An Efficient Sampling Algorithm for Difficult Tree Pairs

Abstract

It is an open question whether there exists a polynomial-time algorithm for computing the rotation distances between pairs of extended ordered binary trees.The problem of computing the rotation distance between an arbitrary pair of trees, (S, T), can be efficiently reduced to the problem of computing the rotation distance between a difficult pair of trees (S', T'), where there is no known first step which is guaranteed to be the beginning of a minimal length path. Of interest, therefore, is how to sample such difficult pairs of trees of a fixed size. We show that it is possible to do so efficiently, and present such an algorithm that runs in time O(n4).

Counting difficult tree pairs with respect to the rotation distance problem

“…Cleary and Maio [6] have a heuristic sampling algorithm for constructing difficult instances of the rotation distance problem of a specified size by a growing method based upon the method of Remy [16] modified to work with tree pairs and which runs in polynomial time. This is preferable to a "test-and-reject" approach particularly when interested in difficult examples of a particular size, as the fractions of success quickly diminish and become quite small for increasing n.…”

Distributions of restricted rotation distances

“…Cleary, Rechnitzer and Wong [8] analyze some properties of the distribution of the resulting sizes of reduced tree pairs. Cleary and Maio [6] have an algorithm which guarantees to produce not only a reduced tree pair of a specified size, but is difficult in an additional sense as well-not having any obvious initial first moves along minimal length paths. Unfortunately, that algorithm does not choose uniformly from among the possible ones.…”