On the DAG Decomposition

3 Yangjun Chen, Yibin Chen 4 5 6 University of Winnipeg 7 8 9 10 11 . 12 ABSTRACT 13 14 In this paper, we propose an efficient algorithm to decompose a directed acyclic graph G into a minimized set of node-disjoint chains, which cover all the nodes of G. For any two nodes u and v on a chain, if u is above v then there is a path from u to v in G. The best algorithm for this problem up to now needs O(n) time, where n is the number of the nodes of G. Our algorithm, however, needs only O(κ⋅n) time, where κ is G’s width, defined to be the size of a largest node subset U of G such that for every pair of nodes x, y ∈ U, there does not exist a path from x to y or from y to x. More importantly, by the existing algorithm, O(n) extra space (besides the space for G itself) is required to maintain the transitive closure of G to do the task while ours needs only O(κ⋅n) extra space. 15