Decomposing DAGs into disjoint chains

Abstract

In this paper, we propose an efficient algorithm to decompose a directed acyclic graph (DAG) into chains, which has a lot of applications in computer science and engineering. Especially, it can be used to store transitive closures of directed graphs in aneconomical way. For a DAG G with n nodes, our algorithm needs O(n2 + bn√b) time to find a minimized set of disjoint chains, where b is G's width, defined to be the largest node subset U of G such that for every pair of nodes u, v E U, there does not exist a path from u of v or from v to u. Accordingly, the transitive closure of G can be stored in O(bn) space, and the reachability can be checked in O(logb) time. The method can also be extended to handle cyclic directed graphs. © Springer-Verlag Berlin Heidelberg 2007.