相关论文: Shotgun Assembly of Random Jigsaw Puzzles

We consider a problem introduced by Mossel and Ross [Shotgun assembly of labeled graphs, arXiv:1504.07682]. Suppose a random $n\times n$ jigsaw puzzle is constructed by independently and uniformly choosing the shape of each "jig" from $q$…

In recent work by Mossel and Ross, it was asked how large $q$ has to be for a random jigsaw puzzle with $q$ different shapes of "jigs" to have exactly one solution. The jigs are assumed symmetric in the sense that two jigs of the same type…

A random jigsaw puzzle is constructed by arranging $n^2$ square pieces into an $n \times n$ grid and assigning to each edge of a piece one of $q$ available colours uniformly at random, with the restriction that touching edges receive the…

In the graph shotgun assembly problem, we are given the balls of radius $r$ around each vertex of a graph and asked to reconstruct the graph. We study the shotgun assembly of the Erd\H{o}s-R\'enyi random graph $\mathcal G(n,p)$ for a wide…

A colouring of the edges of an $n \times n$ grid is said to be \emph{reconstructible} if the colouring is uniquely determined by the multiset of its $n^2$ \emph{tiles}, where the tile corresponding to a vertex of the grid specifies the…

Mossel and Ross (2019) introduce the shotgun assembly problem for random graphs: what radius $R$ ensures that the random graph $G$ can be uniquely recovered from its list of rooted $R$-neighborhoods, with high probability? Here we consider…

We consider the problem of reconstructing graphs or labeled graphs from neighborhoods of a given radius r. Special instances of this problem include the well known: DNA shotgun assembly; the lesser-known: neural network reconstruction; and…

We study the shotgun assembly problem for the lattice labeling model, where i.i.d. uniform labels are assigned to each vertex in a $d$-dimensional box of side length $n$. We wish to recover the labeling configuration on the whole box given…

Mossel and Ross raised the question of when a random colouring of a graph can be reconstructed from local information, namely the colourings (with multiplicity) of balls of given radius. In this paper, we are concerned with random…

Graph shotgun assembly refers to the problem of reconstructing a graph from a collection of local neighborhoods. In this paper, we consider shotgun assembly of \ER random graphs $G(n, p_n)$, where $p_n = n^{-\alpha}$ for $0 < \alpha < 1$.…

In a recent work, Huang and Tikhomirov considered the shotgun assembly for Erd\H os-R\'enyi graphs $\mathcal G(n,p_n)$ with $p_n=n^{-\alpha}$, and showed that the graph is reconstructable if $0<\alpha < \frac{1}{2}$ and not reconstructable…

In the shotgun assembly problem for a graph, we are given the empirical profile for rooted neighborhoods of depth $r$ (up to isomorphism) for some $r\geq 1$ and we wish to recover the underlying graph up to isomorphism. When the underlying…

We prove the computational intractability of rotating and placing $n$ square tiles into a $1 \times n$ array such that adjacent tiles are compatible--either equal edge colors, as in edge-matching puzzles, or matching tab/pocket shapes, as…

In a recent paper [6], J. Gaudio and E. Mossel studied the shotgun assembly of the Erd\H{o}s-R\'enyi graph $\mathcal G(n,p_n)$ with $p_n=n^{-\alpha}$, and showed that the graph is reconstructable form its $1$-neighbourhoods if $0<\alpha <…

We study the complexity of symmetric assembly puzzles: given a collection of simple polygons, can we translate, rotate, and possibly flip them so that their interior-disjoint union is line symmetric? On the negative side, we show that the…

We study coverage processes in which each draw reveals a subset of $[n]$, and the goal is to determine the expected number of draws until all items are seen at least once. A classical example is the Coupon Collector's Problem, where each…

Jigsaw percolation is a nonlocal process that iteratively merges connected clusters in a deterministic "puzzle graph" by using connectivity properties of a random "people graph" on the same set of vertices. We presume the Erdos--Renyi…

Cohn and Umans proposed a framework for developing fast matrix multiplication algorithms based on the embedding computation in certain groups algebras. In subsequent work with Kleinberg and Szegedy, they connected this to the search for…

Jigsaw puzzle solving, the problem of constructing a coherent whole from a set of non-overlapping unordered visual fragments, is fundamental to numerous applications, and yet most of the literature of the last two decades has focused thus…

We present a framework for the design of optimal assembly algorithms for shotgun sequencing under the criterion of complete reconstruction. We derive a lower bound on the read length and the coverage depth required for reconstruction in…