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Edge-guarding Orthogonal Polyhedra Giovanni Viglietta Department of Computer Science, University of Pisa, Italy Rome - July 14 th , 2011 We view things not only from different sides, but with different eyes. Blaise Pascal Edge-guarding

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SLIDE 1

Edge-guarding Orthogonal Polyhedra

Giovanni Viglietta

Department of Computer Science, University of Pisa, Italy

Rome - July 14th, 2011

“We view things not only from different sides, but with different eyes.” Blaise Pascal Edge-guarding Orthogonal Polyhedra

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SLIDE 2

Art Gallery Problem

Planar version: Given a polygon, choose a minimum number

  • f vertices that collectively see its whole interior.

Edge-guarding Orthogonal Polyhedra

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SLIDE 3

Art Gallery Problem

Planar version: Given a polygon, choose a minimum number

  • f vertices that collectively see its whole interior.

Edge-guarding Orthogonal Polyhedra

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SLIDE 4

Art Gallery Problem

Planar version: Given a polygon, choose a minimum number

  • f vertices that collectively see its whole interior.

Edge-guarding Orthogonal Polyhedra

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SLIDE 5

Art Gallery Problem

Planar version: Given a polygon, choose a minimum number

  • f vertices that collectively see its whole interior.

Problem: Generalize to orthogonal polyhedra.

Edge-guarding Orthogonal Polyhedra

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SLIDE 6

Terminology

Polyhedra

genus 0 genus 1 genus 2

Edge-guarding Orthogonal Polyhedra

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SLIDE 7

Terminology

Orthogonal polyhedron Reflex edge

Edge-guarding Orthogonal Polyhedra

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SLIDE 8

Guarding polyhedra

Vertex guards vs. edge guards.

Edge-guarding Orthogonal Polyhedra

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SLIDE 9

Vertex-guarding orthogonal polyhedra

The Art Gallery Problem for vertex guards is unsolvable in some orthogonal polyhedra. Some points in the central region are invisible to all vertices.

Edge-guarding Orthogonal Polyhedra

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SLIDE 10

Point-guarding orthogonal polyhedra

Some orthogonal polyhedra require Ω(n3/2) point guards.

Edge-guarding Orthogonal Polyhedra

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SLIDE 11

Edge guards

Closed edge guards vs. open edge guards.

Edge-guarding Orthogonal Polyhedra

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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.

Edge-guarding Orthogonal Polyhedra

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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?

Edge-guarding Orthogonal Polyhedra

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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.

Edge-guarding Orthogonal Polyhedra

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SLIDE 15

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 bound tight?

Edge-guarding Orthogonal Polyhedra

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Closed vs. open edge guards

Each endpoint of a closed edge guard can be replaced by an adjacent open edge.

Case analysis on all vertex types.

Edge-guarding Orthogonal Polyhedra

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Closed vs. open edge guards

Each endpoint of a closed edge guard can be replaced by an adjacent open edge.

Case analysis on all vertex types.

Hence each closed edge guard can be replaced by 3 open edge guards, and our previous bound is tight.

Edge-guarding Orthogonal Polyhedra

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Bounding edge guards

Most variations of the Art Gallery Problem are NP-hard and APX-hard. Typically, we content ourselves with upper bounds on the minimum number of guards.

Edge-guarding Orthogonal Polyhedra

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Bounding edge guards

Most variations of the Art Gallery Problem are NP-hard and APX-hard. Typically, we content ourselves with upper bounds on the minimum number of guards. Our parameters for bounding edge guards in orthogonal polyhedra are the total number of edges e and the number of reflex edges r.

Edge-guarding Orthogonal Polyhedra

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Lower bound

Asymptotically,

e 12 edge guards may be necessary.

Edge-guarding Orthogonal Polyhedra

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Lower bound

Asymptotically, r

2 edge guards may be necessary.

Edge-guarding Orthogonal Polyhedra

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Upper bound

Observation: Any polyhedron is guarded by the set of its edges.

Upper bound: e.

Edge-guarding Orthogonal Polyhedra

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Upper bound

Observation: Any polyhedron is guarded by the set of its edges.

Upper bound: e.

Observation: Any polyhedron is guarded by the set of its reflex edges.

Upper bound: r.

Edge-guarding Orthogonal Polyhedra

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Upper bound

Observation: Any polyhedron is guarded by the set of its edges.

Upper bound: e.

Observation: Any polyhedron is guarded by the set of its reflex edges.

Upper bound: r.

State of the art (Urrutia) Any orthogonal polyhedron is guardable by e

6 closed edge guards.

Can it be lowered and extended to open edge guards?

Edge-guarding Orthogonal Polyhedra

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Improving the upper bound

Theorem Any orthogonal polyhedron is guardable by e+r

12 open edge guards.

Edge-guarding Orthogonal Polyhedra

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Improving the upper bound

Theorem Any orthogonal polyhedron is guardable by e+r

12 open edge guards.

Proof. We select a coordinate axis X and only place guards on X-parallel edges. There are 8 types of X-parallel edges, and we place guards on the circled ones (X axis pointing toward the audience):

Edge-guarding Orthogonal Polyhedra

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SLIDE 27

Improving the upper bound

There are 4 symmetric ways of picking edge types: α + β′ + δ′, γ + β′ + δ′, β + α′ + γ′, δ + α′ + γ′.

Edge-guarding Orthogonal Polyhedra

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Improving the upper bound

There are 4 symmetric ways of picking edge types: α + β′ + δ′, γ + β′ + δ′, β + α′ + γ′, δ + α′ + γ′. The sum is α + β + γ + δ + 2α′ + 2β′ + 2γ′ + 2δ′ = ex + rx.

Edge-guarding Orthogonal Polyhedra

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SLIDE 29

Improving the upper bound

There are 4 symmetric ways of picking edge types: α + β′ + δ′, γ + β′ + δ′, β + α′ + γ′, δ + α′ + γ′. The sum is α + β + γ + δ + 2α′ + 2β′ + 2γ′ + 2δ′ = ex + rx. Hence, one of the 4 choices picks at most ex+rx

4

edges.

Edge-guarding Orthogonal Polyhedra

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Improving the upper bound

There are 4 symmetric ways of picking edge types: α + β′ + δ′, γ + β′ + δ′, β + α′ + γ′, δ + α′ + γ′. The sum is α + β + γ + δ + 2α′ + 2β′ + 2γ′ + 2δ′ = ex + rx. Hence, one of the 4 choices picks at most ex+rx

4

edges. By selecting the axis X that minimizes the sum ex + rx, we place at most e+r

12 guards.

Edge-guarding Orthogonal Polyhedra

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Improving the upper bound

Indeed, every X-orthogonal section is guarded: For a given p, pick the maximal segment pq and slide it to the left, until it hits a vertex v, which corresponds to a selected edge.

Edge-guarding Orthogonal Polyhedra

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Improving the upper bound

Theorem For every orthogonal polyhedron of genus g, 1 6e + 2g − 2 r 5 6e − 2g − 12

  • holds. Both inequalities are tight for every g.

Edge-guarding Orthogonal Polyhedra

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Improving the upper bound

Theorem For every orthogonal polyhedron of genus g, 1 6e + 2g − 2 r 5 6e − 2g − 12

  • holds. Both inequalities are tight for every g.

Corollary

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

  • rthogonal polyhedron.

Corollary

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

  • rthogonal polyhedron.

Edge-guarding Orthogonal Polyhedra

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Concluding remarks

We showed that closed edge guards are 3 times more powerful than open edge guards, for orthogonal polyhedra. We lowered the upper bound on the number of edge guards from e

6 to 11 72e, whereas the best known lower bound is e 12.

We gave the new upper bound

7 12r, whereas the best known

lower bound is r

2.

Edge-guarding Orthogonal Polyhedra

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Future research

Conjecture Any orthogonal polyhedron is guardable by

e 12 edges and r 2 reflex

edges.

Edge-guarding Orthogonal Polyhedra

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Future research

Conjecture Any orthogonal polyhedron is guardable by

e 12 edges and r 2 reflex

edges. Better analyze our upper bound construction to compute its real efficiency. Study the problem where all selected guards are required to be mutually parallel, and raise the

e 12 lower bound in this case.

Actually use all 3 edge directions to design an improved guarding strategy.

Edge-guarding Orthogonal Polyhedra

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Edge-guarding Orthogonal Polyhedra

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References

  • N. M. Benbernou, E. D. Demaine, M. L. Demaine, A. Kurdia,
  • J. O’Rourke, G. Toussaint, J. Urrutia, G. Viglietta.

Edge-guarding Orthogonal Polyhedra. In Proceedings of the 23rd Canadian Conference on Computational Geometry, 2011. To appear.

  • J. O’Rourke.

Art gallery theorems and algorithms. Oxford University Press, 1987.

  • J. Urrutia.

Art gallery and illumination problems. In J. R. Sack and J. Urrutia, editors, Handbook of Computational Geometry, pages 973–1027. North-Holland, 2000.

Edge-guarding Orthogonal Polyhedra