A Ptolemaic Partitioning Mechanism

Abstract

For many years, exact metric search relied upon the property of triangle inequality to give a lower bound on uncalculated distances. Two exclusion mechanisms derive from this property, generally known as pivot exclusion and hyperplane exclusion. These mechanisms work in any proper metric space and are the basis of many metric indexing mechanisms. More recently, the Ptolemaic and four-point lower bound properties have been shown to give tighter bounds in some subclasses of metric space. Both triangle inequality and the four-point lower bound directly imply straightforward partitioning mechanisms: that is, a method of dividing a finite space according to a fixed partition, in order that one or more classes of the partition can be eliminated from a search at query time. However, up to now, no partitioning principle has been identified for the Ptolemaic inequality, which has been used only as a filtering mechanism. Here, a novel partitioning mechanism for the Ptolemaic lower bound is presented. It is always better than either pivot or hyperplane partitioning. While the exclusion condition itself is weaker than Hilbert (four-point) exclusion, its calculation is cheaper. Furthermore, it can be combined with Hilbert exclusion to give a new maximum for exclusion power with respect to the number of distances measured per query.