Ptolemaic Indexing

TL;DR

This paper introduces a new family of bounds for similarity search based on Ptolemy's inequality, applicable to Euclidean and quadratic form metrics, enhancing filtering power in indexing methods.

Contribution

It presents a novel approach to metric indexing using Ptolemy's inequality, expanding the theoretical foundation and practical efficiency of similarity search algorithms.

Findings

Ptolemaic bounds outperform existing triangular methods in filtering power.

Ptolemaic inequality holds approximately for various important distances.

Combining Ptolemaic and triangular filtering improves indexing performance.

Abstract

This paper discusses a new family of bounds for use in similarity search, related to those used in metric indexing, but based on Ptolemy's inequality, rather than the metric axioms. Ptolemy's inequality holds for the well-known Euclidean distance, but is also shown here to hold for quadratic form metrics in general, with Mahalanobis distance as an important special case. The inequality is examined empirically on both synthetic and real-world data sets and is also found to hold approximately, with a very low degree of error, for important distances such as the angular pseudometric and several Lp norms. Indexing experiments demonstrate a highly increased filtering power compared to existing, triangular methods. It is also shown that combining the Ptolemaic and triangular filtering can lead to better results than using either approach on its own.