Abstract

We introduce a variant of the watchman route problem, which we call the quickest pair-visibility problem. Given two persons standing at points s and t in a simple polygon P with no holes, we want to minimize the distance they travel in order to see each other in P. We solve two variants of this problem, one minimizing the longer distance the two persons travel (min-max) and one minimizing the total travel distance (min-sum), optimally in linear time. We also consider a query version of this problem for the min-max variant. We can preprocess a simple n-gon in linear time so that the minimum of the longer distance the two persons travel can be computed in O(log2⁡n) time for any two query positions s,t where the two persons start.

Highlights

  • In the watchman route problem, a watchman takes a route to guard a given region—that is, any point in the region is visible from at least one point on the route

  • Given two points s and t in a simple polygon P, compute the minimum distance that s and t must travel in order to see each other in P

  • We consider two variants of the quickest pair-visibility problem for a simple polygon: either we want to minimize the maximum length of a traveled path or we want to minimize the sum of the lengths of both traveled paths We give a sweep-line-like approach that “rotates” the lines-of-sight along vertices on the shortest path between the start positions, allowing us to evaluate a linear number of candidate solutions on these lines

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Summary

Introduction

In the watchman route problem, a watchman takes a route to guard a given region—that is, any point in the region is visible from at least one point on the route. This problem may sound similar to the shortest path problem between s and t, in which the objective is to compute the shortest path for s to reach t They differ even for a simple case: for any two points lying in a convex polygon, the distance in the quickest pair-visibility problem is zero while in the shortest path problem it is their Euclidean distance. Khosravi and Ghodsi [13] considered the case for a simple n-gon and presented an algorithm to construct a data structure of O(n2) space so that given a query, it finds the shortest visibility path in O(log n) time.

Our results
Preliminaries
Computing All Events for a Sweep-Line-Like Approach
Algorithm Based on a Sweep-Line-Like Approach
Quickest Pair-Visibility Query Problem
Binary Search for the Path-Events
Binary Search for the Boundary-Events
Binary Search for the Bend-Events
Full Text
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