A strip-like tiling algorithm

AbstractWe extend our previous results on the connection between strip tiling problems and regular grammars by showing that an analogous algorithm is applicable to other tiling problems, not necessarily related to rectangular strips. We find generating functions for monomer and dimer tilings of T- and L-shaped figures, holed and slotted strips, diagonal strips and combinations of them, and show how analogous results can be obtained by using different pieces

Tiling problems, automata, and tiling graphs

This paper continues the investigation of tiling problems via formal languages, which was begun in papers by Merlini, Sprugnoli, and Verri. Those authors showed that certain tiling problems could be encoded by regular languages, which lead automatically to generating functions and other combinatorial information on tilings. We introduce a method of simplifying the DFA's recognizing these language, which leads to bijective proofs of certain tiling identities. We apply these ideas to some other tiling problems, including three-dimensional tilings and tilings with triangles and rhombi. We also study graph-theoretic variations of these tiling problems

Monomer-dimer tatami tilings of square regions

We prove that the number of monomer-dimer tilings of an $n\times n$ square grid, with $m<n$ monomers in which no four tiles meet at any point is $m2^m+(m+1)2^{m+1}$, when $m$ and $n$ have the same parity. In addition, we present a new proof of the result that there are $n2^{n-1}$ such tilings with $n$ monomers, which divides the tilings into $n$ classes of size $2^{n-1}$. The sum of these tilings over all monomer counts has the closed form $2^{n-1}(3n-4)+2$ and, curiously, this is equal to the sum of the squares of all parts in all compositions of $n$. We also describe two algorithms and a Gray code ordering for generating the $n2^{n-1}$ tilings with $n$ monomers, which are both based on our new proof.Comment: Expanded conference proceedings: A. Erickson, M. Schurch, Enumerating tatami mat arrangements of square grids, in: 22nd International Workshop on Combinatorial Al- gorithms (IWOCA), volume 7056 of Lecture Notes in Computer Science (LNCS), Springer Berlin / Heidelberg, 2011, p. 12 pages. More on Tatami tilings at http://alejandroerickson.com/joomla/tatami-blog/collected-resource

Complexity of Two-Dimensional Patterns

In dynamical systems such as cellular automata and iterated maps, it is often useful to look at a language or set of symbol sequences produced by the system. There are well-established classification schemes, such as the Chomsky hierarchy, with which we can measure the complexity of these sets of sequences, and thus the complexity of the systems which produce them. In this paper, we look at the first few levels of a hierarchy of complexity for two-or-more-dimensional patterns. We show that several definitions of ``regular language'' or ``local rule'' that are equivalent in d=1 lead to distinct classes in d >= 2. We explore the closure properties and computational complexity of these classes, including undecidability and L-, NL- and NP-completeness results. We apply these classes to cellular automata, in particular to their sets of fixed and periodic points, finite-time images, and limit sets. We show that it is undecidable whether a CA in d >= 2 has a periodic point of a given period, and that certain ``local lattice languages'' are not finite-time images or limit sets of any CA. We also show that the entropy of a d-dimensional CA's finite-time image cannot decrease faster than t^{-d} unless it maps every initial condition to a single homogeneous state.Comment: To appear in J. Stat. Phy

Recursive tree traversal dependence analysis

While there has been much work done on analyzing and transforming regular programs that operate over linear arrays and dense matrices, comparatively little has been done to try to carry these optimizations over to programs that operate over heap-based data structures using pointers. Previous work has shown that point blocking, a technique similar to loop tiling in regular programs, can help increase the temporal locality of repeated tree traversals. Point blocking, however, has only been shown to work on tree traversals where each traversal is fully independent and would allow parallelization, greatly limiting the types of applications that this transformation could be applied to.^ The purpose of this study is to develop a new framework for analyzing recursive methods that perform traversals over trees, called tree dependence analysis. This analysis translates dependence analysis techniques for regular programs to the irregular space, identifying the structure of dependences within a recursive method that traverses trees. In this study, a dependence test that exploits the dependence structure of such programs is developed, and is shown to be able to prove the legality of several locality— and parallelism-enhancing transformations, including point blocking. In addition, the analysis is extended with a novel path-dependent, conditional analysis to refine the dependence test and prove the legality of transformations for a wider range of algorithms. These analyses are then used to show that several common algorithms that manipulate trees recursively are amenable to several locality— and parallelism-enhancing transformations. This work shows that classical dependence analysis techniques, which have largely been confined to nested loops over array data structures, can be extended and translated to work for complex, recursive programs that operate over pointer-based data structures

Counting With Irrational Tiles

We introduce and study the number of tilings of unit height rectangles with irrational tiles. We prove that the class of sequences of these numbers coincides with the class of diagonals of N-rational generating functions and a class of certain binomial multisums. We then give asymptotic applications and establish connections to hypergeometric functions and Catalan numbers

Proceedings of JAC 2010. Journées Automates Cellulaires

The second Symposium on Cellular Automata “Journ´ees Automates Cellulaires” (JAC 2010) took place in Turku, Finland, on December 15-17, 2010. The first two conference days were held in the Educarium building of the University of Turku, while the talks of the third day were given onboard passenger ferry boats in the beautiful Turku archipelago, along the route Turku–Mariehamn–Turku. The conference was organized by FUNDIM, the Fundamentals of Computing and Discrete Mathematics research center at the mathematics department of the University of Turku. The program of the conference included 17 submitted papers that were selected by the international program committee, based on three peer reviews of each paper. These papers form the core of these proceedings. I want to thank the members of the program committee and the external referees for the excellent work that have done in choosing the papers to be presented in the conference. In addition to the submitted papers, the program of JAC 2010 included four distinguished invited speakers: Michel Coornaert (Universit´e de Strasbourg, France), Bruno Durand (Universit´e de Provence, Marseille, France), Dora Giammarresi (Universit` a di Roma Tor Vergata, Italy) and Martin Kutrib (Universit¨at Gie_en, Germany). I sincerely thank the invited speakers for accepting our invitation to come and give a plenary talk in the conference. The invited talk by Bruno Durand was eventually given by his co-author Alexander Shen, and I thank him for accepting to make the presentation with a short notice. Abstracts or extended abstracts of the invited presentations appear in the first part of this volume. The program also included several informal presentations describing very recent developments and ongoing research projects. I wish to thank all the speakers for their contribution to the success of the symposium. I also would like to thank the sponsors and our collaborators: the Finnish Academy of Science and Letters, the French National Research Agency project EMC (ANR-09-BLAN-0164), Turku Centre for Computer Science, the University of Turku, and Centro Hotel. Finally, I sincerely thank the members of the local organizing committee for making the conference possible. These proceedings are published both in an electronic format and in print. The electronic proceedings are available on the electronic repository HAL, managed by several French research agencies. The printed version is published in the general publications series of TUCS, Turku Centre for Computer Science. We thank both HAL and TUCS for accepting to publish the proceedings.Siirretty Doriast

Complexity and efficient approximability of two dimensional periodically specified problems

The authors consider the two dimensional periodic specifications: a method to specify succinctly objects with highly regular repetitive structure. These specifications arise naturally when processing engineering designs including VLSI designs. These specifications can specify objects whose sizes are exponentially larger than the sizes of the specification themselves. Consequently solving a periodically specified problem by explicitly expanding the instance is prohibitively expensive in terms of computational resources. This leads one to investigate the complexity and efficient approximability of solving graph theoretic and combinatorial problems when instances are specified using two dimensional periodic specifications. They prove the following results: (1) several classical NP-hard optimization problems become NEXPTIME-hard, when instances are specified using two dimensional periodic specifications; (2) in contrast, several of these NEXPTIME-hard problems have polynomial time approximation algorithms with guaranteed worst case performance

Low-rank Based Algorithms for Rectification, Repetition Detection and De-noising in Urban Images

In this thesis, we aim to solve the problem of automatic image rectification and repeated patterns detection on 2D urban images, using novel low-rank based techniques. Repeated patterns (such as windows, tiles, balconies and doors) are prominent and significant features in urban scenes. Detection of the periodic structures is useful in many applications such as photorealistic 3D reconstruction, 2D-to-3D alignment, facade parsing, city modeling, classification, navigation, visualization in 3D map environments, shape completion, cinematography and 3D games. However both of the image rectification and repeated patterns detection problems are challenging due to scene occlusions, varying illumination, pose variation and sensor noise. Therefore, detection of these repeated patterns becomes very important for city scene analysis. Given a 2D image of urban scene, we automatically rectify a facade image and extract facade textures first. Based on the rectified facade texture, we exploit novel algorithms that extract repeated patterns by using Kronecker product based modeling that is based on a solid theoretical foundation. We have tested our algorithms in a large set of images, which includes building facades from Paris, Hong Kong and New York