DeepAI AI Chat
Log In Sign Up

A Ptolemaic Partitioning Mechanism

08/19/2022
by   Richard Connor, et al.
University of St Andrews
0

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.

READ FULL TEXT

page 1

page 2

page 3

page 4

09/27/2021

Counting colorings of triangle-free graphs

By a theorem of Johansson, every triangle-free graph G of maximum degree...
07/08/2021

A Triangle Inequality for Cosine Similarity

Similarity search is a fundamental problem for many data analysis techni...
10/18/2022

Capacitated Vehicle Routing in Graphic Metrics

We study the capacitated vehicle routing problem in graphic metrics (gra...
02/14/2020

An Inductive Bias for Distances: Neural Nets that Respect the Triangle Inequality

Distances are pervasive in machine learning. They serve as similarity me...
02/22/2023

Lower Bounds for Intersection Reporting among Flat Objects

Recently, Ezra and Sharir [ES22a] showed an O(n^3/2+σ) space and O(n^1/2...
09/01/2020

Rank-one partitioning: formalization, illustrative examples, and a new cluster enhancing strategy

In this paper, we introduce and formalize a rank-one partitioning learni...
06/12/2019

A New Proof of Hopf's Inequality Using a Complex Extension of the Hilbert Metric

It is well known from the Perron-Frobenius theory that the spectral gap ...