Computing a rectilinear shortest path amid splinegons in plane

TL;DR

This paper presents a reduction method to compute rectilinear shortest paths amid splinegons by transforming the problem into a polygonal domain, with efficient algorithms for disjoint splinegons and concave-in cases.

Contribution

It introduces a novel reduction technique from splinegonal to polygonal domains and provides specialized algorithms for disjoint and concave-in splinegons, along with path property characterizations.

Findings

Reduction algorithm runs in O(n + h log n) time

Efficient algorithms for disjoint splinegons and concave-in splinegons

Characterized properties of rectilinear shortest paths amid splinegons

Abstract

We reduce the problem of computing a rectilinear shortest path between two given points s and t in the splinegonal domain \calS to the problem of computing a rectilinear shortest path between two points in the polygonal domain. As part of this, we define a polygonal domain \calP from \calS and transform a rectilinear shortest path computed in \calP to a path between s and t amid splinegon obstacles in \calS. When \calS comprises of h pairwise disjoint splinegons with a total of n vertices, excluding the time to compute a rectilinear shortest path amid polygons in \calP, our reduction algorithm takes O(n + h \lg{n}) time. For the special case of \calS comprising of concave-in splinegons, we have devised another algorithm in which the reduction procedure does not rely on the structures used in the algorithm to compute a rectilinear shortest path in polygonal domain. As part of these, we have characterized few of the properties of rectilinear shortest paths amid splinegons which could be of independent interest.