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Shotgun assembly of random jigsaw puzzles
Author(s) -
Bordenave Charles,
Feige Uriel,
Mossel Elchanan
Publication year - 2020
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20899
Subject(s) - jigsaw , vertex (graph theory) , grid , combinatorics , mathematics , computer science , algorithm , geometry , graph , mathematics education
We consider the shotgun assembly problem for a random jigsaw puzzle, introduced by Mossel and Ross (2015). Their model consists of a puzzle—an n × n grid, where each vertex is viewed as a center of a piece. Each of the four edges adjacent to a vertex is assigned one of q colors (corresponding to “jigs,” or cut shapes) uniformly at random. Unique assembly refers to there being only one puzzle (the original one) that is consistent with the collection of individual pieces. We show that for any ε >0, if q ≥ n 1+ ε , then unique assembly holds with high probability. The proof uses an algorithm that assembles the puzzle in time n Θ(1/ ε ) .22