All-pairs ancestor problems in weighted dags

This work studies (lowest) common ancestor problems in (weighted) directed acyclic graphs. We improve previous algorithms for the all-pairs representative LCA problem to O(n(2.575)) by using fast rectangular matrix multiplication. We prove a first non-trivial upper bound of O(min{n(2)m, n(3.575)}) for the all-pairs all lowest common ancestors problem. Furthermore, classes of dags are identified for which the problem can be solved considerably faster. Our algorithms scale with the maximal number of LCAs for one pair and-based on the famous Dilworth's theorem-with the size of a maximum antichain (i.e., width) of the dag. We extend and generalize previous results on computing shortest ancestral distances. It is shown that finding shortest distance common ancestors in weighted dags is not harder than computing all-pairs shortest distances, up to a polylogarithmic factor. Finally, we present a solution for the general all-pairs shortest distance LCA problem based on computing all-pairs all LCAs.