On the transitive closure representation and adjustable compression

Abstract

A composite object represented as a directed graph (digraph for short) is an important data structure that requires efficient support in CAD/CAM, CASE, office systems, software management, web databases, and document databases. It is cumbersome to handle such objects in relational database systems when they involve ancestor-descendant relationships (or say, recursive relationships). In this paper, we present a new encoding method to label a digraph, which reduces the footprints of all previous strategies. This method is based on a tree labeling method and the concept of branchings that are used in 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. On the one hand, the proposed encoding scheme achieves the smallest space requirements among all previously published strategies for recognizing recursive relationships. On the other hand, it leads to a new algorithm for computing transitive closures for DAGs (directed acyclic graph) in O(e·b) time and O(n·b) space, where n represents the number of the nodes of a DAG, e the numbers of the edges, and b the DAG's breadth. The method can also be extended to graphs containing cycles. Especially, based on this encoding method, a multi-level compression is developed, by means of which the space for the representation of a transitive closure can be reduced to O((b/dk)·n), where k is the number of compression levels and d is the average outdegree of the nodes.