A fixed-parameter algorithm for guarding 1.5D terrains

A 1.5D terrain is a region on a plane determined by an x-monotone polygonal chain. A set G of points on the terrain is called a guarding set if every point on the terrain is seen by some point in G. In the 1.5D terrain guarding problem we are given a 1.5D terrain and the goal is to find the minimum...

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Bibliographic Details
Published in:Theoretical computer science Vol. 595; pp. 130 - 142
Main Authors: Khodakarami, Farnoosh, Didehvar, Farzad, Mohades, Ali
Format: Journal Article
Language:English
Published: Elsevier B.V 30.08.2015
Subjects:
1.5D terrain
Fixed-parameter algorithm
Guarding
Onion peeling
Tree decomposition
1.5D terrain
Tree decomposition
Onion peeling
Guarding
Fixed-parameter algorithm
ISSN:0304-3975, 1879-2294
Online Access:Get full text
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Summary:A 1.5D terrain is a region on a plane determined by an x-monotone polygonal chain. A set G of points on the terrain is called a guarding set if every point on the terrain is seen by some point in G. In the 1.5D terrain guarding problem we are given a 1.5D terrain and the goal is to find the minimum guarding set for the given input terrain. It is proved that this problem is NP-hard and the only previous theoretical results for this problem involve approximation. In this paper, we turn to fixed-parameter tractability. We present “depth of terrain onion peeling” as a new geometric parameter. Based on this parameter, we give an upper bound for the treewidth of the terrain visibility graph. By presenting a dynamic programming algorithm, we show that the 1.5D terrain guarding problem is fixed-parameter tractable with respect to this parameter. •“the depth of the terrain onion peeling” is introduced as a parameter.•Tree decomposition of the visibility graph of a 1.5D terrain is constructed and its width is bounded by this parameter.•“terrain path decomposition” is introduced.•A dynamic programming algorithm is presented to produce a solution to the 1.5D terrain guarding problem.•1.5D terrain guarding problem is fixed-parameter tractable with respect to “the depth of terrain onion peeling”.
ISSN:0304-3975
1879-2294
DOI:10.1016/j.tcs.2015.06.028