Graph Encoding and Transitive Closure Representation

Abstract

Composite objects represented as directed graphs are an important data structure that require efficient support Web and document databases (Abiteboul, Cluet, Christophides, Milo, Moerkotte, & Simon, 1997; Chen & Aberer, 1998, 1999; Mendelzon, Mihaila, & Milo, 1997; Zhang, Naughton, Dewitt, Luo, & Lohman, 2001), CAD/ CAM, CASE, office systems, and software management. It is cumbersome to handle such objects in relational database systems when they involve ancestor-descendant relations (or say, reachability relations). In this article, we present a new graph encoding based on a tree labeling method and the concept of branchings that are used in the graph theory for finding the shortest connection networks. A branching is a subgraph of a given digraph that is in fact a forest, but covers all the nodes of the graph. Concretely, for a DAG G (directed acyclic graph) of n nodes, the space needed for storing its transitive closure can be reduced to O(b·n), where b is the number of the leaf nodes of G’s branching. Such a compression is, however, at the expense of querying time. Theoretically, it takes O(logb) time to check whether a node is reachable from another. The method can also be extended to digraphs containing cycles.