On the uniform generation of modular diagrams

Abstract

In this paper we present an algorithm that generates $k$-noncrossing, $σ$-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 $σ$ is a maximal sequence $((i,j),(i+1,j-1),\dots,(i+(σ-1),j-(σ-1)))$. A diagram is $σ$-modular if any arc is contained in a stack of length at least $σ$. Our algorithm generates after $O(n^k)$ preprocessing time, $k$-noncrossing, $σ$-modular diagrams in $O(n)$ time and space complexity.