Abstract
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.
Highlights
Dominance drawings of directed acyclic graphs (DAGs) are very important in many areas of research, including graph drawing [1], computational geometry [2], and information visualization [3], even in very large databases [4], just to mention a few
We show that the compressed transitive closure of G given Sc can be seen as a k-dimensional dominance drawing of G
Dominance Drawing is a concept that can be very useful for answering reachability queries, that is, checking the existence of a path connecting two vertices in graph databases.Recall that given a dominance drawing of a DAG G having k dimensions, it is possible to answer to a reachability query very efficiently in O(k) time by using O(kn) space
Summary
Dominance drawings of directed acyclic graphs (DAGs) are very important in many areas of research, including graph drawing [1], computational geometry [2], and information visualization [3], even in very large databases [4], just to mention a few. A linear-time algorithm that constructs two-dimensional dominance drawings of upward planar graphs is described in [2] (see [6]). Hua, and Zhou considered high-dimensional weak dominance drawings in order to obtain efficient solutions to the reachability query problem [10] Their experimental results show that their algorithms compute weak dominance drawings using few dimensions and having few fips. These results suggest that a possible direction for this line of research is to use directly dominance drawings instead of weak dominance drawings This technique is based on the concept of compressed transitive closure, which is a data structure introduced in [12] that can be used to answer any reachability query in O(1) and requires O(kn) space, where k is a parameter less than or equal to n. We discuss the option of using weak dominance more extensively and close our paper by presenting our conclusions and future research challenges
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