Multidimensional Dominance Drawings and Their Applications

In a dominance drawing Γ of a directed acyclic graph (DAG) G, a vertex v is reachable from a vertex u if, and only if all the coordinates of v are greater than or equal to the coordinates of u in Γ. Dominance drawings of DAGs are very important in many areas of research. They combine the aspect of drawing a DAG on the grid with the fact that the transitive closure of the DAG is apparently obvious by the dominance relation between grid points associated with the vertices. The smallest number d for which a given DAG G has a d-dimensional dominance drawing is called dominance drawing dimension, and it is NP-hard to compute. In this paper, we present efficient algorithms for computing dominance drawings of G with a number of dimensions respecting theoretical bounds. We first describe a simple algorithm that shows how to compute a dominance drawing of G from its compressed transitive closure. Next, we describe a more complicated algorithm, which is based on the concept of modular decomposition of G, and obtaining dominance drawings with a lower number of dimensions. Finally, we consider the concept of weak dominance, a relaxed version of the dominance, and we discuss interesting experimental results.