Querying Two Boundary Points for Shortest Paths in a Polygonal Domain

TL;DR

This paper introduces a data structure for efficiently answering shortest path queries between boundary points in a polygonal domain, achieving logarithmic query time with high preprocessing costs, and offers a space-time tradeoff.

Contribution

It presents a novel approach connecting Davenport-Schinzel sequences to shortest path queries, enabling fast boundary-to-boundary shortest path computations with high preprocessing and flexible space-time tradeoffs.

Findings

Logarithmic query time with O~(n^5) preprocessing and space.

A tradeoff allows sublinear query time with O(n^{3+ε}) space.

Extension to shortest paths between points on line segments.

Abstract

We consider a variant of two-point Euclidean shortest path query problem: given a polygonal domain, build a data structure for two-point shortest path query, provided that query points always lie on the boundary of the domain. As a main result, we show that a logarithmic-time query for shortest paths between boundary points can be performed using O~ (n^5) preprocessing time and O(n^5) space where n is the number of corners of the polygonal domain and the O~ notation suppresses the polylogarithmic factor. This is realized by observing a connection between Davenport-Schinzel sequences and our problem in the parameterized space. We also provide a tradeoff between space and query time; a sublinear time query is possible using O(n^{3+epsilon}) space. Our approach also extends to the case where query points should lie on a given set of line segments.