Output polynomial enumeration of all fixed-cardinality ideals of a poset, respectively all fixed-cardinality subtrees of a tree

TL;DR

This paper presents an efficient algorithm for enumerating all fixed-cardinality ideals of a poset and subtrees of a tree, using wildcards for compact display, with specific time bounds for each case.

Contribution

It introduces a novel algorithm that enumerates fixed-cardinality ideals and subtrees efficiently using wildcards, improving over naive enumeration methods.

Findings

Enumeration of poset ideals in O(Nw^3) time

Enumeration of tree subtrees in O(Nw^5) time

Compact display of all order ideals using wildcards

Abstract

The N cardinality k ideals of any w-element poset (w, k variable) can be enumerated in time O(Nw^3). The corresponding bound for k-element subtrees of a w-element tree is O(Nw^5). An algorithm is described that by the use of wildcards displays all order ideals of a poset in a compact manner, i.e. not one by one.