On the random generation and counting of weak order extensions of a poset with given class cardinalities

Abstract

In previous work, we have proposed a simple algorithm to generate random linear extensions of a partially ordered set (poset). A closely related problem is the random generation of so-called weak order extensions of a poset. Such an extension can be informally characterized as a linear order on the equivalence classes of a partition of the poset, not contradicting the underlying poset order. The generation of linear extensions can then be seen as a special case of the generation of weak order extensions where each equivalence class degenerates into a singleton. If no a priori knowledge about the underlying partition is available, time complexity increases tremendously. In first instance, we therefore restrict to the generation of weak order extensions with given class cardinalities, a problem encountered in the context of ranking algorithms. It will be shown that a first random weak order extension can be generated in O ( w 2 ( P ) · | I ( P ) | ) time, while every subsequent extension with the same class cardinalities can be obtained in O ( w ( P ) · | P | ) time, where | I ( P ) | denotes the number of ideals of the poset P, and w( P) the width of the poset P. Additionally, the number of weak order extensions obeying the specified class cardinalities can also be obtained in the stated O ( w 2 ( P ) · | I ( P ) | ) time.