On the uniform generation of modular diagrams

TL;DR

This paper introduces an efficient algorithm for uniformly generating $k$-noncrossing, $\sigma$-modular diagrams, which are specific labeled graphs with constraints on crossings and stacks, with applications in combinatorics and graph theory.

Contribution

The authors present a novel algorithm that efficiently generates $k$-noncrossing, $\sigma$-modular diagrams with uniform probability, improving on previous methods in terms of time and space complexity.

Findings

Preprocessing time is $O(n^k)$.

Generation time and space complexity are $O(n)$.

Algorithm produces diagrams uniformly at random.

Abstract

In this paper we present an algorithm that generates $k$-noncrossing, $\sigma$-modular diagrams with uniform probability. A diagram is a labeled graph of degree $\le 1$ over $n$ vertices drawn in a horizontal line with arcs $(i,j)$ in the upper half-plane. A $k$-crossing in a diagram is a set of $k$ distinct arcs $(i_1, j_1), (i_2, j_2),\ldots,(i_k, j_k)$ with the property $i_1 < i_2 < \ldots < i_k < j_1 < j_2 < \ldots< j_k$. A diagram without any $k$-crossings is called a $k$-noncrossing diagram and a stack of length $\sigma$ is a maximal sequence $((i,j),(i+1,j-1),\dots,(i+(\sigma-1),j-(\sigma-1)))$. A diagram is $\sigma$-modular if any arc is contained in a stack of length at least $\sigma$. Our algorithm generates after $O(n^k)$ preprocessing time, $k$-noncrossing, $\sigma$-modular diagrams in $O(n)$ time and space complexity.