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

Page 90 of 349

  • A Linear Threshold for Uniqueness of Solutions to Random Jigsaw Puzzles

  • Part of
  • ANDERS MARTINSSON
  • Journal:
    Combinatorics, Probability and Computing / Volume 28 / Issue 2 / March 2019
    Published online by Cambridge University Press:
    08 January 2019, pp. 287-302
    • Article
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  • We consider a problem introduced by Mossel and Ross (‘Shotgun assembly of labeled graphs’, arXiv:1504.07682). Suppose a random n × n jigsaw puzzle is constructed by independently and uniformly choosing the shape of each ‘jig’ from q possibilities. We are given the shuffled pieces. Then, depending on q, what is the probability that we can reassemble the puzzle uniquely? We say that two solutions of a puzzle are similar if they only differ by a global rotation of the puzzle, permutation of duplicate pieces, and rotation of rotationally symmetric pieces. In this paper, we show that, with high probability, such a puzzle has at least two non-similar solutions when 2 ⩽ q ⩽ 2e−1/2n, all solutions are similar when q ⩾ (2+ϵ)n, and the solution is unique when q = ω(n).

Kernelization
  • Theory of Parameterized Preprocessing
  • Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, Meirav Zehavi
  • Published online:
    03 January 2019
    Print publication:
    10 January 2019
  • Preprocessing, or data reduction, is a standard technique for simplifying and speeding up computation. Written by a team of experts in the field, this book introduces a rapidly developing area of preprocessing analysis known as kernelization. The authors provide an overview of basic methods and important results, with accessible explanations of the most recent advances in the area, such as meta-kernelization, representative sets, polynomial lower bounds, and lossy kernelization. The text is divided into four parts, which cover the different theoretical aspects of the area: upper bounds, meta-theorems, lower bounds, and beyond kernelization. The methods are demonstrated through extensive examples using a single data set. Written to be self-contained, the book only requires a basic background in algorithmics and will be of use to professionals, researchers and graduate students in theoretical computer science, optimization, combinatorics, and related fields.

Page 90 of 349