A Coreset for Approximate Furthest-Neighbor Queries in a Simple Polygon

Core Concepts

The author presents a method to construct a coreset for approximate furthest-neighbor queries in a simple polygon using geodesic distances, ensuring efficient query responses with minimal storage requirements.

Abstract

The content discusses the construction of a coreset for approximate furthest-neighbor queries in a simple polygon using geodesic distances. The method involves placing points strategically to ensure accurate approximations while minimizing storage needs. Lemmas and proofs are provided to support the effectiveness of the approach. Key Points: Introduction to furthest-neighbor queries in various settings. Explanation of exact and approximate data structures for nearest-neighbor queries. Detailed analysis of constructing an ε-coreset for furthest-neighbor queries inside a simple polygon. Utilization of geodesic distance and Voronoi diagrams for efficient query processing. Lemmas and proofs demonstrating the effectiveness of the proposed coreset construction method.

Stats

The coreset can be constructed in O(1/ε(n log(1/ε) + (n + m) log(n + m))) time. A set C ⊂ P of size O(1/ε2) is used for answering ε-approximate furthest neighbor queries.

Quotes

"We prove that there exists, for any ε > 0, a set C ⊂ P of size O(1/ε2) such that...the geodesic distance from q to its furthest neighbor in C is at least 1−ε times..." "The coreset can be constructed in O(1/ε(n log(1/ε) + (n + m) log(n + m))) time."

Key Insights Distilled From

A Coreset for Approximate Furthest-Neighbor Queries in a Simple Polygon

by Mark de Berg... at arxiv.org 03-08-2024

https://arxiv.org/pdf/2403.04513.pdf

Deeper Inquiries

How does the proposed coreset construction method compare to existing approaches

The proposed coreset construction method in the context provided offers a novel approach to addressing approximate furthest-neighbor queries within simple polygons. By introducing the concept of an ε-coreset, the research provides a set C that allows for efficient answering of ε-approximate furthest-neighbor queries with minimal storage requirements. This method stands out from existing approaches by offering a solution that is independent of the size and complexity of the polygon P, providing a constant-size coreset regardless of these factors. The use of canonical directions, pocket edges, and intermediate edges in defining the points placed into Ci showcases a detailed and systematic way to construct an effective coreset for such queries.

What implications does this research have on computational geometry beyond simple polygons

This research on constructing coresets for approximate furthest-neighbor queries in simple polygons has broader implications for computational geometry beyond this specific application. The development of efficient data structures with minimal storage requirements can have significant impacts on various geometric algorithms and problems. For instance, similar techniques could be applied to other types of spatial data structures or geometric optimization problems where reducing storage while maintaining query efficiency is crucial. Additionally, insights gained from studying geodesic distances within polygons can potentially be extended to more complex geometries or even non-Euclidean spaces, opening up new avenues for algorithm design in diverse fields.

How might advancements in geodesic distance calculations impact other areas of algorithm design

Advancements in geodesic distance calculations as demonstrated in this research can have far-reaching effects on algorithm design across different domains. Improved methods for computing geodesic distances efficiently not only benefit geometric algorithms but also impact areas like network analysis, robotics path planning, geographic information systems (GIS), and computer graphics. By enhancing our ability to measure distances along curved paths or surfaces accurately and quickly, advancements in geodesic distance calculations enable more precise modeling and analysis in various applications requiring pathfinding or proximity evaluation based on real-world geometries or terrains.