The Art Gallery Problem for polyhedra
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1 Carleton Algorithms Seminar Giovanni Viglietta School of Computer Science, Carleton University February 8, 2013
2 Art Gallery Problem Planar version (Klee, 1973): Given a polygon, choose a minimum number of points that collectively see its whole interior.
3 Art Gallery Problem Planar version (Klee, 1973): Given a polygon, choose a minimum number of points that collectively see its whole interior.
4 Art Gallery Problem Planar version (Klee, 1973): Given a polygon, choose a minimum number of points that collectively see its whole interior.
5 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.
6 Fisk s solution: sufficiency For polygons with n vertices, n 3 vertex guards are sufficient.
7 Fisk s solution: sufficiency For polygons with n vertices, n 3 vertex guards are sufficient.
8 Fisk s solution: sufficiency For polygons with n vertices, n 3 vertex guards are sufficient.
9 Fisk s solution: sufficiency For polygons with n vertices, n 3 vertex guards are sufficient.
10 Fisk s solution: necessity n 3 guards may be necessary.
11 Guarding orthogonal polygons: O Rourke s solution n 4 vertex guards are sufficient and occasionally necessary.
12 Guarding orthogonal polygons: O Rourke s solution n 4 vertex guards are sufficient and occasionally necessary.
13 Guarding orthogonal polygons: O Rourke s solution n 4 vertex guards are sufficient and occasionally necessary.
14 Guarding orthogonal polygons: O Rourke s solution n 4 vertex guards are sufficient and occasionally necessary. In terms of the number of reflex vertices, r
15 Guarding triangulated terrains n 2 vertex guards are sufficient and occasionally necessary (Bose et al.).
16 Guarding triangulated terrains n 2 vertex guards are sufficient and occasionally necessary (Bose et al.). edge guards are sufficient (Everett et al.). n 3 4n 4 13 edge guards are occasionally necessary (Bose et al.).
17 Guarding triangulated terrains n 2 vertex guards are sufficient and occasionally necessary (Bose et al.). edge guards are sufficient (Everett et al.). n 3 4n 4 13 edge guards are occasionally necessary (Bose et al.). All proofs are essentially combinatorial (i.e., not geometric).
18 Terminology Polyhedra genus 0 genus 1 genus 2
19 Terminology Orthogonal polyhedron Reflex edge
20 Generalizing guards Vertex guards vs. edge guards.
21 Generalizing guards Vertex guards vs. edge guards. (Face guards?)
22 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).
23 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).
24 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.
25 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.
26 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?
27 Edge-guarding orthogonal polyhedra Any polyhedron is guarded by the set of its edges. Upper bound: e edge guards.
28 Edge-guarding orthogonal polyhedra Any polyhedron is guarded by the set of its edges. Upper bound: e edge guards. Lower bound: e 12 edge guards.
29 Edge-guarding orthogonal polyhedra Any polyhedron is guarded by the set of its reflex edges. Upper bound: r reflex edge guards.
30 Edge-guarding orthogonal polyhedra Any polyhedron is guarded by the set of its reflex edges. Upper bound: r reflex edge guards. Lower bound: r reflex edge guards.
31 Open edge guards Closed edge guards vs. open edge guards.
32 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.
33 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?
34 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.
35 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?
36 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 D B C A E
37 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 D B C A E Our previous bound is tight for orthogonal polyhedra.
38 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.
39 Polyhedral faces: a poor model for patroling guards The top face guard cannot be replaced by o(n 2 ) patroling guards lying on it.
40 Polyhedral faces: a poor model for patroling guards The top face guard cannot be replaced by o(n 2 ) patroling guards lying on it.
41 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.
42 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. For orthogonal polyhedra (c = 3): f 6 face guards. For 4-oriented polyhedra: f 4 face guards. For general polyhedra (c = f ): f 2 1 face guards.
43 Face-guarding orthogonal polyhedra f 7 closed face guards are occasionally necessary.
44 Face-guarding orthogonal polyhedra f 6 open face guards are always sufficient and occasionally necessary.
45 Face-guarding 4-oriented polyhedra f 5 closed face guards are occasionally necessary.
46 Face-guarding 4-oriented polyhedra f 4 open face guards are always sufficient and occasionally necessary.
47 Guarding 2-reflex orthogonal polyhedra Problem: Optimally guarding with reflex edge guards an orthogonal polyhedron having reflex edges in only two directions.
48 Guarding 2-reflex orthogonal polyhedra Problem: Optimally guarding with reflex edge guards an orthogonal polyhedron having reflex edges in only two directions. Goal: Show that r g reflex edge guards are sufficient (g is the genus).
49 Guarding 2-reflex orthogonal polyhedra Problem: Optimally guarding with reflex edge guards an orthogonal polyhedron having reflex edges in only two directions. Goal: Show that r g reflex edge guards are sufficient (g is the genus). Each horizontal cross section is a collection of rectangles. The polyhedron is naturally partitioned into cuboidal bricks. It is safe to separate two bricks if they share at least two reflex edges, or if the operation reduces the polyhedron s genus.
50 Guarding 2-reflex orthogonal polyhedra Odd cuts are safe, too. The problem reduces to guarding double castles.
51 Guarding 2-reflex orthogonal polyhedra Odd cuts are safe, too. The problem reduces to guarding double castles. In turns, each double castle can be partitioned into pieces that require only one guard (cf. O Rourke s L-shaped pieces).
52 Guarding 2-reflex orthogonal polyhedra Theorem Any 2-reflex orthogonal polyhedron with r > 0 reflex edges and genus g is guardable by r g reflex edge guards.
53 Guarding 2-reflex orthogonal polyhedra Theorem Any 2-reflex orthogonal polyhedron with r > 0 reflex edges and genus g is guardable by r g reflex edge guards. The same construction yields also an upper bound in terms of e. Theorem Any 2-reflex orthogonal polyhedron with e edges and genus g is guardable by e g reflex edge guards.
54 Guarding 2-reflex orthogonal polyhedra Theorem Any 2-reflex orthogonal polyhedron with r > 0 reflex edges and genus g is guardable by r g reflex edge guards. The same construction yields also an upper bound in terms of e. Theorem Any 2-reflex orthogonal polyhedron with e edges and genus g is guardable by e g reflex edge guards. Valid for both open and closed edge guards. Guard locations can be computed in O(n log n) time.
55 Guarding with parallel edges Problem: Orthogonally guard an orthogonal polyhedron with mutually parallel edge guards.
56 Guarding with parallel edges Problem: Orthogonally guard an orthogonal polyhedron with mutually parallel edge guards. Motivations: Point location, tracking, navigation.
57 Guarding with parallel edges Problem: Orthogonally guard an orthogonal polyhedron with mutually parallel edge guards. Motivations: Point location, tracking, navigation. Strategy: Pick parallel cross sections and solve infinitely many 2D problems. Solutions in neighboring sections should be as similar as possible. Efficiently construct a 3D solution.
58 Guarding with parallel edges Solution: In any cross section, pick vertices of only 3 types. This selection remains consistent as the plane shifts.
59 Guarding with parallel edges The selected edges form a guarding set. v v q q q v p p p
60 Guarding with parallel edges The selected edges form a guarding set. v v q q q v p p p Do the math... Theorem Any orthogonal polyhedron is orthogonally guardable by e+r 12 mutually parallel edge guards.
61 Guarding with parallel edges Can we express our bound in terms of e or r only?
62 Guarding with parallel edges Can we express our bound in terms of e or r only? Lemma For every orthogonal polyhedron of genus g, 1 6 e + 2g 2 r 5 e 2g Both inequalities are tight for every g.
63 Guarding with parallel edges Corollary e g 6 1 parallel edge guards are sufficient to guard any orthogonal polyhedron. Corollary 7 12 r g + 1 parallel edge guards are sufficient to guard any orthogonal polyhedron.
64 Guarding with parallel edges Corollary e g 6 1 parallel edge guards are sufficient to guard any orthogonal polyhedron. Corollary 7 12 r g + 1 parallel edge guards are sufficient to guard any orthogonal polyhedron. Additionally: Guards are mutually parallel. Guards can be open or closed. Polyhedra are orthogonally guarded.
65 Edge-guarding 4-oriented polyhedra Problem: Edge-guarding polyhedra with faces oriented in 4 different directions. Orthogonal polyhedra come as a subclass.
66 Edge-guarding 4-oriented polyhedra Problem: Edge-guarding polyhedra with faces oriented in 4 different directions. Orthogonal polyhedra come as a subclass. The lower bound raises to e 6 (from e 12 ).
67 Edge-guarding 4-oriented polyhedra Solution: Again, consider parallel cross sections and pick 11 types of vertices (out of the 24 possible types).
68 Edge-guarding 4-oriented polyhedra The selected edges form a guarding set. v q v q v q v q p p p p v q v q v q p p p
69 Edge-guarding 4-oriented polyhedra The selected edges form a guarding set. v v v v q q q q p p p p v q v q v q p p p Theorem Any 4-oriented polyhedron is guardable by e+r 6 edge guards.
70 Edge-guarding general polyhedra Distinguish 4 edge classes.
71 Edge-guarding general polyhedra Distinguish 4 edge classes. Lemma If a point does not see any vertex, it sees edges in at least 2 classes.
72 Edge-guarding general polyhedra Solution: Pick an edge set that covers all vertices and, of the remaining edges, pick those in the 3 smallest classes. Lemma In any polyhedron, the vertex set is covered by 3e 8 edges. If all faces are triangles, the bound improves to 2e 9.
73 Edge-guarding general polyhedra Solution: Pick an edge set that covers all vertices and, of the remaining edges, pick those in the 3 smallest classes. Lemma In any polyhedron, the vertex set is covered by 3e 8 edges. If all faces are triangles, the bound improves to 2e 9. Do the math... Theorem (Cano Tóth Urrutia) Any polyhedron is guardable by 27e 32 closed edge guards. If all faces are triangles, the bound improves to 29e 36. Not valid for open edge guards (need endpoints to cover vertices).
74 Edge-guarding polyhedra: summary Open edge guards e-lower b. e-upper b. r-lower b. r-upper b. 1-reflex e/12 e/12 r/2 r/2 2-reflex e/12 e/8 r/2 r/2 3-reflex e/12 11e/72 r/2 7r/12 4-oriented e/6 e/3 r/2 r General e/6 e r r Closed edge guards e-lower b. e-upper b. r-lower b. r-upper b. 1-reflex e/16 e/12 r/3 r/2 2-reflex e/12 e/8 r/3 r/2 3-reflex e/12 11e/72 r/3 7r/12 4-oriented e/6 e/3 r/3 r General e/6 27e/32 r/2 r
75 Open questions Are Ω(n 2 ) point guards necessary for general polyhedra? Are n 4 point guards sufficient for every orthogonal terrain? Are r reflex edge guards sufficient for every orthogonal polyhedron? Are f 7 closed face guards sufficient for every orthogonal polyhedron? Are e 6 edge guards sufficient for every polyhedron?
76 References J. O Rourke Art gallery theorems and algorithms Oxford University Press, 1987 G. Viglietta Guarding and seraching polyhedra Ph.D. Thesis, University of Pisa, 2012 J. Cano, C. D. Tóth, and J. Urrutia Edge guards for polyhedra in 3-space CCCG 2012
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