Global k-Level Crossing Reduction

Abstract

Directed graphs are commonly drawn by a four phase framework introduced by Sugiyama et al. in 1981. The vertices are placed on parallel horizontal levels. The edge routing between consecutive levels is computed by solving one-sided 2-level crossing minimization problems, which are repeated in up and down sweeps over all levels. Crossing minimization problems are generally N P-hard.We introduce a global crossing reduction, which at any particular time considers all crossings between all levels. Our approach is based on the sifting technique. It yields an improvement of 5 -10% in the number of crossings over the level-by-level one-sided 2-level crossing reduction heuristics. In addition, it avoids type 2 conflicts which are crossings between edges whose endpoints are dummy vertices. This helps straightening long edges spanning many levels. Finally, the global crossing reduction approach can directly be extended to cyclic, radial, and clustered level graphs achieving similar improvements. The running time is quadratic in the size of the input graph, whereas the common level-by-level approaches are faster but operate on larger graphs with many dummy vertices for long edges.

Grid sifting

“…PROOF. A horizontal sifting step of a block B needs O(|E| · deg(B)) time [Bachmaier et al 2011]. Hence, (horizontally) sifting an edge block takes O(|E|) time, as its degree is 2.…”

“…We extend our approach of global sifting [Bachmaier et al 2011] and try to find optimal positions for each (vertex) block on all levels. We use a two-dimensional grid on which we place each vertex block (see Figure 4(a)).…”

“…As the level assignment φ does not change during a horizontal sifting step, the algorithm basically equals the sifting step introduced in global sifting [Bachmaier et al 2011]. …”

“…Moreover, metaheuristics have been suggested in literature, for example, genetic algorithms [Utech et al 1998;Kuntz et al 2006], tabu search [Laguna et al 1997], or windows optimization [Eschbach et al 2002]. We have presented a global sifting algorithm [Bachmaier et al 2011], which aligns long edges to blocks and tries to find an apt position for each block as a whole.…”

“…We extend global sifting [Bachmaier et al 2011] to grid sifting such that the blocks are not only sifted "horizontally" on their levels but also "vertically" on all suitable levels and prioritize the crossing reduction over the leveling. The algorithm yields better results than traditional heuristics.…”