Fast Lowest Common Ancestor Computations in Dags

This work studies lowest common ancestor problems in directed acyclic graphs. We present fast algorithms for solving the All-Pairs Representative LCA and AllPairs All LCA problems with expected running time of O(n2 log n) and O(n3 log log n) respectively. The speed-ups over recently developed methods are achieved by applying transitive reduction on the input dags. The algorithms are experimentally evaluated against previous approaches demonstrating a significant improvement. On the purely theoretical side, we improve the upper bound for All-Pairs Representative LCA to O(n2.575) and the upper bound for All-Pairs All LCA to O(n3.3399). We give first fully dynamic algorithms for both All-Pairs Representative LCA and All-Pairs All LCA . Here, the update complexities are O(n2.5) and O(n3) respectively, with constant query times.