FAST ALGORITHMS FOR FINDING NEAREST COMMON ANCESTORS.

The authors consider the following problem: Given a collection of rooted trees, answer on-line queries of the form, 'What is the nearest common ancestor of vertices x and y? ' They show that any pointer machine that solves this problem requires OMEGA (log log n) time per query in the worst case, where n is the total number of vertices in the trees. On the other hand, they present an algorithm for a random access machine with uniform cost measure (and a bound of O(log n) on the number of bits per word) that requires O(1) time per query and O(n) preprocessing time, assuming that the collection of trees is static. For a version of the problem in which the trees can change between queries, they obtain an almost-linear-time (and linear-space) algorithm.