Elsevier

Information Sciences

Volume 177, Issue 1, 1 January 2007, Pages 220-230
Information Sciences

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

https://doi.org/10.1016/j.ins.2006.04.003Get rights and content

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(w2(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(w2(P)·|I(P)|) time.

Section snippets

Introduction and overview

In a classification problem, to each element of a set of objects a label must be assigned (see e.g., [15], [18]). The objects are usually identified with attribute vectors. In a multi-criteria decision making context, the attributes represent criteria and their domains are usually linearly ordered; the vector representations equipped with the product order form a partially ordered set or poset for short (see e.g., [6], [10]). Moreover, in this context, the labels can be seen as quality

Preliminaries

A partially ordered set, or poset for short, is a set P equipped with a partial order ⩽P, i.e., a reflexive, antisymmetric and transitive relation on P. The poset will be denoted by (P, P) or simply P if no distinction between order relations has to be made. Two elements x and y of P are called comparable if x P y or x P y; otherwise they are called incomparable, and we write xy. A chain of a poset P is a subset of P in which every two elements are comparable. Dually, an antichain of a poset P

Standardization of weak order extensions

Let us consider for a given poset P on the one hand all linear extensions of P, and on the other hand all weak order extensions of P with given class cardinalities (c1, c2,  , ck). Let us use the same class cardinalities to partition any linear extension such that the first c1 elements in the linear order belong to the first class, the next c2 elements to the second class, etc. In this way, we associate with each linear extension exactly one weak order extension. However, note that different

The algorithm

By taking the ideal lattice as the basic data structure, Corollary 4 now puts us in the position to design an appropriate algorithm for generating random weak order extensions with prescribed class cardinalities. Hence, we will assume that the ideal lattice of the poset is at our disposal. Note that, if this is not the case, it can be quite easily generated from the covering relation of the poset using the algorithm suggested by Habib et al. [11] running in constant amortized time, i.e., in O(w(

Complexity of the algorithm

By careful observation, it is immediately clear that the time complexity of the algorithm Assign is O(w2(P)·|I(P)|). The algorithm will visit O(w(P)·|I(P)|) edges before it terminates. For each edge, up to O(w(P)) immediate successors will have to be examined to see whether they should be visited. From this observation, the stated complexity follows. Furthermore, it should be noted that |I(P)| could be exponential in the size of the poset. Consider the extreme case of an antichain. The number

Conclusion

An algorithm has been proposed for generating random weak order extensions with given class cardinalities by modifying some essential steps in our previously established algorithm for the random generation of linear extensions. The main difference is that here an auxiliary numbering is performed of ideal lattice edges instead of ideal lattice nodes at the expense of a higher time complexity. Due to the time complexity of the algorithm, its application is presently restricted to posets

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