Simple and efficient bilayer cross counting

Abstract

We consider the problem of counting the interior edge crossings when a bipartite graph G = (V, E) with node set V and edge set E is drawn such that the nodes of the two shores of the bipartition are drawn as distinct points on two parallel lines and the edges as straight line segments. The efficient solution of this problem is important in layered graph drawing. Our main observation is that it can be reduced to counting the inversions of a certain sequence. This leads to an O(|E| + |C|) algorithm, where C denotes the set of pairwise interior edge crossings, as well as to a simple O(|E| log | Vsmall|) algorithm, where Vsmall is the smaller cardinality node set in the bipartition of the node set V of the graph. We present the algorithms and the results of computational experiments with these and other algorithms on a large collection of instances. © Springer-Verlag Berlin Heidelberg 2002.