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A Linear Threshold for Uniqueness of Solutions to Random Jigsaw Puzzles

Part of: Graph theory Combinatorial probability

Published online by Cambridge University Press:  08 January 2019

ANDERS MARTINSSON
ANDERS MARTINSSON*
Affiliation:
Institute of Theoretical Computer Science, ETH Zürich, 8092 Zürich, Switzerland (e-mail: [email protected])
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Abstract

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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).

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 05C10: Planar graphs; geometric and topological aspects of graph theory
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 2 , March 2019 , pp. 287 - 302
Copyright
Copyright © Cambridge University Press 2019 

References

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