TY - GEN
T1 - An art gallery approach to ensuring that landmarks are distinguishable
AU - Erickson, Lawrence H.
AU - Lavalle, Steven M
PY - 2012
Y1 - 2012
N2 - How many different classes of partially distinguishable landmarks are needed to ensure that a robot can always see a landmark without simultaneously seeing two of the same class? To study this, we introduce the chromatic art gallery problem. A guard set S ⊂ P is a set of points in a polygon P such that for all p ∈ P, there exists an s ∈ S such that s and p are mutually visible. Suppose that two members of a finite guard set S ⊂ P must be given different colors if their visible regions overlap. What is the minimum number of colors required to color any guard set (not necessarily a minimal guard set) of a polygon P? We call this number, χG(P), the chromatic guard number of P. We believe this problem has never been examined before, and it has potential applications to robotics, surveillance, sensor networks, and other areas. We show that for any spiral polygon Pspi, χG(Pspi) ≤ 2, and for any staircase polygon (strictly monotone orthogonal polygon) Psta, χG(Psta) ≤ 3. For lower bounds, we construct a polygon with 4k vertices that requires k colors. We also show that for any positive integer k, there exists a monotone polygon Mk with 3k2 vertices such that χG(Mk) ≥ k, and for any odd integer k, there exists an orthogonal polygon Rk with 4k2 + 10k + 10 vertices such that χG(Rk) ≥ k.
AB - How many different classes of partially distinguishable landmarks are needed to ensure that a robot can always see a landmark without simultaneously seeing two of the same class? To study this, we introduce the chromatic art gallery problem. A guard set S ⊂ P is a set of points in a polygon P such that for all p ∈ P, there exists an s ∈ S such that s and p are mutually visible. Suppose that two members of a finite guard set S ⊂ P must be given different colors if their visible regions overlap. What is the minimum number of colors required to color any guard set (not necessarily a minimal guard set) of a polygon P? We call this number, χG(P), the chromatic guard number of P. We believe this problem has never been examined before, and it has potential applications to robotics, surveillance, sensor networks, and other areas. We show that for any spiral polygon Pspi, χG(Pspi) ≤ 2, and for any staircase polygon (strictly monotone orthogonal polygon) Psta, χG(Psta) ≤ 3. For lower bounds, we construct a polygon with 4k vertices that requires k colors. We also show that for any positive integer k, there exists a monotone polygon Mk with 3k2 vertices such that χG(Mk) ≥ k, and for any odd integer k, there exists an orthogonal polygon Rk with 4k2 + 10k + 10 vertices such that χG(Rk) ≥ k.
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M3 - Conference contribution
AN - SCOPUS:84891859671
SN - 9780262517799
T3 - Robotics: Science and Systems
SP - 81
EP - 88
BT - Robotics
A2 - Durrant-Whyte, Hugh
A2 - Abbeel, Pieter
A2 - Roy, Nicholas
PB - MIT Press Journals
T2 - International Conference on Robotics Science and Systems, RSS 2011
Y2 - 27 June 2011 through 1 July 2011
ER -