary process obtained as a limit of word frequencies in the Linus sequence and infinite average value of the Sally sequence. In addition we make a number of conjectures about both sequences.

  • Preface

  • Devdatt P. Dubhashi, Chalmers University of Technology, Gothenberg, Alessandro Panconesi, Università degli Studi di Roma 'La Sapienza', Italy
  • Book:
    Concentration of Measure for the Analysis of Randomized Algorithms
    Published online:
    19 October 2009
    Print publication:
    15 June 2009, pp xi-xvi

  • Summary

    The aim of this book is to provide a body of tools for establishing concentration of measure that is accessible to researchers working in the design and analysis of randomized algorithms.

    Concentration of measure refers to the phenomenon that a function of a large number of random variables tends to concentrate its values in a relatively narrow range (under certain conditions of smoothness of the function and under certain conditions of the dependence amongst the set of random variables). Such a result is of obvious importance to the analysis of randomized algorithms: for instance, the running time of such an algorithm can then be guaranteed to be concentrated around a pre-computed value. More generally, various other parameters measuring the performance of randomized algorithms can be provided tight guarantees via such an analysis.

    In a sense, the subject of concentration of measure lies at the core of modern probability theory as embodied in the laws of large numbers, the central limit theorem and, in particular, the theory of large deviations [26]. However, these results are asymptotic: they refer to the limit as the number of variables n goes to infinity, for example. In the analysis of algorithms, we typically require quantitative estimates that are valid for finite (though large) values of n. The earliest such results can be traced back to the work of Azuma, Chernoff and Hoeffding in the 1950s. Subsequently, there have been steady advances, particularly in the classical setting of martingales. In the last couple of decades, these methods have taken on renewed interest, driven by applications in algorithms and optimisation. Also several new techniques have been developed.

Page 191 of 351