Computing Lowest Common Ancestors in Directed Acyclic Graphs

Finding the lowest common ancestor of a given pair of nodes is a fundamental algorithmic problem. In this paper we study the lowest common ancestor(LCA) problem on directed acyclic graphs (DAGs). A lowest common ancestor of two nodes a and b is a node c which is a common ancestor of a and b and no other node is both a common ancestor of a and b and a proper descendant of c. LCA on trees have been studied extensively. LCA on DAGs has also been investigated by several researchers. Many problems that require finding LCA cannot be solved using tree-LCA algorithms because the structure of DAGs are quite different from that of trees. Nykänen and Ukkonen [6] give a linear-time preprocessing, constant-time-query algorithm for the LCA in arbitrarily directed trees. They ask whether it is possible to preprocess a DAG in o(n3) time to support Θ(k)-time set-LCA queries, where a set-LCA query returns all k lowest common ancestors of the given pair. Ait-Kaci et al. [2] considered the problem of LCA on lattices and lower semi-lattices (where a node pair has a unique LCA). Bender et al. gave an algorithm [3] for all-pairs-representative LCA in DAGs in O(n2.688) time. In this paper we show that all-pairs-representative LCA in DAGs can be computed in O(n2.575) time. This time complexity coincides with the current time complexity for computing all-pairs shortest paths for directed unweighted graphs [7]. Research supported in part by NSF grant 0310245.