Search results for "Polyominoes"

Computation of the area in the discrete plane: Green’s theorem revisited

2017

International audience; The detection of the contour of a binary object is a common problem; however, the area of a region, and its moments, can be a significant parameter. In several metrology applications, the area of planar objects must be measured. The area is obtained by counting the pixels inside the contour or using a discrete version of Green's formula. Unfortunately, we obtain the area enclosed by the polygonal line passing through the centers of the pixels along the contour. We present a modified version of Green's theorem in the discrete plane, which allows for the computation of the exact area of a two-dimensional region in the class of polyominoes. Penalties are introduced and …

A reconstruction algorithm for L-convex polyominoes

2006

AbstractWe give an algorithm that uniquely reconstruct an L-convex polyomino from the size of some special paths, called bordered L-paths.

Tomographical aspects of L-convex polyominoes

2007

An Efficient Algorithm for the Generation of Z-Convex Polyominoes

2014

We present a characterization of Z-convex polyominoes in terms of pairs of suitable integer vectors. This lets us design an algorithm which generates all Z-convex polyominoes of size n in constant amortized time.

Combinatorial aspects of L-convex polyominoes

2007

We consider the class of L-convex polyominoes, i.e. those polyominoes in which any two cells can be connected with an ''L'' shaped path in one of its four cyclic orientations. The paper proves bijectively that the number f"n of L-convex polyominoes with perimeter 2(n+2) satisfies the linear recurrence relation f"n"+"2=4f"n"+"1-2f"n, by first establishing a recurrence of the same form for the cardinality of the ''2-compositions'' of a natural number n, a simple generalization of the ordinary compositions of n. Then, such 2-compositions are studied and bijectively related to certain words of a regular language over four letters which is in turn bijectively related to L-convex polyominoes. In …

Recognizable picture languages and polyominoes

2007

We consider the problem of recognizability of some classes of polyominoes in the theory of picture languages. In particular we focus our attention oil the problem posed by Matz of finding a non-recognizable picture language for which his technique for proving the non-recognizability of picture languages fails. We face the problem by studying the family of L-convex polyominoes and some closed families that are similar to the recognizable family of all polyominoes but result to be non-recognizable. Furthermore we prove that the family of L-convex polyominoes satisfies the necessary condition given by Matz for the recognizability and we conjecture that the family of L-convex polyominoes is non…

Reconstruction of L-convex Polyominoes.

2003

Abstract We introduce the family of L-convex polyominoes, a subset of convex polyominoes whose elements satisfy a special convexity property. We develop an algorithm that reconstructs an L-convex polyomino from the set of its maximal L-polyominoes.

Enumeration of L-convex polyominoes by rows and columns

2005

In this paper, we consider the class of L-convex polyominoes, i.e. the convex polyominoes in which any two cells can be connected by a path of cells in the polyomino that switches direction between the vertical and the horizontal at most once.Using the ECO method, we prove that the number fn of L-convex polyominoes with perimeter 2(n + 2) satisfies the rational recurrence relation fn = 4fn-1 - 2fn-2, with f0 = 1, f1 = 2, f2 = 7. Moreover, we give a combinatorial interpretation of this statement. In the last section, we present some open problems.

On the exhaustive generation of k-convex polyominoes

2017

The degree of convexity of a convex polyomino P is the smallest integer k such that any two cells of P can be joined by a monotone path inside P with at most k changes of direction. In this paper we present a simple algorithm for computing the degree of convexity of a convex polyomino and we show how it can be used to design an algorithm that generates, given an integer k, all k-convex polyominoes of area n in constant amortized time, using space O(n). Furthermore, by applying few changes, we are able to generate all convex polyominoes whose degree of convexity is exactly k.

Disconnected Graph Layout and the Polyomino Packing Approach

2002

Conference name: GD: International Symposium on Graph Drawing 9th International Symposium Date of Conference: 23–26 September 2001 We review existing algorithms and present a new approach for layout of disconnected graphs. The new approach is based on polyomino representation of components as opposed to rectangles. The parameters of our algorithm and their influence on the drawings produced as well as a variation of the algorithm for multiple pages are discussed. We also analyze our algorithm both theoretically and experimentally and compare it with the existing ones. The new approach produces much more compact and uniform drawings than previous methods. © Springer-Verlag Berlin Heidelberg …