Random generation of trees and other combinatorial objects

Abstract

In this paper, we present a general method for the random generation of some classes of combinatorial objects. Our basic idea is to translate ECO method (Enumerating Combinatorial Objects) from a method for the enumeration of combinatorial objects into a random generation method. The algorithms we illustrate are based on the concepts of succession rule and generating tree: the former is a law that predicts the combinatorial object class growth according to a given parameter. The generating tree related to a given succession rule is a particular labelled plane tree that represents the rule in an extensive way. Each node of a generating tree can also be seen as a particular combinatorial object and so a random path in the generating tree coincides with the random generation of that combinatorial object. The generation is uniform if we take the probability of each branch to be selected into account when the path is generated. We also give the formulae evaluating complexity. Finally, we take the class of m-ary trees into consideration in order to illustrate our general method. In this case, the average time complexity of the generating algorithm can be estimated as O( mn).