Fast lowest common ancestor computations in dags

Abstract

This work studies lowest common ancestor computations in directed acyclic graphs. We present fast algorithms for solving the ALL-PAIRS REPRESENTATIVE LCA and ALL-PAIRS ALL LCA problems with expected running time of O(n2 log n) and O(n3 log log n) respectively, where the expectation is taken over a distribution of input graphs. 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 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 non-trivial update complexities are O(n 2.5) and O(n3) respectively, with constant query times. © Springer-Verlag Berlin Heidelberg 2007.