Altitude Terrain Guarding and Guarding Uni-Monotone Polygons

TL;DR

This paper introduces an optimal linear-time algorithm for terrain guarding with guards on a line, proving the equivalence of guard sets and witness points, and extends results to monotone mountain polygons, establishing a new class of perfect polygons.

Contribution

It presents the first linear-time algorithm for terrain guarding with guards on a line and proves the perfectness of monotone mountain polygons.

Findings

Optimal linear-time algorithm for terrain guarding.

Guarding number equals witness number for monotone mountains.

First non-trivial class of perfect polygons established.

Abstract

We present an optimal, linear-time algorithm for the following version of terrain guarding: given a 1.5D terrain and a horizontal line, place the minimum number of guards on the line to see all of the terrain. We prove that the cardinality of the minimum guard set coincides with the cardinality of a maximum number of ``witnesses'', i.e., terrain points, no two of which can be seen by a single guard. We show that our results also apply to the Art Gallery problem in ``monotone mountains'', i.e., $x$-monotone polygons with a single edge as one of the boundary chains. This means that any monotone mountain is ``perfect'' (its guarding number is the same as its witness number); we thus establish the first non-trivial class of perfect polygons.