5,362 research outputs found

    Parallel O(log(n)) time edge-colouring of trees and Halin graphs

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    We present parallel O(log(n))-time algorithms for optimal edge colouring of trees and Halin graphs with n processors on a a parallel random access machine without write conflicts (P-RAM). In the case of Halin graphs with a maximum degree of three, the colouring algorithm automatically finds every Hamiltonian cycle of the graph

    Optimally edge-colouring outerplanar graphs is in NC

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    We prove that every outerplanar graph can be optimally edge-coloured in polylogarithmic time using a polynomial number of processors on a parallel random access machine without write conflicts (P-RAM)

    Local algorithms in (weakly) coloured graphs

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    A local algorithm is a distributed algorithm that completes after a constant number of synchronous communication rounds. We present local approximation algorithms for the minimum dominating set problem and the maximum matching problem in 2-coloured and weakly 2-coloured graphs. In a weakly 2-coloured graph, both problems admit a local algorithm with the approximation factor (Δ+1)/2(\Delta+1)/2, where Δ\Delta is the maximum degree of the graph. We also give a matching lower bound proving that there is no local algorithm with a better approximation factor for either of these problems. Furthermore, we show that the stronger assumption of a 2-colouring does not help in the case of the dominating set problem, but there is a local approximation scheme for the maximum matching problem in 2-coloured graphs.Comment: 14 pages, 3 figure

    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(n1−2/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

    The Semantics of Graph Programs

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    GP (for Graph Programs) is a rule-based, nondeterministic programming language for solving graph problems at a high level of abstraction, freeing programmers from handling low-level data structures. The core of GP consists of four constructs: single-step application of a set of conditional graph-transformation rules, sequential composition, branching and iteration. We present a formal semantics for GP in the style of structural operational semantics. A special feature of our semantics is the use of finitely failing programs to define GP's powerful branching and iteration commands
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