The Art Gallery Problem for polyhedra Carleton Algorithms Seminar - - PowerPoint PPT Presentation

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

For polygons with n vertices, n

3

• vertex guards are sufficient.

The Art Gallery Problem for polyhedra

Fisk’s solution: sufficiency

For polygons with n vertices, n

3

• vertex guards are sufficient.

The Art Gallery Problem for polyhedra

Fisk’s solution: sufficiency

For polygons with n vertices, n

3

• vertex guards are sufficient.

The Art Gallery Problem for polyhedra

Fisk’s solution: sufficiency

For polygons with n vertices, n

3

• vertex guards are sufficient.

The Art Gallery Problem for polyhedra

Fisk’s solution: necessity

n

3

• guards may be necessary.

The Art Gallery Problem for polyhedra

Guarding orthogonal polygons: O’Rourke’s solution

n

4

• vertex guards are sufficient and occasionally necessary.

The Art Gallery Problem for polyhedra

Guarding orthogonal polygons: O’Rourke’s solution

n

4

• vertex guards are sufficient and occasionally necessary.

The Art Gallery Problem for polyhedra

Guarding orthogonal polygons: O’Rourke’s solution

n

4

• vertex guards are sufficient and occasionally necessary.

The Art Gallery Problem for polyhedra

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

2

• + 1.

The Art Gallery Problem for polyhedra

Guarding triangulated terrains

n

2

• vertex guards are sufficient and occasionally necessary

(Bose et al.).

The Art Gallery Problem for polyhedra

Guarding triangulated terrains

n

2

• vertex guards are sufficient and occasionally necessary

(Bose et al.). n

3

• edge guards are sufficient (Everett et al.).

4n−4

13

• edge guards are occasionally necessary (Bose et al.).

The Art Gallery Problem for polyhedra

Guarding triangulated terrains

n

2

• vertex guards are sufficient and occasionally necessary

(Bose et al.). n

3

• edge guards are sufficient (Everett et al.).

4n−4

13

• edge guards are occasionally necessary (Bose et al.).

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

• btained 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(n3/2) point guards.

• uter view

cross section

Every orthogonal polyhedron yields a BSP tree of size O(n3/2) (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(r2) 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(r2) 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.

Lower bound:

e 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.

Lower bound: r

2

• + 1 reflex edge guards.

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.

A B C D E F

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.

A B C D E F

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(n2) 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(n2) 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.

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.

The Art Gallery Problem for polyhedra

Face-guarding orthogonal polyhedra

f

7

• closed face guards are occasionally necessary.

The Art Gallery Problem for polyhedra

Face-guarding orthogonal polyhedra

f

6

• pen face guards are always sufficient and occasionally

necessary.

The Art Gallery Problem for polyhedra

Face-guarding 4-oriented polyhedra

f

5

• closed face guards are occasionally necessary.

The Art Gallery Problem for polyhedra

Face-guarding 4-oriented polyhedra

f

4

• pen face guards are always sufficient and occasionally

necessary.

The Art Gallery Problem for polyhedra

Guarding 2-reflex orthogonal polyhedra

Problem: Optimally guarding with reflex edge guards an

• rthogonal polyhedron having reflex edges in only two

directions.

The Art Gallery Problem for polyhedra

Guarding 2-reflex orthogonal polyhedra

Problem: Optimally guarding with reflex edge guards an

• rthogonal polyhedron having reflex edges in only two

directions. Goal: Show that r−g

2

• + 1 reflex edge guards are sufficient

(g is the genus).

The Art Gallery Problem for polyhedra

Guarding 2-reflex orthogonal polyhedra

Problem: Optimally guarding with reflex edge guards an

• rthogonal polyhedron having reflex edges in only two

directions. Goal: Show that r−g

2

• + 1 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.

The Art Gallery Problem for polyhedra

Guarding 2-reflex orthogonal polyhedra

Odd cuts are safe, too. The problem reduces to guarding double castles.

The Art Gallery Problem for polyhedra

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).

The Art Gallery Problem for polyhedra

Guarding 2-reflex orthogonal polyhedra

Theorem Any 2-reflex orthogonal polyhedron with r > 0 reflex edges and genus g is guardable by r−g

2

• + 1 reflex edge guards.

The Art Gallery Problem for polyhedra

Guarding 2-reflex orthogonal polyhedra

Theorem Any 2-reflex orthogonal polyhedron with r > 0 reflex edges and genus g is guardable by r−g

2

• + 1 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−4

8

• + g reflex edge guards.

The Art Gallery Problem for polyhedra

Guarding 2-reflex orthogonal polyhedra

Theorem Any 2-reflex orthogonal polyhedron with r > 0 reflex edges and genus g is guardable by r−g

2

• + 1 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−4

8

• + g reflex edge guards.

Valid for both open and closed edge guards. Guard locations can be computed in O(n log n) time.

The Art Gallery Problem for polyhedra

Guarding with parallel edges

Problem: Orthogonally guard an orthogonal polyhedron with mutually parallel edge guards.

The Art Gallery Problem for polyhedra

Guarding with parallel edges

Problem: Orthogonally guard an orthogonal polyhedron with mutually parallel edge guards. Motivations: Point location, tracking, navigation.

The Art Gallery Problem for polyhedra

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.

The Art Gallery Problem for polyhedra

Guarding with parallel edges

Solution: In any cross section, pick vertices of only 3 types. This selection remains “consistent” as the plane shifts.

The Art Gallery Problem for polyhedra

Guarding with parallel edges

The selected edges form a guarding set.

p p p v v v q q q

The Art Gallery Problem for polyhedra

Guarding with parallel edges

The selected edges form a guarding set.

p p p v v v q q q

Do the math... Theorem Any orthogonal polyhedron is orthogonally guardable by e+r

12

mutually parallel edge guards.

The Art Gallery Problem for polyhedra

Guarding with parallel edges

Can we express our bound in terms of e or r only?

The Art Gallery Problem for polyhedra

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 6e + 2g − 2 r 5 6e − 2g − 12. Both inequalities are tight for every g.

The Art Gallery Problem for polyhedra

Guarding with parallel edges

Corollary

11 72e − g 6 − 1 parallel edge guards are sufficient to guard any

• rthogonal polyhedron.

Corollary

7 12r − g + 1 parallel edge guards are sufficient to guard any

• rthogonal polyhedron.

The Art Gallery Problem for polyhedra

Guarding with parallel edges

Corollary

11 72e − g 6 − 1 parallel edge guards are sufficient to guard any

• rthogonal polyhedron.

Corollary

7 12r − g + 1 parallel edge guards are sufficient to guard any

• rthogonal polyhedron.

Additionally:

Guards are mutually parallel. Guards can be open or closed. Polyhedra are orthogonally guarded.

The Art Gallery Problem for polyhedra

Edge-guarding 4-oriented polyhedra

Problem: Edge-guarding polyhedra with faces oriented in 4 different directions.

Orthogonal polyhedra come as a subclass.

The Art Gallery Problem for polyhedra

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).

The Art Gallery Problem for polyhedra

Edge-guarding 4-oriented polyhedra

Solution: Again, consider parallel cross sections and pick 11 types of vertices (out of the 24 possible types).

The Art Gallery Problem for polyhedra

Edge-guarding 4-oriented polyhedra

The selected edges form a guarding set.

p q v p q v p q v p q v p q v p q v p q v

The Art Gallery Problem for polyhedra

Edge-guarding 4-oriented polyhedra

The selected edges form a guarding set.

p q v p q v p q v p q v p q v p q v p q v

Theorem Any 4-oriented polyhedron is guardable by e+r

6

edge guards.

The Art Gallery Problem for polyhedra

Edge-guarding general polyhedra

Distinguish 4 edge classes.

The Art Gallery Problem for polyhedra

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.

The Art Gallery Problem for polyhedra

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

• .

The Art Gallery Problem for polyhedra

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–T´

• th–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).

The Art Gallery Problem for polyhedra

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

The Art Gallery Problem for polyhedra

Open questions

Are Ω(n2) point guards necessary for general polyhedra? Are n

4

• point guards sufficient for every orthogonal terrain?

Are r

2

• + 1 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?

The Art Gallery Problem for polyhedra

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. T´
• th, and J. Urrutia

Edge guards for polyhedra in 3-space CCCG 2012

The Art Gallery Problem for polyhedra