We show that the problem of guarding an x-monotone terrain from an altitude line and the problem of guarding a uni-monotone polygon are equivalent. We present a polynomial time algorithm for both problems, and show that the cardinality of a minimum guard set and the cardinality of a maximum witness set coincide.Thus, uni-monotone polygons are perfect; this result also extends to monotone mountains .
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 .
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show that the problem of guarding an x-monotone terrain from an altitude line and the problem of guarding a uni-monotone polygon are equivalent. We present a polynomial time algorithm for both problems, and show
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
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witness set coincide.Thus, uni-monotone polygons are perfect; this result also extends to monotone mountains
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