Theta-3 is connected Oswin Aichholzer 1 Sang Won Bae 2 Luis Barba 34 - - PowerPoint PPT Presentation

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Oct 18, 2023 426 likes •885 views

Theta-3 is connected Oswin Aichholzer 1 Sang Won Bae 2 Luis Barba 34 Prosenjit Bose 3 Matias Korman 5 e van Renssen 3 Andr Perouz Taslakian 6 Sander Verdonschot 3 1 Graz University of Technology 2 Kyonggi University 3 Carleton University 4

SLIDE 1

Theta-3 is connected

Oswin Aichholzer1 Sang Won Bae2 Luis Barba34 Prosenjit Bose3 Matias Korman5 Andr´ e van Renssen3 Perouz Taslakian6 Sander Verdonschot3

1Graz University of Technology 2Kyonggi University 3Carleton University 4Universit´

e Libre de Bruxelles

5Universitat Polit´

ecnica de Catalunya

6American University of Armenia

25th Canadian Conference on Computational Geometry

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 1 / 16

SLIDE 2

θ-graphs

Partition plane into cones Add edge to ‘closest’ vertex in each cone

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 2 / 16

SLIDE 3

θ-graphs

Partition plane into cones Add edge to ‘closest’ vertex in each cone

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 2 / 16

SLIDE 4

Geometric Spanners

Graphs with short detours between vertices For every u and w, there is a path with length ≤ t · |uw|

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 3 / 16

SLIDE 5

Previous Work

Clarkson 1987 θ-graphs with > 8 cones are spanners Keil 1988 Ruppert & Seidel 1991 θ-graphs with > 6 cones are spanners

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 4 / 16

SLIDE 6

Previous Work

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 5 / 16

SLIDE 7

Previous Work

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 6 / 16

SLIDE 8

Previous Work

Clarkson 1987 θ-graphs with > 8 cones are spanners Keil 1988 Ruppert & Seidel 1991 θ-graphs with > 6 cones are spanners El Molla 2009 θ2 and θ3 are not spanners

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 7 / 16

SLIDE 9

Previous Work

Clarkson 1987 θ-graphs with > 8 cones are spanners Keil 1988 Ruppert & Seidel 1991 θ-graphs with > 6 cones are spanners El Molla 2009 θ2 and θ3 are not spanners Bonichon et al. 2010 θ6 is a planar 2-spanner Barba et al. 2013 θ4 and θ5 are spanners Bose et al.

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 7 / 16

SLIDE 10

Connectedness

Even θ-graphs

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 8 / 16

SLIDE 11

Connectedness

Odd θ-graphs

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 9 / 16

SLIDE 12

Connectedness

Odd θ-graphs

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 9 / 16

SLIDE 13

Connectedness

Odd θ-graphs

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 9 / 16

SLIDE 14

Connectedness

Theta-routing does not work in θ3

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 10 / 16

SLIDE 15

Properties

Edges in the same cone cannot cross

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 11 / 16

SLIDE 16

Properties

Edges in the same cone cannot cross Edges cannot cross empty cones

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 11 / 16

SLIDE 17

Paths

Unique up-path from each vertex

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 12 / 16

SLIDE 18

Paths

Paths can form barriers

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 12 / 16

SLIDE 19

Paths

Paths can form barriers

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 12 / 16

SLIDE 20

Paths

Up-paths cannot cross up-barriers

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 12 / 16

SLIDE 21

Paths

Up-paths cannot cross up-barriers

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 12 / 16

SLIDE 22

Paths

Up-paths cannot cross up-barriers

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 12 / 16

SLIDE 23

Paths

Up-paths cannot cross up-barriers

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 12 / 16

SLIDE 24

Paths

Up-paths cannot cross up-barriers Other paths can be forced to cross up-barriers

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 12 / 16

SLIDE 25

Lemma

Special configuration of up-sinks

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 13 / 16

SLIDE 26

Lemma

Special configuration of up-sinks

. . . 0

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 13 / 16

SLIDE 27

Lemma

Special configuration of up-sinks

. . . 0

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 13 / 16

SLIDE 28

Lemma

Special configuration of up-sinks ⇒ they are connected

. . . 0

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 13 / 16

SLIDE 29

Lemma

Special configuration of up-sinks ⇒ they are connected

. . . k

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 13 / 16

SLIDE 30

Lemma

Special configuration of up-sinks ⇒ they are connected

. . . k

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 13 / 16

SLIDE 31

Lemma

Special configuration of up-sinks ⇒ they are connected

. . . k

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 13 / 16

SLIDE 32

Lemma

Special configuration of up-sinks ⇒ they are connected

. . . k − 1

Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 13 / 16

SLIDE 33

Lemma

Special configuration of up-sinks ⇒ they are connected