# Simple and Efficient Bilayer Cross Counting

Barth, Wilhelm and Jünger, Michael and Mutzel, Petra (2002) Simple and Efficient Bilayer Cross Counting. In: Graph Drawing 10th International Symposium, GD 2002, August 26-28, 2002, Irvine, CA, USA , pp. 130-141 (Official URL: http://dx.doi.org/10.1007/3-540-36151-0_13).

Full text not available from this repository.

## 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 $O(\vert E\vert\log\vert V_{\rm small}\vert)$ algorithm, where $V_{\rm small}$ 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 w ith these and other algorithms on a large collection of instances.

Item Type: Conference Paper 10.1007/3-540-36151-0_13 M Methods > M.500 LayeredP Styles > P.720 Straight-lineG Algorithms and Complexity > G.420 CrossingsP Styles > P.480 Layered 252

### Available Versions of this Item

Repository Staff Only: item control page

References

B. Chazelle, Reporting and counting segment intersections. Journal of Computer and System Sciences 32 (1986) 156-182.

B. Chazelle and H. Edelsbrunner, An optimal algorithm for intersecting line segments in the plane. Journal of the ACM 39 (1992) 1-54.

T.H. Cormen, C. E. Leiserson, and R.L. Rivest, Introduction to algorithms. MIT Press, Cambridge, MA, 1990.

P. Eades and N. Wormland, Edge crossings in drawings of bipartite graphs. Algorithmica 11 (1994) 379-403.

C. Gutwenger, M. Jünger, G.W. Klau, S. Leipert, and P. Mutzel, Graph Drawing Algorithm Engineering with AGD. in: S. Diehl (ed.), Software Visualization, International Dagstuhl Seminar on Software Visualization 2001, Lecture Notes in Computer Science 2269, Springer, 2002, pp. 307-323, see also: http://www.mpi-sb.mpg.de/AGD/

M. Jünger and P. Mutzel, 2-layer straight line crossing minimization: performance of exact and heuristic algorithms. Journal of Graph Algorithms and Applications 1 (1997) 1-25.

D.E. Knuth, The Stanford GraphBase: A platform for combinatorial computing. Addison-Wesley, Reading, Massachusets, 1993.

G.S. Lueker, A data structure for orthogonal range queries. Proceedings of the 19th IEEE Symposium on Foundations of Computer Science, 1978, pp. 28-34.

G. Sander, Graph Layout through the VCG Tool. in: R. Tamassia and I.G. Tollis (eds): Graph Drawing 1994, Lecture Notes in Computer Science 894, Springer, 1995, pp. 194-205, see also: http://rw4.cs.uni-sb.de/users/sander/html/gsvcg1.html

G. Sander, Visualisierungstechniken für den Compilerbau. Pirrot Verlag & Druck, Saarbrücken, 1996.

K. Sugiyama, S. Tagawa, and M. Toda, Methods for visual understanding of hierarchical system structures. IEEE Transactions on Systems, Man, and Cybernetics 11 (1981) 109-125.

V. Waddle and A. Malhotra, An E log E line crosssing algorithm for levelled graphs. in: J. Kratochvíl (ed.) Graph Drawing 1999, Lecture Notes in Computer Science 1731, Springer, 1999, pp. 59-70.