On the Decomposition of Posets into Minimum Set Node-Disjoint Chains

Abstract

One of the most famous results in the theory of partially ordered sets is due to Dilworth (1950) who showed that the size of a minimum decomposition (into chains) of a partially ordered set S is equal to the size of a maximum antichain, which is a subset of pairwise incomparable elements. However, up to now, the bestalgorithm to decompose S into a minimum set of chains needs O(n 3) time, where n is the number of the elements in S. In this paper, we address this problem and propose an algorithm which produces a minimum decomposition in O(n 2) time and O(m + n)space, where is the size of a maximum anti chain and m is the number of relations between elements (i.e., the number of pairs (a, c) such that ac).In general, is much smaller than n.

On the Graph Decomposition

“…However, in the method discussed in [5] each node is associated with a large data structure and requires O(Nn 2 ) space in the worst case. By [6], the generated chains may contain some newly created nodes, but how to remove such nodes are not discussed at all. Different from the above strategies, the algorithm discussed in [9] is to find a maximum k-chain in a planar point set M N u N, where N = {0, 1, ..., n -1} and is defined by establishing (i´, j´) (i, j) iff i´ > i and j´ > j.…”

“…But in some cases it fails to find a minimum set of chains since when generating chains, only part of reachability information is considered. This problem is removed by [5] and [6] both with the same time complexity O(Nn 2 ). However, in the method discussed in [5] each node is associated with a large data structure and requires O(Nn 2 ) space in the worst case.…”