Random generation of Q-convex sets

Sara Brunetti (1) , Alain Daurat (2)

1 DSMI - Department of Mathematics and Computer Science / Dipartimento di Scienze Matematiche e Informatiche "Roberto Magari"
2 LSIIT - Laboratoire des Sciences de l'Image, de l'Informatique et de la Télédétection

The problem of randomly generating Q-convex sets is considered. We present two generators. The first one uses the Q-convex hull of a set of random points in order to generate a Q-convex set included in the square [0,n)2. This generator is very simple, but is not uniform and its performance is quadratic in n. The second one exploits a coding of the salient points, which generalizes the coding of a border of polyominoes. It is uniform, and is based on the method by rejection. Experimentally, this algorithm works in linear time in the length of the word coding the salient points. Besides, concerning the enumeration problem, we determine an asymptotic formula for the number of Q-convex sets according to the size of the word coding the salient points in a special case, and in general only an experimental estimation.

Mots clés: Uniform generator; Lattice sets; Convexity; Salient points

Domaines: Mathématique discrète [cs.DM]; Combinatoire [math.CO]

Sara Brunetti, Alain Daurat. Random generation of Q-convex sets. Theoretical Computer Science, 2005, 347, pp.393-414. ⟨10.1016/j.tcs.2005.06.033⟩. ⟨hal-00023088⟩