Large Crossing Angles in Circular Layouts Quan Nguyen 1 , Peter Eades 1 , Seok-Hee Hong 1 , and Weidong Huang 2 1 School of Information Technologies, University of Sydney {qnguyen,peter,shhong}@it.usyd.edu.au 2 CSIRO ICT Centre, Australia tony.huang@csiro.au 1 Introduction Recent empirical research has shown that increasing the angle of crossings re- duces the effect of crossings and improves human readability [5]. In this paper, we introduce a post-processing algorithm, namely MAXCIR, that aims to increase crossing angles of circular layouts by using Quadratic Programming. Experi- mental results indicate that our method significantly increases crossing angles compared to the traditional equal-spacing algorithm, and that the running time is fairly negligible. 2 Algorithm The post-processing approach MAXCIR in this paper aims to increase the cross- ing angles after a circular ordering of the vertices is given, such as by a crossing reduction algorithm [3]. With a fixed ordering of the vertices, all of the crossing pairs (e,e ′ ) can be pre-determined in linear time. Let Ω denote the set of pairs of crossing edges. We aim to minimise F =  (e,e ′ )∈Ω  α (e,e ′ ) - π 2  2 , (1) where α (e,e ′ ) denotes the angle at which edges e and e ′ cross. The circular layout is a function θ : V → [0 .. 2π) that associates an angle θ(u) with each vertex u. Suppose that the ordering of the vertices around the circle is (u 0 ,u 1 ,...,u n-1 ). We denote θ(u i ) by θ i . For every edge e of G, let e m and e M denote the end vertices such that θ(e m ) <θ(e M ). For a pair of crossing edges e and e ′ , their crossing angle is given by α (e,e ′ ) = 1 2 (θ(e M ) - θ(e ′ M )+ θ(e m ) - θ(e ′ m )) where θ(e ′ m ) <θ(e m ) <θ(e ′ M ) < θ(e M ). In practice an angular gap g between vertices needs to be preserved, e.g., for avoiding overlapping node labels. The circular ordering and the gap lead to the following constraints: 0 ≤ θ 0 ; θ i + g<θ i+1 , ∀i =0..n - 2; θ n-1 + g< 2π. (2) Minimizing F in equation (1) subject to the constraints (2) defines a quadratic program. U. Brandes and S. Cornelsen (Eds.): GD 2010, LNCS 6502, pp. 397–399, 2011. c  Springer-Verlag Berlin Heidelberg 2011