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

On the random generation of monotone data sets

“…First, we draw 15 attribute vectors (n = 15) uniformly at random from a 4-dimensional (m = 4) discrete vector space 4, 7, 4) and (c) (3, 5, 7). Next, for each of the label distributions the exact number of weak order extensions is computed (see Table 1) with the algorithm described in [12]. Note that the number n has been chosen such that this remains feasible.…”

“…In order to generate weak order extensions uniformly at random, one can use the approach outlined in [12]. The first and most time consuming phase consists of constructing the so-called lattice of ideals corresponding to the given poset A, followed by some graph counting operations on this lattice.…”

“…Since any weak order extension can be characterized by means of a unique linear extension once an initial linear order on the elements is fixed [13,12,11], one could suggest to use one of the algorithms to almost uniformly generate a linear extension. Each time a linear extension is generated that does not correspond to the canonical representation of a weak order extension, it is skipped and a next one is generated.…”

Weak Ratio Rules

“…The concept ideal is also called downset (Bezrukov & Leck, 2005;Loof et al, 2007), lower set (Gierz et al, 1980), lower segment (Burris & Sankappanavar, 1981).…”