Tree Encoding and Transitive Closure Compression

Tree encoding is a very useful mechanism to check ancestor-descendant relationships of the nodes in a tree structure, by which each node is associated with a pair of integers that can be used to characterize the reachability. Recently, this method is extended to directed graphs (digraph for short) G by associating each node in G with a pair sequence. However, no approach is reported to minimize such pair sequences. In this paper, we address this issue and propose an efficient algorithm that is always able to generate minimized pair sequences. The algorithm runs in O(bmiddote + bmiddotnmiddotradicb) time and O(bmiddot n), where n and e are the numbers of the nodes and edges of a DAG, respectively; and b is the DAG's width.