On the uniform generation of modular diagrams

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 $i_1 &lt; i_2 &lt; \ldots &lt; i_k &lt; j_1 &lt; j_2 &lt; \ldots&lt; j_k$. 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 $O(n<sup>k)$ preprocessing time, $k$-noncrossing, $\sigma$-modular diagrams in $O(n)$ time and space complexity.