The Art Gallery Problem for polyhedra Carleton Algorithms Seminar Giovanni Viglietta School of Computer Science, Carleton University February 8, 2013 The Art Gallery Problem for polyhedra
Art Gallery Problem Planar version (Klee, 1973): Given a polygon, choose a minimum number of points that collectively see its whole interior. The Art Gallery Problem for polyhedra
Art Gallery Problem Planar version (Klee, 1973): Given a polygon, choose a minimum number of points that collectively see its whole interior. The Art Gallery Problem for polyhedra
Art Gallery Problem Planar version (Klee, 1973): Given a polygon, choose a minimum number of points that collectively see its whole interior. The Art Gallery Problem for polyhedra
Art Gallery Problem Planar version (Klee, 1973): Given a polygon, choose a minimum number of points that collectively see its whole interior. Research problem: Generalize to polyhedra . The Art Gallery Problem for polyhedra
Fisk’s solution: sufficiency � n � For polygons with n vertices, vertex guards are sufficient. 3 The Art Gallery Problem for polyhedra
Fisk’s solution: sufficiency � n � For polygons with n vertices, vertex guards are sufficient. 3 The Art Gallery Problem for polyhedra
Fisk’s solution: sufficiency � n � For polygons with n vertices, vertex guards are sufficient. 3 The Art Gallery Problem for polyhedra
Fisk’s solution: sufficiency � n � For polygons with n vertices, vertex guards are sufficient. 3 The Art Gallery Problem for polyhedra
Fisk’s solution: necessity � n � guards may be necessary. 3 The Art Gallery Problem for polyhedra
Guarding orthogonal polygons: O’Rourke’s solution � n � vertex guards are sufficient and occasionally necessary. 4 The Art Gallery Problem for polyhedra
Guarding orthogonal polygons: O’Rourke’s solution � n � vertex guards are sufficient and occasionally necessary. 4 The Art Gallery Problem for polyhedra
Guarding orthogonal polygons: O’Rourke’s solution � n � vertex guards are sufficient and occasionally necessary. 4 The Art Gallery Problem for polyhedra
Guarding orthogonal polygons: O’Rourke’s solution � n � vertex guards are sufficient and occasionally necessary. 4 � r � In terms of the number of reflex vertices, + 1 . 2 The Art Gallery Problem for polyhedra
Guarding triangulated terrains � n � vertex guards are sufficient and occasionally necessary 2 (Bose et al.). The Art Gallery Problem for polyhedra
Guarding triangulated terrains � n � vertex guards are sufficient and occasionally necessary 2 (Bose et al.). � n � edge guards are sufficient (Everett et al.). 3 � 4n − 4 � edge guards are occasionally necessary (Bose et al.). 13 The Art Gallery Problem for polyhedra
Guarding triangulated terrains � n � vertex guards are sufficient and occasionally necessary 2 (Bose et al.). � n � edge guards are sufficient (Everett et al.). 3 � 4n − 4 � edge guards are occasionally necessary (Bose et al.). 13 All proofs are essentially combinatorial (i.e., not geometric). The Art Gallery Problem for polyhedra
Terminology Polyhedra genus 0 genus 1 genus 2 The Art Gallery Problem for polyhedra
Terminology Orthogonal polyhedron Reflex edge The Art Gallery Problem for polyhedra
Generalizing guards Vertex guards vs. edge guards. The Art Gallery Problem for polyhedra
Generalizing guards Vertex guards vs. edge guards. (Face guards?) The Art Gallery Problem for polyhedra
Computational complexity All known 2D variations of the Art Gallery Problem are NP-hard and APX-hard. By tweaking the 2D constructions, similar results can be obtained for all types of 3D guards. No 2D variation is known to be in APX. The Art Gallery Problem for polygons with holes is as hard to approximate as SET COVER. This extends to simply connected polyhedra (holes can become “pillars” that almost reach the ceiling). The Art Gallery Problem for polyhedra
Vertex-guarding orthogonal polyhedra The Art Gallery Problem for vertex guards is unsolvable, even for orthogonal polyhedra. Some points in the central region are invisible to all vertices (hence this polyhedron is not tetrahedralizable ). The Art Gallery Problem for polyhedra
Point-guarding orthogonal polyhedra Some orthogonal polyhedra require Ω ( n 3 / 2 ) point guards. outer view cross section Every orthogonal polyhedron yields a BSP tree of size O ( n 3 / 2 ) (Paterson, Yao), hence the bound is tight. The Art Gallery Problem for polyhedra
Point-guarding general polyhedra For polygons with r reflex edges, there exists a partition into O ( r 2 ) convex parts (Chazelle), hence this many point guards are sufficient. The Art Gallery Problem for polyhedra
Point-guarding general polyhedra For polygons with r reflex edges, there exists a partition into O ( r 2 ) convex parts (Chazelle), hence this many point guards are sufficient. Open question: Do Chazelle’s polyhedra provide a tight lower bound? The Art Gallery Problem for polyhedra
Edge-guarding orthogonal polyhedra Any polyhedron is guarded by the set of its edges. Upper bound: e edge guards. The Art Gallery Problem for polyhedra
Edge-guarding orthogonal polyhedra Any polyhedron is guarded by the set of its edges. Upper bound: e edge guards. e Lower bound: 12 edge guards. The Art Gallery Problem for polyhedra
Edge-guarding orthogonal polyhedra Any polyhedron is guarded by the set of its reflex edges. Upper bound: r reflex edge guards. The Art Gallery Problem for polyhedra
Edge-guarding orthogonal polyhedra Any polyhedron is guarded by the set of its reflex edges. Upper bound: r reflex edge guards. � r � Lower bound: + 1 reflex edge guards. 2 The Art Gallery Problem for polyhedra
Open edge guards Closed edge guards vs. open edge guards. The Art Gallery Problem for polyhedra
Open edge guards Closed edge guards vs. open edge guards. Motivation for open edge guards: Each illuminated point receives light from a non-degenerate subsegment of a guard. The Art Gallery Problem for polyhedra
Open edge guards Closed edge guards vs. open edge guards. Motivation for open edge guards: Each illuminated point receives light from a non-degenerate subsegment of a guard. Problem: How much more powerful are closed edge guards? The Art Gallery Problem for polyhedra
Closed vs. open edge guards Closed edge guards are at least 3 times more powerful. No open edge can see more than one red dot. The Art Gallery Problem for polyhedra
Closed vs. open edge guards Closed edge guards are at least 3 times more powerful. No open edge can see more than one red dot. Is this lower bound tight? The Art Gallery Problem for polyhedra
Closed vs. open edge guards In orthogonal polyhedra, each endpoint of a closed edge guard can be replaced by an adjacent open edge. Case-by-case analysis on all vertex types. F E B C D A The Art Gallery Problem for polyhedra
Closed vs. open edge guards In orthogonal polyhedra, each endpoint of a closed edge guard can be replaced by an adjacent open edge. Case-by-case analysis on all vertex types. F E B C D A Our previous bound is tight for orthogonal polyhedra. The Art Gallery Problem for polyhedra
Edge guards as patroling guards Model for patroling guards. An edge guard cannot be replaced by finitely many point guards lying on it. The right endpoint must be a limit point for the guarding set. The Art Gallery Problem for polyhedra
Polyhedral faces: a poor model for patroling guards The top face guard cannot be replaced by o ( n 2 ) patroling guards lying on it. The Art Gallery Problem for polyhedra
Polyhedral faces: a poor model for patroling guards The top face guard cannot be replaced by o ( n 2 ) patroling guards lying on it. The Art Gallery Problem for polyhedra
Face-guarding polyhedra: upper bound Theorem Every c-oriented polyhedron with f faces is guardable by � f � 2 − f c (open or closed) face guards. The Art Gallery Problem for polyhedra
Face-guarding polyhedra: upper bound Theorem Every c-oriented polyhedron with f faces is guardable by � f � 2 − f c (open or closed) face guards. � f � For orthogonal polyhedra ( c = 3): face guards. 6 � f � For 4-oriented polyhedra: face guards. 4 � f � For general polyhedra ( c = f ): − 1 face guards. 2 The Art Gallery Problem for polyhedra
Face-guarding orthogonal polyhedra � f � closed face guards are occasionally necessary. 7 The Art Gallery Problem for polyhedra
Face-guarding orthogonal polyhedra � f � open face guards are always sufficient and occasionally 6 necessary. The Art Gallery Problem for polyhedra
Face-guarding 4-oriented polyhedra � f � closed face guards are occasionally necessary. 5 The Art Gallery Problem for polyhedra
Face-guarding 4-oriented polyhedra � f � open face guards are always sufficient and occasionally 4 necessary. The Art Gallery Problem for polyhedra
Guarding 2-reflex orthogonal polyhedra Problem: Optimally guarding with reflex edge guards an orthogonal polyhedron having reflex edges in only two directions. The Art Gallery Problem for polyhedra
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