Posets

Abstract

When an N-key sorting network is given an unordered array of N keys, the network performs a number of comparisons and re-arrangements until the array is totally ordered. At any point in the middle of the sorting network , the array of keys is partially-ordered. A set of items where an ordering relation is established between some of the pairs of items is called a partially-ordered set or poset for short. The partial-ordering of the keys at any point in the sorting network can be illustrated with a Haase diagram. Some partial-orderings of keys established in earlier steps of a sorting network might be lost by a comparator in a later step. If corresponding keys in two similar parts of a Haase diagram are compared, then the relations within each part are preserved. Thus, one needs to be careful as they pick the pairs of keys to be compared in each step of the sorting network .