Giusi Castiglione

Reconstruction of L-convex Polyominoes

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.

A tomographical characterization of l-convex polyominoes

Our main purpose is to characterize the class of L-convex polyominoes introduced in [3] by means of their horizontal and vertical projections. The achieved results allow an answer to one of the most relevant questions in tomography i.e. the uniqueness of discrete sets, with respect to their horizontal and vertical projections. In this paper, by giving a characterization of L-convex polyominoes, we investigate the connection between uniqueness property and unimodality of vectors of horizontal and vertical projections. In the last section we consider the continuum environment; we extend the definition of L-convex set, and we obtain some results analogous to those for the discrete case.

Enumeration of L-convex polyominoes by rows and columns

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.

Ordering and convex polyominoes

We introduce a partial order on pictures (matrices), denoted by ≼ that extends to two dimensions the subword ordering on words. We investigate properties of special families of discrete sets (corresponding to {0,1}-matrices) with respect to this partial order. In particular we consider the families of polyominoes and convex polyominoes and the family, recently introduced by the authors, of L-convex polyominoes. In the first part of the paper we study the closure properties of such families with respect to the order. In particular we obtain a new characterization of L-convex polyominoes: a discrete set P is a L-convex polyomino if and only if all the elements Q≼P are polyominoes. In the second part of the paper we investigate whether the partial orderings introduced are well-orderings. Since our order extends the subword ordering, which is a well-ordering (Higmans theorem), the problem is whether there exists some extension of Higmans theorem to two dimensions. A negative answer is given in the general case, and also if we restrict ourselves to polyominoes and even to convex polyominoes. However we prove that the restriction to the family of L-convex polyominoes is a well-ordering. This is a further result that shows the interest of the notion of L-convex polyomino.

On extremal cases of Hopcroft's algorithm

In this paper we consider the problem of minimization of deterministic finite automata (DFA) with reference to Hopcrofts algorithm. Hopcrofts algorithm has several degrees of freedom, so there can exist different executions that can lead to different sequences of refinements of the set of the states up to the final partition. We find an infinite family of binary automata for which such a process is unique, whatever strategy is chosen. Some recent papers (cf. Berstel and Carton (2004) [3], Castiglione et al. (2008) [6] and Berstel et al. (2009) [1]) have been devoted to find families of automata for which Hopcrofts algorithm has its worst execution time. They are unary automata associated with circular words. However, automata minimization can be achieved also in linear time when the alphabet has only one letter (cf. Paige et al. (1985) [14]), but such a method does not seem to extend to larger alphabet. So, in this paper we face the tightness of Hopcrofts algorithm when the alphabet contains more than one letter. In particular we define an infinite family of binary automata representing the worst case of Hopcrofts algorithm, for each execution. They are automata associated with particular trees and we deepen the connection between the refinement process of Hopcrofts algorithm and the combinatorial properties of such trees.

A reconstruction algorithm for L-convex polyominoes

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

Recognizable picture languages and polyominoes

An irrigation system includes an irrigation line and a wind brace assembly which can be automatically disengaged from the irrigation line preparatory to moving the irrigation line across a field to a new position. The assembly is automatically engaged with the irrigation line in the new position to prevent wind from moving the irrigation line.

A challenging family of automata for classical minimization algorithms

In this paper a particular family of deterministic automata that was built to reach the worst case complexity of Hopcrofts state minimization algorithm is considered. This family is also challenging for the two other classical minimization algorithms: it achieves the worst case for Moores algorithm, as a consequence of a result by Berstel et al., and is of at least quadratic complexity for Brzozowskis solution, which is our main contribution. It therefore constitutes an interesting family, which can be useful to measure the efficiency of implementations of well-known or new minimization algorithms.

On Extremal Cases of Hopcroft's Algorithm

In this paper we consider the problem of minimization of deterministic finite automata (DFA) with reference to Hopcrofts algorithm. Hopcrofts algorithm has several degrees of freedom, so there can exist different sequences of refinements of the set of the states that lead to the final partition. We find an infinite family of binary automata for which such a process is unique. Some recent papers (cf. [3,7,1]) have been devoted to find families of automata for which Hopcrofts algorithm has its worst execution time. They are unary automata associated to circular words. However, automata minimization can be achieved also in linear time when the alphabet has only one letter (cf. [14]), so in this paper we face the tightness of the algorithm when the alphabet contains more than one letter. In particular we define an infinite family of binary automata representing the worst case of Hopcrofts algorithm. They are automata associated to particular trees and we deepen the connection between the refinement process of Hopcrofts algorithm and the combinatorial properties of such trees.

Patterns in words and languages

A word p, over the alphabet of variables E, is a pattern of a word w over A if there exists a non-erasing morphism h from E* to A* such that h(p)=w. If we take E=A, given two words u, v ∈ A*, we write u ≤ v if u is a pattern of v. The restriction of ≤ to aA*, where A is the binary alphabet {a, b}, is a partial order relation. We introduce, given a word v, the set P(v) of all words u such that u ≤ v. P(v), with the relation ≤, is a poset and it is called the pattern poset of v. The first part of the paper is devoted to investigate the relationships between the structure of the poset P(v) and the combinatorial properties of the word v. In the last section, for a given language L, we consider the language P(L) of all patterns of words in L. The main result of this section shows that, if L is a regular language, then P(L) is a regular language too.