On the Uniform Generation of Modular Diagrams

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 ≤ 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 (i1, j1), (i2, j2), . . . , (ik, jk) with the property i1 < i2 < . . . < ik < j1 < j2 < . . . < jk. 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), . . . , (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(nk) preprocessing time, k-noncrossing, σ-modular diagrams in O(n) time and space complexity.