Randomized Incremental Construction of Net-Trees

TL;DR

This paper presents a simple randomized algorithm for constructing net-trees efficiently on doubling metrics, introduces a new linear-size net-tree variant, and connects these structures to approximate Voronoi diagrams to simplify point location.

Contribution

It introduces a new randomized algorithm with $O(n ext{log}n)$ expected time for net-tree construction and a linear-size net-tree variant that simplifies analysis and algorithms.

Findings

Constructs net-trees in expected $O(n ext{log}n)$ time.

Defines a linear-size net-tree variant simplifying analysis.

Establishes a connection between net-trees and approximate Voronoi diagrams.

Abstract

Net-trees are a general purpose data structure for metric data that have been used to solve a wide range of algorithmic problems. We give a simple randomized algorithm to construct net-trees on doubling metrics using $O(n\log n)$ time in expectation. Along the way, we define a new, linear-size net-tree variant that simplifies the analyses and algorithms. We show a connection between these trees and approximate Voronoi diagrams and use this to simplify the point location necessary in net-tree construction. Our analysis uses a novel backwards analysis that may be of independent interest.