Renzo Pinzani

Reconstructing convex polyominoes from horizontal and vertical projections

In [1], we studied the problem of reconstructing a discrete set 5 from its horizontal and vertical projections. We defined an algorithm that establishes the existence of a convex polyomino Λ whose horizontal and vertical projections are equal to a pair of assigned vectors (H,V), with H ∈ ℕ m and V ∈ ℕ n . Its computational cost is O(n4m4). In this paper, we introduce some operations for recontructing convex polyominoes by means of vectors Hs and Vs partial sums. These operations allows us to define a new algorithm whose complexity is less than O(n2m2).

The number of convex polyominoes reconstructible from their orthogonal projections

Abstract Many problems of computer-aided tomography, pattern recognition, image processing and data compression involve a reconstruction of bidimensional discrete sets from their projections. [3–5,10,12,16,17]. The main difficulty involved in reconstructing a set Λ starting out from its orthogonal projections ( V,H ) is the ‘ambiguity’ arising from the fact that, in some cases, many different sets have the same projections ( V,H ). In this paper, we study this problem of ambiguity with respect to convex polyominoes, a class of bidimensional discrete sets that satisfy some connection properties similar to those used by some reconstruction algorithms. We determine an upper and lower bound to the maximum number of convex polyominoes having the same orthogonal projections ( V,H ), with V ∈ N n and H ∈ N m . We prove that under these connection conditions, the ambiguity is sometimes exponential. We also define a construction in order to obtain some convex polyominoes having the same orthogonal projections.

An algebraic characterization of the set of succession rules

“Qui dedit beneficium taceat; narret qui accepit” (Seneca) Merci Maurice In this paper we will give a formal description of succession rules in terms of linear operators satisfying certain conditions. This representation allows us to introduce a system of well-defined operations into the set of succession rules and then to tackle problems of combinatorial enumeration simply by using operators instead of generating functions. Finally, we will suggest several open problems whose solution should lead to an algebraic characterization of the set of succession rules.

Medians of polyominoes: A property for reconstruction

In a previous report, we studied the problem of reconstructing a discrete set 𝒮 from its horizontal and vertical projections. We defined an algorithm that decides whether there is a convex polyomino 𝒮 whose horizontal and vertical projections are given by (H, V), with H ∈ ℕm and V ∈ ℕn. If there is at least one convex polyomino with these projections, the algorithm reconstructs one of them in O(n4m4) time. In this article, we introduce the geometrical concept of a discrete sets medians. Starting out from this geometric property, we define some operations for reconstructing convex polyominoes from their projections (H, V). We are therefore able to define a new algorithm whose complexity is less than O(n2m2). Hence, this algorithm is much faster than the previous one. At the moment, however, we only have experimental evidence that this algorithm decides if there is a convex polyomino whose projections are equal to (H, V), for all (H, V) instances.

Jumping succession rules and their generating functions

We study a generalization of the concept of succession rule, called jumping succession rule, where each label is allowed to produce its sons at different levels, according to the production of a fixed succession rule. By means of suitable linear algebraic methods, we obtain simple closed forms for the numerical sequences determined by such rules and give applications concerning classical combinatorial structures. Some open problems are proposed at the end of the paper.

The random generation of directed animals

Abstract In this paper, we propose an algorithm to randomly generate a directed animal. Directed animals are well-known combinatorial objects and have been widely used for modelling the physical phenomenon of percolation. The algorithm consists of three steps, and we prove that each of them is performed in linear time. Finally, we report the results of our experiments made by means of appropriate computer programs in order to give empirical evidence that our algorithm really works.

X-rays characterizing some classes of discrete sets

In this paper, we study the problem of determining discrete sets by means of their X-rays. An X-ray of a discrete set F in a direction u counts the number of points in F on each line parallel to u. A class F of discrete sets is characterized by the set U of directions if each element in F is determined by its X-rays in the directions of U. By using the concept of switching component introduced by Chang and Ryser [Comm. ACM 14 (1971) 21; Combinatorial Mathematics, The Carus Mathematical Monographs, No. 14, The Mathematical Association of America, Rahway, 1963] and extended in [Discrete Comput. Geom. 5 (1990) 223], we prove that there are some classes of discrete sets that satisfy some connectivity and convexity conditions and that cannot be characterized by any set of directions. Gardner and Gritzmann [Trans. Amer. Math. Soc. 349 (1997) 2271] show that any set U of four directions having cross ratio that does not belong to {4/3,3/2,2,3,4}, characterizes the class of convex sets. We prove the converse, that is, if Us cross ratio is in {4/3,3/2,2,3,4}, then the hv-convex sets cannot be characterized by U. We show that if the horizontal and vertical directions do not belong to U, Gardner and Gritzmanns result cannot be extended to hv-convex polyominoes. If the horizontal and vertical directions belong to U and Us cross ratio is not in {4/3,3/2,2,3,4}, we believe that U characterizes the class of hv-convex polyominoes. We give experimental evidence to support our conjecture. Moreover, we prove that there is no number δ such that, if |U|⩾δ, then U characterizes the hv-convex polyominoes. This number exists for convex sets and is equal to 7 (see [Trans. Amer. Math. Soc. 349 (1997) 2271]).

From Motzkin to Catalan permutations

For every integer j>1, we dene a class of permutations in terms of certain forbidden subsequences. For j=1, the corresponding permutations are counted by the Motzkin numbers, and for j= 1 (dened in the text), they are counted by the Catalan numbers. Each value of j>1 gives rise to a counting sequence that lies between the Motzkin and the Catalan numbers. We compute the generating function associated to these permutations according to several parameters. For every j>1, we show that only this generating function is algebraic according to the length of the permutations. c 2000 Elsevier Science B.V. All rights reserved.

A New Approach to Cross-Bifix-Free Sets

Cross-bifix-free sets are sets of words such that no prefix of any word is a suffix of any other word. In this paper, we introduce a general constructive method for the sets of cross-bifix-free binary words of fixed length. It enables us to determine a cross-bifix-free words subset which has the property to be non-expandable.

A general exhaustive generation algorithm for Gray structures

Starting from a succession rule for Catalan numbers, we define a procedure for encoding and listing the objects enumerated by these numbers such that two consecutive codes of the list differ only by one digit. The Gray code we obtain can be generalized to all the succession rules with the stability property: each label (k) has in its productions two labels c1 and c2, always in the same position, regardless of k. Because of this link, we define Gray structures as the sets of those combinatorial objects whose construction can be encoded by a succession rule with the stability property. This property is a characteristic that can be found among various succession rules, such as the finite, factorial or transcendental ones. We also indicate an algorithm which is a very slight modification of Walsh’s one, working in O(1) worst-case time per word for generating Gray codes.