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6974 results in Algorithmics, Complexity, Computer Algebra, Computational Geometry

Page 81 of 349

Network Flow Algorithms
  • David P. Williamson
  • Published online:
    21 August 2019
    Print publication:
    05 September 2019
  • Network flow theory has been used across a number of disciplines, including theoretical computer science, operations research, and discrete math, to model not only problems in the transportation of goods and information, but also a wide range of applications from image segmentation problems in computer vision to deciding when a baseball team has been eliminated from contention. This graduate text and reference presents a succinct, unified view of a wide variety of efficient combinatorial algorithms for network flow problems, including many results not found in other books. It covers maximum flows, minimum-cost flows, generalized flows, multicommodity flows, and global minimum cuts and also presents recent work on computing electrical flows along with recent applications of these flows to classical problems in network flow theory.

  • Expansion of Percolation Critical Points for Hamming Graphs

  • Part of
  • Lorenzo Federico, Remco Van Der Hofstad, Frank Den Hollander, Tim Hulshof
  • Journal:
    Combinatorics, Probability and Computing / Volume 29 / Issue 1 / January 2020
    Published online by Cambridge University Press:
    05 August 2019, pp. 68-100

  • The Hamming graph H(d, n) is the Cartesian product of d complete graphs on n vertices. Let ${m=d(n-1)}$ be the degree and $V = n^d$ be the number of vertices of H(d, n). Let $p_c^{(d)}$ be the critical point for bond percolation on H(d, n). We show that, for $d \in \mathbb{N}$ fixed and $n \to \infty$,

    $$p_c^{(d)} = {1 \over m} + {{2{d^2} - 1} \over {2{{(d - 1)}^2}}}{1 \over {{m^2}}} + O({m^{ - 3}}) + O({m^{ - 1}}{V^{ - 1/3}}),$$
    which extends the asymptotics found in [10] by one order. The term $O(m^{-1}V^{-1/3})$ is the width of the critical window. For $d=4,5,6$ we have $m^{-3} =O(m^{-1}V^{-1/3})$, and so the above formula represents the full asymptotic expansion of $p_c^{(d)}$. In [16] we show that this formula is a crucial ingredient in the study of critical bond percolation on H(d, n) for $d=2,3,4$. The proof uses a lace expansion for the upper bound and a novel comparison with a branching random walk for the lower bound. The proof of the lower bound also yields a refined asymptotics for the susceptibility of a subcritical Erdös–Rényi random graph.

  • 2 - Monoidal Finite-State Automata

  • from Part I - Formal Background
  • Stoyan Mihov, Klaus U. Schulz, Ludwig-Maximilians-Universität Munchen
  • Book:
    Finite-State Techniques
    Published online:
    29 July 2019
    Print publication:
    01 August 2019, pp 23-42

  • Summary

    As a starting point we study finite-state automata, which represent the simplest devices for recognizing languages. The theory of finite-state automata has been described in numerous textbooks both from a computational and an algebraic point of view. Here we immediately look at the more general concept of a monoidal finite-state automaton, and the focus of this chapter is general constructions and results for finite-state automata over arbitrary monoids and monoidal languages. Refined pictures for the special (and more standard) cases where we only consider free monoids or Cartesian products of monoids will be given later.

  • 1 - Formal Preliminaries

  • from Part I - Formal Background
  • Stoyan Mihov, Klaus U. Schulz, Ludwig-Maximilians-Universität Munchen
  • Book:
    Finite-State Techniques
    Published online:
    29 July 2019
    Print publication:
    01 August 2019, pp 3-22

  • Summary

    The aim of this chapter is twofold. First, we recall a collection of basic mathematical notions that are needed for the discussions of the following chapters. Second, we have a first, still purely mathematical, look at the central topics of the book: languages, relations and functions between strings, as well as important operations on languages, relations and functions. We also introduce monoids, a class of algebraic structures that gives an abstract view on strings, languages, and relations.

Page 81 of 349