4,614 research outputs found

    Online Multi-Coloring with Advice

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    We consider the problem of online graph multi-coloring with advice. Multi-coloring is often used to model frequency allocation in cellular networks. We give several nearly tight upper and lower bounds for the most standard topologies of cellular networks, paths and hexagonal graphs. For the path, negative results trivially carry over to bipartite graphs, and our positive results are also valid for bipartite graphs. The advice given represents information that is likely to be available, studying for instance the data from earlier similar periods of time.Comment: IMADA-preprint-c

    The Sampling-and-Learning Framework: A Statistical View of Evolutionary Algorithms

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    Evolutionary algorithms (EAs), a large class of general purpose optimization algorithms inspired from the natural phenomena, are widely used in various industrial optimizations and often show excellent performance. This paper presents an attempt towards revealing their general power from a statistical view of EAs. By summarizing a large range of EAs into the sampling-and-learning framework, we show that the framework directly admits a general analysis on the probable-absolute-approximate (PAA) query complexity. We particularly focus on the framework with the learning subroutine being restricted as a binary classification, which results in the sampling-and-classification (SAC) algorithms. With the help of the learning theory, we obtain a general upper bound on the PAA query complexity of SAC algorithms. We further compare SAC algorithms with the uniform search in different situations. Under the error-target independence condition, we show that SAC algorithms can achieve polynomial speedup to the uniform search, but not super-polynomial speedup. Under the one-side-error condition, we show that super-polynomial speedup can be achieved. This work only touches the surface of the framework. Its power under other conditions is still open

    Algebraic Methods in the Congested Clique

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    In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an O(n12/ω)O(n^{1-2/\omega}) round matrix multiplication algorithm, where ω<2.3728639\omega < 2.3728639 is the exponent of matrix multiplication. In conjunction with known techniques from centralised algorithmics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include: -- triangle and 4-cycle counting in O(n0.158)O(n^{0.158}) rounds, improving upon the O(n1/3)O(n^{1/3}) triangle detection algorithm of Dolev et al. [DISC 2012], -- a (1+o(1))(1 + o(1))-approximation of all-pairs shortest paths in O(n0.158)O(n^{0.158}) rounds, improving upon the O~(n1/2)\tilde{O} (n^{1/2})-round (2+o(1))(2 + o(1))-approximation algorithm of Nanongkai [STOC 2014], and -- computing the girth in O(n0.158)O(n^{0.158}) rounds, which is the first non-trivial solution in this model. In addition, we present a novel constant-round combinatorial algorithm for detecting 4-cycles.Comment: This is work is a merger of arxiv:1412.2109 and arxiv:1412.266
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