Continuous Terrain Guarding with Two-Sided Guards

TL;DR

This paper introduces an optimal algorithm for the continuous 1.5D terrain guarding problem with two-sided guards, transforming it into a discrete problem and ensuring all points are guarded by minimal vertex guards.

Contribution

It presents a novel transformation from continuous to discrete terrain guarding, enabling an optimal solution for two-sided guarding in 1.5D terrains.

Findings

Transformation from continuous to discrete terrain guarding

Optimal algorithm for two-sided guarding

Guarantees all points are guarded with minimal guards

Abstract

Herein, we consider the continuous 1.5-dimensional(1.5D) terrain guarding problem with two-sided guarding. We provide an x-monotone chain T and determine the minimal number of vertex guards such that all points of T have been two-sided guarded. A point p is two-sided guarded if there exist two vertices vi (left of p) and (right of p) that both see p. A vertex vi sees a point p on T if the line segment connecting vi to p is on or above T. We demonstrate that the continuous 1.5D terrain guarding problem can be transformed to the discrete terrain guarding problem with a finite point set X and that if X is two-sided guarded, then T is also two-sided guarded. Through this transformation, we achieve an optimal algorithm that solves the continuous 1.5D terrain guarding problem under two-sided guarding.