On the random generation and counting of weak order extensions of a poset with given class cardinalities
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 x∥y. 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
Complexity of the algorithm
By careful observation, it is immediately clear that the time complexity of the algorithm Assign is . The algorithm will visit edges before it terminates. For each edge, up to 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 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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Random generation of k-interactive capacities
2022, Fuzzy Sets and SystemsCitation Excerpt :The selected linear extension corresponds to one simplex in the simplicial partition of the (appropriately scaled) order polytope P [31], and hence we focus on selecting linear extensions uniformly. A closely related problem is generating random monotone data sets for testing and benchmarking monotone classification and approximation algorithms was considered in a series of papers [9–12]. There weak order extensions (as opposed to linear extensions in [8]) were generated uniformly by using lattices of ideals of the posets together with Markov chains.
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