Simple and Efficient Bilayer Cross Counting

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 directly to an O(|E| log |V |) algorithm based on merge sorting. We present an even simpler O(|E| log |Vsmall|) algorithm, where Vsmall is the smaller cardinality node set in the bipartition of the node set V of the graph. This algorithm is very easy to implement. Our computational experiments on a large collection of instances show that it performs well in comparison to previously published algorithms, which are much more complicated to understand and implement. Article Type Communicated by Submitted Revised Regular Paper X. He March 2003 December 2003 Partially supported by the Future and Emerging Technologies Programme of the European Union under contract number IST-1999-14186 (ALCOM-FT). W. Barth et al., Bilayer Cross Counting , JGAA, 8(2) 179–194 (2004) 180