Shotgun assembly of random jigsaw puzzles. Issue 4 (22nd January 2020)

• Record Type: Journal Article
• Authors: Bordenave, Charles; Feige, Uriel; Mossel, Elchanan
• Is Part Of: Random structures & algorithms. Volume 56:Issue 4(2020)
• Page Start: 998
• Page End: 1015
• Publication Date: 2020-01-22
• Subjects: Unique reconstruction -- Planted solution -- Exact recovery; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20899
• Languages: English
• ISSNs: 1042-9832

Abstract

Abstract : 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