An o(n log n) set refinement algorithm with applications

Given two partitions A and B of a totally ordered set S, of size n, corresponding to the equivalence relations RA and RB respectively, the refinement problem calls for the computation of the partition C corresponding to the relation RA and RB We present an O(n log n) algorithm for this problem. As an application, we present an algorithm that converts an inverted lists structure to a multiple attribute tree for a set of multi-dimensional points. Using the idea of the proposed algorithm we present an alternate O(n log n) algorithm for an existing partition algorithm. The alternate algorithm is very simple to implement and direct to prove.