相关论文: An optimal algorithm to generate tilings
The move from hand-designed features to learned features in machine learning has been wildly successful. In spite of this, optimization algorithms are still designed by hand. In this paper we show how the design of an optimization algorithm…
The lattice $A_n^*$ is an important lattice because of its covering properties in low dimensions. Clarkson \cite{Clarkson1999:Anstar} described an algorithm to compute the nearest lattice point in $A_n^*$ that requires $O(n\log{n})$…
Mining and exploring databases should provide users with knowledge and new insights. Tiles of data strive to unveil true underlying structure and distinguish valuable information from various kinds of noise. We propose a novel Boolean…
We introduce an algorithm for computing closure systems derived from a family of implications on a set. Semilattices presentations are explored and used in conjunction with the algorithm to compute various types of lattices freely generated…
Many problems in computational geometry are not stated in graph-theoretic terms, but can be solved efficiently by constructing an auxiliary graph and performing a graph-theoretic algorithm on it. Often, the efficiency of the algorithm…
We give a generalized definition of stretch that simplifies the efficient construction of low-stretch embeddings suitable for graph algorithms. The generalization, based on discounting highly stretched edges by taking their $p$-th power for…
This paper proposes a fully-automatic, text-guided generative method for producing perfectly-repeating, periodic, tile-able 2D imagery, such as the one seen on floors, mosaics, ceramics, and the work of M.C. Escher. In contrast to square…
The reduction of covering decision systems is an important problem in data mining, and covering-based rough sets serve as an efficient technique to process the problem. Geometric lattices have been widely used in many fields, especially…
We present a lattice algorithm specifically designed for some classical applications of lattice reduction. The applications are for lattice bases with a generalized knapsack-type structure, where the target vectors are boundably short. For…
A covering code is a set of codewords with the property that the union of balls, suitably defined, around these codewords covers an entire space. Generally, the goal is to find the covering code with the minimum size codebook. While most…
We generalize the Hamiltonian Monte Carlo algorithm with a stack of neural network layers and evaluate its ability to sample from different topologies in a two dimensional lattice gauge theory. We demonstrate that our model is able to…
Domino tileability is a classical problem in Discrete Geometry, famously solved by Thurston for simply connected regions in nearly linear time in the area. In this paper, we improve upon Thurston's height function approach to a nearly…
Human reasoning can distill principles from observed patterns and generalize them to explain and solve novel problems. The most powerful artificial intelligence systems lack explainability and symbolic reasoning ability, and have therefore…
The Wang tiling is a classical problem in combinatorics. A major theoretical question is to find a (small) set of tiles which tiles the plane only aperiodically. In this case, resulting tilings are rather restrictive. On the other hand,…
In this paper we give the generalization of lifted codes over any finite chain ring. This has been done by using the construction of finite chain rings from $p$-adic fields. Further we propose a lattice construction from linear codes over…
We calculate the generating functions for the number of tilings of rectangles of various widths by the right tromino, the $L$ tetromino, and the $T$ tetromino. This allows us to place lower bounds on the entropy of tilings of the plane by…
This work presents a novel lattice-based methodology for incorporating multidimensional constraints into continuous decision variables within a genetic algorithm (GA) framework. The proposed approach consolidates established transcription…
The girth of a matrix is the least number of linearly dependent columns, in contrast to the rank which is the largest number of linearly independent columns. This paper considers the construction of {\it high-girth} matrices, whose…
Generating functions for plane overpartitions are obtained using various methods such as nonintersecting paths, RSK type algorithms and symmetric functions. We extend some of the generating functions to cylindric partitions. Also, we show…
It has been well known for a long time that the height function of random lozenge tilings of large domains follow a law of large number and possible limits called dimer limit shapes are well understood. For the next order, it is expected…