Fr\'echet Distance Problems in Weighted Regions

TL;DR

This paper presents approximation algorithms for computing the Fréchet distance in weighted planar subdivisions and 3D obstacle environments, addressing two variants involving weighted lengths and shortest paths.

Contribution

It introduces (1+epsilon)-approximation algorithms for the Fréchet distance in weighted regions and in 3D obstacle spaces, extending previous work to more complex weighted and 3D scenarios.

Findings

Algorithms achieve (1+epsilon)-approximation for weighted regions.

Extends Fréchet distance approximation to 3D obstacle environments.

Provides computational methods for complex weighted and obstacle scenarios.

Abstract

We discuss two versions of the Fr\'echet distance problem in weighted planar subdivisions. In the first one, the distance between two points is the weighted length of the line segment joining the points. In the second one, the distance between two points is the length of the shortest path between the points. In both cases, we give algorithms for finding a (1+epsilon)-factor approximation of the Fr\'echet distance between two polygonal curves. We also consider the Fr\'echet distance between two polygonal curves among polyhedral obstacles in R^3 and present a (1+epsilon)-factor approximation algorithm.