A Linear Threshold for Uniqueness of Solutions to Random Jigsaw Puzzles

Martinsson, Anders

doi:10.1017/s0963548318000391

Cited by 9 publications

(11 citation statements)

References 6 publications

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“…Finally in two independent recent works [3,9], it is shown that if q ≥ (2 + )n then the puzzle has unique edge assembly and if q ≥ (n), it has unique vertex assembly. The proofs of Balister, Bollobás, and Narayanan [3] and Martinsson [9] build on the ideas presented in the current paper. In particular the sharper results are obtained by a better analysis of k × k windows.…”

Section: Concurrent and Follow Up Workmentioning

confidence: 98%

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Shotgun assembly of random jigsaw puzzles

Bordenave

Feige

Mossel³

2020

Random Struct Algorithms

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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 > 0, if q ≥ n 1+ , then unique assembly holds with high probability. The proof uses an algorithm that assembles the puzzle in time n Θ(1∕ ) .

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Section: Concurrent and Follow Up Workmentioning

confidence: 98%

“…In particular the sharper results are obtained by a better analysis of k × k windows. However, since the results of Balister, Bollobás, and Narayanan [3], Martinsson [9] involve polynomially large windows they do not provide a polynomial time algorithm for assembly.…”

Section: Concurrent and Follow Up Workmentioning

confidence: 99%

Shotgun assembly of random jigsaw puzzles

Bordenave

Feige

Mossel³

2020

Random Struct Algorithms

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“…We remark that this approach of combining probabilistic and structural elements is very natural and has been applied before, for example, in previous work (see [2,6,16,21]) considering the jigsaw puzzle. For jigsaws, a general approach has been to consider partial reconstructions of the edge-coloring in large "windows" of the form v…”

Section: Theorem 18 Let S(n) Be a Function With S(n) → ∞ And S(n) = O(n) As N → ∞ Then There Exists A Constant C (Which May Depend On S(nmentioning

confidence: 99%

“…Mossel and Ross also considered the problem of reconstructing randomly colored trees, randomly colored lattices in any fixed number of dimensions, and the random jigsaw puzzle problem, in which the edges of the n × n square lattice are randomly colored with q colors, and the problem is to determine for which q it is possible to reconstruct the original jigsaw from the collection of 1-balls. The random jigsaw puzzle problem has since been studied by Bordenave, Feige, and Mossel [6], Nenadov, Pfister, and Steger [21], Balister, Bollobás, and Narayanan [2], and by Martinsson [16].…”

Section: Introductionmentioning

confidence: 99%

Shotgun reconstruction in the hypercube

Przykucki

Roberts

Scott

2021

Random Struct Algorithms

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Mossel and Ross raised the question of when a random coloring of a graph can be reconstructed from local information, namely, the colorings (with multiplicity) of balls of given radius. In this article, we are concerned with random 2-colorings of the vertices of the n-dimensional hypercube, or equivalently random Boolean functions. In the worst case, balls of diameter Ω(n) are required to reconstruct. However, the situation for random colorings is dramatically different: we show that almost every 2-coloring can be reconstructed from the multiset of colorings of balls of radius 2. Furthermore, we show that for q ≥ n 2+𝜖 , almost every q-coloring can be reconstructed from the multiset of colorings of 1-balls.

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“…Remark. After the results in this paper were proved (in November 2016), but before this paper was completed, Martinsson [8], working independently, also announced (in January 2017) a proof of a result analogous to Theorem 1.2 in a very closely related model (and with a more reasonable constant in the 1-statement). We briefly point out that while the respective 0-statements are established in essentially the same fashion both here and in [8], the estimates needed to prove the respective 1-statements are established by quite different approaches.…”

Section: Introductionmentioning

confidence: 96%

Reconstructing Random Jigsaws

Balister

Bollobás

Narayanan³

2018

Multiplex and Multilevel Networks

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A colouring of the edges of an n×n grid is said to be reconstructible if the colouring is uniquely determined by the multiset of its n 2 tiles, where the tile corresponding to a vertex of the grid specifies the colours of the edges incident to that vertex in some fixed order. In 2015, Mossel and Ross asked the following question: if the edges of an n × n grid are coloured independently and uniformly at random using q = q(n) different colours, then is the resulting colouring reconstructible with high probability? From below, Mossel and Ross showed that such a colouring is not reconstructible when q = o(n 2/3 ) and from above, Bordenave, Feige and Mossel and Nenadov, Pfister and Steger independently showed, for any fixed ε > 0, that such a colouring is reconstructible when q ≥ n 1+ε . Here, we improve on these results and prove the following: there exist absolute constants C, c > 0 such that, as n → ∞, the probability that a random colouring as above is reconstructible tends to 1 if q ≥ Cn and to 0 if q ≤ cn.Proof of the 0-statement in Theorem 1.2. Recall that J (n, q) is the set of all (n, q)jigsaws, D(n, q) is the family of all multisets of size n 2 whose elements are chosen from [q] 4 , and D : J (n, q) → D(n, q) is the map sending a jigsaw J to its deck D(J).Let J R (n, q) ⊂ J (n, q) denote the set of all reconstructible jigsaws, i.e., jigsaws J such that D −1 (D(J)) = {J}. Since D : J R (n, q) → D(n, q) is an injection, |J R (n, q)| ≤ |D(n, q)|. Consequently, we have P(J(n, q) is reconstructible) = |J R (n, q)|/|J (n, q)| ≤ |D(n, q)|/|J (n, q)|. 6Proposition 4.7 . If (A, h) is a large v-template of type (δ, r 1 , r 2 ), then we have γ(h) ≥ δ/20 and γ(h) ≥ 2r 1 + r 2 /2 − 2r 1 /(2k + 1).Proof. We shall use Proposition 4.2 to bound γ(h) from below. We will estimate the size of both the vertex set and the edge set of G h .Since G h contains one edge for each h-split edge, it is easy to see that the edge set of G h has size at least A * , so |E(G h )| ≥ |A * | = |A| − (8k + 4) as |∂[−k, k] 2 | = 8k + 4. Now, since A contains an edge incident to (0, 0), is dual-connected and also contains 13

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A Linear Threshold for Uniqueness of Solutions to Random Jigsaw Puzzles

Cited by 9 publications

References 6 publications

Shotgun assembly of random jigsaw puzzles

Shotgun assembly of random jigsaw puzzles

Shotgun reconstruction in the hypercube

Reconstructing Random Jigsaws

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