How to Layer a Directed Acyclic Graph

Abstract

We consider the problem of partitioning a directed acyclic graph into layers such that all edges point unidirectionally. We perform an experimental analysis of some of the existing layering algorithms and then propose a new algorithm that is more realistic in the sense that it is possible to incorporate specific information about node and edge widths into the algorithm. The goal is to minimize the total sum of edge spans subject to dimension constraints on the drawing. We also present some preliminary results from experiments we have conducted using our layering algorithm on over 5900 example directed acyclic graphs.

In search for efficient heuristics for minimum-width graph layering with consideration of dummy nodes

“…It also finds layerings with the minimum height. However, it performs very poorly in terms of drawing area, number of dummy nodes, and edge density [Healy and Nikolov 2002b]. The longest-path layerings tend to be very wide at the bottom layers.…”

Applying Ant Colony Optimization Metaheuristic to the DAG Layering Problem

“…The attractiveness of this method is that it has linear time complexity (because the graph is acyclic) and it uses the minimum number of layers possible. The disadvantage of the LPL method is that its layerings tend to be too wide [6]. Because the area occupied by the final drawing depends on both its width and its height the Longest-Path Layering is not the best choice if minimal layering area is the main priority.…”

Upward Planarization Layout

“…Healy and Nikolov give an experimental analysis of existing layering algorithms for DAGs. They also give an ILP formulation which computes a layering with minimum number of dummy nodes with a given upper bound on the width and height of the layering and an branch and cut algorithm to solve it [14,13]. However, a major drawback of Sugiyama's framework could not be solved by any of these modifications: Since layer assignment and crossing reduction are realized as independent steps, the resulting drawing might have many unnecessary crossings caused by an unfortunate layer assignment.…”

“…[6]. (a) input DAG G augmented to G via the artificial super sourceŝ; (b) embedded feasible subgraph U of G obtained by deleting the arcs (10, 13), (2,14), (4,3), (6,5); (c) upward planar representation R of G after reinserting the deleted arcs (dashed line). R contains five crossing dummies.…”