Constant delay Gray code enumeration of ideals and antichains in posets

TL;DR

This paper introduces a constant delay Gray code enumeration algorithm for ideals and antichains in posets, solving a long-standing open problem and presenting a new analysis framework called the Pyramid method.

Contribution

It provides the first constant delay Gray code enumeration algorithms for ideals and antichains in posets, along with a novel potential-based analysis framework.

Findings

Algorithms enumerate ideals and antichains with at most three-element differences.

The Pyramid method generalizes the Push-out method of Uno.

The approach improves upon previous algorithms by Pruesse, Ruskey, and others.

Abstract

We present an algorithm that enumerates all ideals of an input poset with constant delay in Gray code order, i.e., such that consecutively visited ideals differ in at most three elements. This answers a long-standing open problem posed by Pruesse and Ruskey, and improves upon previous algorithms by Pruesse and Ruskey, Squire, Habib, Medina, Nourine and Steiner, as well as Abdo. Using the same techniques, we also obtain an algorithm that enumerates all antichains of an input poset with constant delay such that successively visited antichains differ in at most three elements. As a key technical ingredient, we introduce a new potential-based analysis framework for recursive algorithms, which we call the Pyramid method. We show that this method subsumes the Push-out method of Uno. Beyond the present application, the Pyramid method is a general framework to analyze recursive algorithms and may thus be of independent interest.