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6914 results in Algorithmics, Complexity, Computer Algebra, Computational Geometry

Page 36 of 346

200 Problems on Languages, Automata, and Computation
  • Edited by Filip Murlak, Damian Niwiński, Wojciech Rytter
  • Published online:
    14 April 2023
    Print publication:
    20 April 2023
  • Formal languages and automata have long been fundamental to theoretical computer science, but students often struggle to understand these concepts in the abstract. This book provides a rich source of compelling exercises designed to help students grasp the subject intuitively through practice. The text covers important topics such as finite automata, regular expressions, push-down automata, grammars, and Turing machines via a series of problems of increasing difficultly. Problems are organised by topic, many with multiple follow-ups, and each section begins with a short recap of the basic notions necessary to make progress. Complete solutions are given for all exercises, making the book well suited for self-study as well as for use as a course supplement. Developed over the course of the editors' two decades of experience teaching the acclaimed Automata, Formal Languages, and Computation course at the University of Warsaw, it is an ideal resource for students and instructors alike.

  • 2 - Basic Tools

  • from Part I - Preliminaries
  • Alan Frieze, Carnegie Mellon University, Pennsylvania, Michał Karoński, Adam Mickiewicz University, Poznań, Poland
  • Book:
    Random Graphs and Networks: A First Course
    Published online:
    02 March 2023
    Print publication:
    09 March 2023, pp 8-26

  • Summary

    Before reading and studying the results on random graphs included in the text one should become familiar with the basic rules of asymptotic computation, find leading terms in combinatorial expressions, choose suitable bounds for the binomials, as well as get acquainted with probabilistic tools needed to study tail bounds, i.e., the probability that a random variable exceeds (or is smaller than) some real value. This chapter offers the reader a short description of these important technical tools used throughout the text.

  • 7 - Small Subgraphs

  • from Part II - Erdős–Rényi–Gilbert Model
  • Alan Frieze, Carnegie Mellon University, Pennsylvania, Michał Karoński, Adam Mickiewicz University, Poznań, Poland
  • Book:
    Random Graphs and Networks: A First Course
    Published online:
    02 March 2023
    Print publication:
    09 March 2023, pp 85-92

  • Summary

    In this chapter, we see how many random edges are required to have a particular fixed size subgraph w.h.p. In addition, we will consider the distribution of the number of copies of strictly balanced subgraphs. From these general results, one can deduce thresholds for small trees, stars, cliques, bipartite cliques, and many other small subgraphs which play an important role in the analysis of the properties not only of classic random graphs but also in the interpretation of characteristic features of real-world networks. Computing the frequency of small subgraphs is a fundamental problem in network analysis, used across diverse domains: bioinformatics, social sciences, and infrastructure networks studies.

  • 5 - Vertex Degrees

  • from Part II - Erdős–Rényi–Gilbert Model
  • Alan Frieze, Carnegie Mellon University, Pennsylvania, Michał Karoński, Adam Mickiewicz University, Poznań, Poland
  • Book:
    Random Graphs and Networks: A First Course
    Published online:
    02 March 2023
    Print publication:
    09 March 2023, pp 64-77

  • Summary

    In this chapter, we study some typical properties of the degree sequence of a random graph. We begin by discussing the typical degrees in a sparse random graph, i.e., one with cn/2 edges for some positive constant c. We prove some results on the asymptotic distribution of degrees. We continue by looking at the typical values of the minimum and maximum degrees in dense random graphs, i.e., when edge probability p is constant. Given these properties of the degree sequence of dense graphs, we can then describe a simple canonical labeling algorithm that enables one to solve the graph isomorphism problem on a dense random graph.

  • 3 - Uniform and Binomial Random Graphs

  • from Part II - Erdős–Rényi–Gilbert Model
  • Alan Frieze, Carnegie Mellon University, Pennsylvania, Michał Karoński, Adam Mickiewicz University, Poznań, Poland
  • Book:
    Random Graphs and Networks: A First Course
    Published online:
    02 March 2023
    Print publication:
    09 March 2023, pp 29-44

  • Summary

    In this chapter, we formally introduce both Erdős–Rényi–Gilbert’s models, study their relationships, and establish conditions for their asymptotic equivalence. We also define and study the basic features of the asymptotic behavior of random graphs, i.e., the existence of thresholds for monotone graph properties.

Page 36 of 346