The nearest common ancestor in a dynamic tree

Abstract

We consider the problem of determining the nearest common ancestor of two given nodes x and y (denoted by nca(x,y)) in a dynamic arbitrary tree T. We present an implementation of T by a pointer machine which needs linear space, performs m arbitrary insertions and deletions in the initially empty tree T in time O(m) and a query about nca(x,y) can be answered on-line in time O(log(min{depth(x), depth(y)})+α(k,k)), where the second factor is amortized over k queries, α is a functional inverse of Ackermann's function and depth(x) the distance from node x to the root of T.

We also consider the static version of the problem and give an optimal pointer machine algorithm which needs O(n) preprocessing time, space O(n) and answers a query on-line in time O(log log n).