DAGmaps and dominance relationships

Abstract

In [2] we use the term DAGmap to describe space filling visualizations of DAGs according to constraints that generalize treemaps and we show that deciding whether or not a DAG admits a DAGmap drawing is NP-complete. Let G = (V,E) be a DAG with a single source s. A component st-graph G u,v of G is a subgraph of G with a single source u and a single sink v that contains at least two edges and that is connected with the rest of G through vertex u and/or vertex v. A vertex w dominates a vertex v if every path from s to v passes through w. The dominance relation in G can be represented in compact form as a tree T, called the dominator tree of G, in which the dominators of a vertex v are its ancestors. Vertex w is the immediate dominator of v if w is the parent of v in T. A simple and fast algorithm to compute T has been proposed by Cooper et al. [1]. The post-dominators of G are defined as the dominators in the graph obtained from G by reversing all directed edges and assuming that all vertices are reachable from a (possibly artificial) vertex t. Using the definition of DAGmaps, it is easy to prove that in a DAGmap of G the rectangle of a vertex u includes the rectangles of all vertices that are dominated (resp. post-dominated) by u. Therefore when vertex u dominates vertex v and vertex v post-dominates vertex u then the rectangles Ru and Rv of u and v coincide. Based on this observation, we propose a heuristic algorithm that transforms a DAG G into a DAG G′ that admits a DAGmap. When G contains component st-graphs then our algorithm performs significantly fewer duplications than the transformation of G into a tree. © 2010 Springer-Verlag.