On Plane Constrained Bounded-Degree Spanners Prosenjit Bose, Rolf - - PowerPoint PPT Presentation

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On Plane Constrained Bounded-Degree Spanners Prosenjit Bose, Rolf - - PowerPoint PPT Presentation

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On Plane Constrained Bounded-Degree Spanners Prosenjit Bose, Rolf Fagerberg, Andr e van Renssen and Sander Verdonschot Carleton University, University of Southern Denmark April 15, 2012 Sander Verdonschot (Carleton University) Constrained

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SLIDE 1

On Plane Constrained Bounded-Degree Spanners

Prosenjit Bose, Rolf Fagerberg, Andr´ e van Renssen and Sander Verdonschot

Carleton University, University of Southern Denmark

April 15, 2012

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 1 / 15

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SLIDE 2

Geometric Spanners

Given: Set of points in the plane Goal: Approximate the complete Euclidean graph

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 2 / 15

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SLIDE 3

Geometric Spanners

Given: Set of points in the plane Goal: Approximate the complete Euclidean graph

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 2 / 15

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SLIDE 4

Geometric Spanners

Given: Set of points in the plane Goal: Approximate the complete Euclidean graph shortest path ≤ k · Euclidean distance

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 2 / 15

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SLIDE 5

Geometric Spanners

Small spanning ratio Planarity Bounded degree Small number of hops Low total edge length

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 3 / 15

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SLIDE 6

Geometric Spanners

Small spanning ratio Planarity Bounded degree Small number of hops Low total edge length

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 4 / 15

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SLIDE 7

Plane Spanners

Empty square (L1) Delaunay triangulation ≤ 3.16 (Chew - 1986) = 2.61 (Bonichon et al. - 2012) Empty circle (L2) Delaunay triangulation ≤ 5.08 (Dobkin et al. - 1987) ≤ 2.42 (Keil, Gutwin - 1992) Empty equilateral triangle Delaunay triangulation = 2 (Chew - 1989) Equivalent to half-θ6-graph (Bonichon et al. - 2010)

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 5 / 15

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SLIDE 8

Plane Bounded-Degree Spanners

Degree k Authors 27 10.02 Bose et al. - 2005 23 7.79 Li, Wang - 2004 17 28.54 Bose et al. - 2009 14 3.53 Kanj, Perkovi´ c - 2008 6 98.91 Bose et al. - 2012 6 6 Bonichon et al. - 2010

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 6 / 15

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SLIDE 9

Constrained Geometric Spanners

Given: Set of points in the plane V Set of constraints ⊆ V × V Goal: Approximate visibility graph

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 7 / 15

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SLIDE 10

Constrained Geometric Spanners

Given: Set of points in the plane V Set of constraints ⊆ V × V Goal: Approximate visibility graph

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 7 / 15

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SLIDE 11

Constrained Geometric Spanners

Given: Set of points in the plane V Set of constraints ⊆ V × V Goal: Approximate visibility graph

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 7 / 15

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SLIDE 12

Constrained Geometric Spanners

Given: Set of points in the plane V Set of constraints ⊆ V × V Goal: Approximate visibility graph

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 7 / 15

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SLIDE 13

Constrained Geometric Spanners

k B.D. Plane Authors Graph 1 + ǫ Clarkson - 1987 1 + ǫ

  • Das - 1997

5.08

  • Karavelas - 2001

Delaunay triangulation 2.42

  • Bose, Keil - 2006

Delaunay triangulation 2

  • Our result

Half-θ6-graph 6

  • Our result

Half-θ6-graph

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 8 / 15

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SLIDE 14

Half-θ6-graph

6 Cones around each vertex: 3 positive, 3 negative

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 9 / 15

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SLIDE 15

Half-θ6-graph

Connect to ‘closest’ vertex in each positive cone

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 9 / 15

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SLIDE 16

Half-θ6-graph

Connect to ‘closest’ vertex in each positive cone

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 9 / 15

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SLIDE 17

Half-θ6-graph

Connect to ‘closest’ vertex in each positive cone

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 9 / 15

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SLIDE 18

Constrained Half-θ6-graph

Connect to ‘closest’ visible vertex in each positive cone

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 10 / 15

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SLIDE 19

Constrained Half-θ6-graph

Connect to ‘closest’ visible vertex in each positive cone

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 10 / 15

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SLIDE 20

Constrained Half-θ6-graph

Connect to ‘closest’ visible vertex in each positive subcone

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 10 / 15

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SLIDE 21

Constrained Half-θ6-graph

Connect to ‘closest’ visible vertex in each positive subcone

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 10 / 15

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SLIDE 22

Constrained Half-θ6-graph

Connect to ‘closest’ visible vertex in each positive subcone

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 10 / 15

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SLIDE 23

Spanning ratio

Theorem

The constrained half-θ6-graph is a 2-spanner of the visibility graph

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 11 / 15

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SLIDE 24

Spanning ratio

Theorem

The constrained half-θ6-graph is a 2-spanner of the visibility graph Proof by induction on the area of the equilateral triangle

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 11 / 15

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SLIDE 25

Spanning ratio

Induction hypothesis: there is a path of length at most one side plus the longer top segment

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 12 / 15

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SLIDE 26

Spanning ratio

Induction hypothesis: there is a path of length at most one side plus the longer top segment If the larger side is empty, the length is at most one side plus the shorter top segment

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 12 / 15

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SLIDE 27

Spanning ratio

Induction hypothesis: there is a path of length at most one side plus the longer top segment If the larger side is empty, the length is at most one side plus the shorter top segment

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 12 / 15

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SLIDE 28

Spanning ratio

Induction hypothesis: there is a path of length at most one side plus the longer top segment If the larger side is empty, the length is at most one side plus the shorter top segment

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 12 / 15

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SLIDE 29

Spanning ratio

Induction hypothesis: there is a path of length at most one side plus the longer top segment If the larger side is empty, the length is at most one side plus the shorter top segment

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 12 / 15

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SLIDE 30

Spanning ratio

Induction hypothesis: there is a path of length at most one side plus the longer top segment If the larger side is empty, the length is at most one side plus the shorter top segment

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 12 / 15

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SLIDE 31

Spanning ratio

Induction hypothesis: there is a path of length at most one side plus the longer top segment If the larger side is empty, the length is at most one side plus the shorter top segment

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 12 / 15

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SLIDE 32

Spanning ratio

Induction hypothesis: there is a path of length at most one side plus the longer top segment If the larger side is empty, the length is at most one side plus the shorter top segment

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 12 / 15

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SLIDE 33

Spanning ratio

Induction hypothesis: there is a path of length at most one side plus the longer top segment If the larger side is empty, the length is at most one side plus the shorter top segment

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 12 / 15

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SLIDE 34

Spanning ratio

Induction hypothesis: there is a path of length at most one side plus the longer top segment If the larger side is empty, the length is at most one side plus the shorter top segment

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 12 / 15

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SLIDE 35

Spanning ratio

Induction hypothesis: there is a path of length at most one side plus the longer top segment If the larger side is empty, the length is at most one side plus the shorter top segment

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 12 / 15

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SLIDE 36

Bounded-Degree subgraph

Theorem

The constrained half-θ6-graph has a bounded degree subgraph that is a 6-spanner of the visibility graph

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 13 / 15

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SLIDE 37

Bounded-Degree subgraph

Theorem

The constrained half-θ6-graph has a bounded degree subgraph that is a 6-spanner of the visibility graph

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 13 / 15

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SLIDE 38

Bounded-Degree subgraph

Theorem

The constrained half-θ6-graph has a bounded degree subgraph that is a 6-spanner of the visibility graph

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 13 / 15

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SLIDE 39

Bounded-Degree subgraph

Theorem

The constrained half-θ6-graph has a bounded degree subgraph that is a 6-spanner of the visibility graph

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 13 / 15

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SLIDE 40

Bounded-Degree subgraph

Theorem

The constrained half-θ6-graph has a bounded degree subgraph that is a 6-spanner of the visibility graph

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 13 / 15

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SLIDE 41

Bounded-Degree subgraph

Theorem

The constrained half-θ6-graph has a bounded degree subgraph that is a 6-spanner of the visibility graph

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 13 / 15

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SLIDE 42

Bounded-Degree subgraph

Theorem

The constrained half-θ6-graph has a bounded degree subgraph that is a 6-spanner of the visibility graph

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 13 / 15

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SLIDE 43

Bounded-Degree subgraph

Theorem

The constrained half-θ6-graph has a bounded degree subgraph that is a 6-spanner of the visibility graph

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 13 / 15

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SLIDE 44

Bounded-Degree subgraph

Theorem

The constrained half-θ6-graph has a bounded degree subgraph that is a 6-spanner of the visibility graph

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 13 / 15

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SLIDE 45

Bounded-Degree subgraph

Theorem

The constrained half-θ6-graph has a bounded degree subgraph that is a 6-spanner of the visibility graph

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 13 / 15

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SLIDE 46

Bounded-Degree subgraph

A modification of the previous graph gives maximum degree 6 + c

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 14 / 15

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SLIDE 47

Bounded-Degree subgraph

A modification of the previous graph gives maximum degree 6 + c

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 14 / 15

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SLIDE 48

Conclusion

Improved the spanning ratio of the best known plane constrained spanner to 2 Introduced the first plane constrained bounded-degree spanner, with a maximum degree of 6 + c Main open problem: Can we do better than 6 + c?

Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 15 / 15