Counting edge crossings in a 2-layered drawing

• Published in: Information processing letters Vol. 91; no. 5; pp. 221 - 225
• Main Authors: Nagamochi, Hiroshi, Yamada, Nobuyasu
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 15.09.2004; Elsevier Science; Elsevier Sequoia S.A
• Subjects: Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Bipartite graphs; Calculus of variations and optimal control; Combinatorics; Combinatorics. Ordered structures; Computer science; Computer science; control theory; systems; Counting; Designs and configurations; Divide-and-conquer; Dynamic programming; Edge crossings; Exact sciences and technology; Graph algorithms; Graph drawing; Graph theory; Graphs; Mathematical analysis; Mathematics; Sciences and techniques of general use; Studies; Theoretical computing; Edge crossings; Graph drawing; Graph algorithms; Dynamic programming; Divide-and-conquer; Bipartite graphs; Counting; Computer theory; Line segment; Information processing; Space time; Edge crossing; Bipartite graph; Edge set; Algorithm analysis; Graph algorithm
• ISSN: 0020-0190; 1872-6119
• DOI: 10.1016/j.ipl.2004.05.001

Given a bipartite graph G=( V, W, E) with a bipartition { V, W} of a vertex set and an edge set E, a 2-layered drawing of G in the plane means that the vertices of V and W are respectively drawn as distinct points on two parallel lines and the edges as straight line segments. We consider the problem of counting the number of edge crossings. In this paper, we design two algorithms to this problem based on the dynamic programming and divide-and-conquer approaches. These algorithms run in O( n 1 n 2) time and O( m) space and in O(min{ n 1 n 2,| E|log(min{| V|,| W|})}) time and O( m) space, respectively. Our algorithms outperform the previously fastest Θ(| E|log(min{| V|,| W|})) time algorithm for dense graphs.