New and Improved Spanning Ratios for Yao Graphs Luis Barba 12 - - PowerPoint PPT Presentation

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New and Improved Spanning Ratios for Yao Graphs Luis Barba 12 - - PowerPoint PPT Presentation

Jan 02, 2023 38 likes •559 views

New and Improved Spanning Ratios for Yao Graphs Luis Barba 12 Prosenjit Bose 1 Mirela Damian 3 Rolf Fagerberg 4 Wah Loon Keng 5 Joseph ORourke 6 e van Renssen 1 Andr Perouz Taslakian 7 Sander Verdonschot 1 Ge Xia 5 1 Carleton University 2

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SLIDE 1

New and Improved Spanning Ratios for Yao Graphs

Luis Barba12 Prosenjit Bose1 Mirela Damian3 Rolf Fagerberg4 Wah Loon Keng5 Joseph O’Rourke6 Andr´ e van Renssen1 Perouz Taslakian7 Sander Verdonschot1 Ge Xia5

1Carleton University 2Universit´

e Libre de Bruxelles

3Villanova University 4University of Southern Denmark 5Lafayette College 6Smith College 7American University of Armenia

30th Annual Symposium on Computational Geometry

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 1 / 15

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Yao-graphs

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

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 2 / 15

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SLIDE 3

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) Spanning Ratios for Yao Graphs SoCG 2014 3 / 15

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SLIDE 4

Previous Work

k > 8 (1 + ε) (Alth¨

  • fer et al., 1993)

k > 8 1 cos θ − sin θ (Bose et al., 2004) k > 6 1 1 − 2 sin θ

2

(Bose et al., 2010)

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 4 / 15

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SLIDE 5

Previous Work

k > 6 1/(1 − 2 sin(θ/2)) (Bose et al., 2010) k ≥ 5 and odd 1/(1 − 2 sin(3θ/8)) Y6 ?? Y5 ?? Y4 ?? Y3 ?? Y2 ??

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 5 / 15

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SLIDE 6

Previous Work

k > 6 1/(1 − 2 sin(θ/2)) (Bose et al., 2010) k ≥ 5 and odd 1/(1 − 2 sin(3θ/8)) Y6 ?? Y5 ?? Y4 ?? Y3

×

(El Molla, 2009) Y2

×

(El Molla, 2009)

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 5 / 15

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Previous Work

k > 6 1/(1 − 2 sin(θ/2)) (Bose et al., 2010) k ≥ 5 and odd 1/(1 − 2 sin(3θ/8)) Y6 ?? Y5 ?? Y4 663 (Bose et al., 2012) Y3

×

(El Molla, 2009) Y2

×

(El Molla, 2009)

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 5 / 15

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SLIDE 8

Previous Work

k > 6 1/(1 − 2 sin(θ/2)) (Bose et al., 2010) k ≥ 5 and odd 1/(1 − 2 sin(3θ/8)) Y6 17.7 (Damian & Raudonis, 2012) Y5 ?? Y4 663 (Bose et al., 2012) Y3

×

(El Molla, 2009) Y2

×

(El Molla, 2009)

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 5 / 15

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Our Results

k > 6 1/(1 − 2 sin(θ/2)) (Bose et al., 2010) k > 3 and odd 1/(1 − 2 sin(3θ/8)) Y6 17.7 (Damian & Raudonis, 2012) Y5 ?? Y4 663 (Bose et al., 2012) Y3

×

(El Molla, 2009) Y2

×

(El Molla, 2009)

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 6 / 15

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SLIDE 10

Our Results

k > 6 1/(1 − 2 sin(θ/2)) (Bose et al., 2010) k > 3 and odd 1/(1 − 2 sin(3θ/8)) Y6 17.7 (Damian & Raudonis, 2012) Y5 10.9 Y4 663 (Bose et al., 2012) Y3

×

(El Molla, 2009) Y2

×

(El Molla, 2009)

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 6 / 15

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Our Results

k > 6 1/(1 − 2 sin(θ/2)) (Bose et al., 2010) k > 3 and odd 1/(1 − 2 sin(3θ/8)) Y6 17.7 (Damian & Raudonis, 2012) Y5 10.9 3.74 Y4 663 (Bose et al., 2012) Y3

×

(El Molla, 2009) Y2

×

(El Molla, 2009)

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 6 / 15

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Our Results

k > 6 1/(1 − 2 sin(θ/2)) (Bose et al., 2010) k > 3 and odd 1/(1 − 2 sin(3θ/8)) Y6 17.7 5.8 (Damian & Raudonis, 2012) Y5 10.9 3.74 Y4 663 (Bose et al., 2012) Y3

×

(El Molla, 2009) Y2

×

(El Molla, 2009)

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 6 / 15

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Odd Yao graphs

Basic lemma (used for Yao graphs with k > 6)

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 7 / 15

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Odd Yao graphs

Basic lemma (used for Yao graphs with k > 6)

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 7 / 15

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Odd Yao graphs

Basic lemma (used for Yao graphs with k > 6)

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 7 / 15

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Odd Yao graphs

Even number of cones: Increasing one angle also increases the other

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 7 / 15

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Odd Yao graphs

Even number of cones: Increasing one angle also increases the other

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 7 / 15

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Odd Yao graphs

Odd number of cones: Increasing one angle decreases the other

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 7 / 15

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Odd Yao graphs

Odd number of cones: Worst case occurs for 3θ/4

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 7 / 15

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Our Results

k > 6 1/(1 − 2 sin(θ/2)) (Bose et al., 2010) k > 3 and odd 1/(1 − 2 sin(3θ/8)) Y6 17.7 (Damian & Raudonis, 2012) Y5 10.9 Y4 663 (Bose et al., 2012) Y3

×

(El Molla, 2009) Y2

×

(El Molla, 2009)

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 8 / 15

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Our Results

First constant upper bound for Y5 ⇒ Yk is a constant spanner iff k > 3

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 9 / 15

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Our Results

First constant upper bound for Y5 ⇒ Yk is a constant spanner iff k > 3 Can we do better for Y5?

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 9 / 15

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Our Results

First constant upper bound for Y5 ⇒ Yk is a constant spanner iff k > 3 Can we do better for Y5? Always apply basic lemma

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 9 / 15

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Our Results

First constant upper bound for Y5 ⇒ Yk is a constant spanner iff k > 3 Can we do better for Y5? Always Strategically apply basic lemma Handle remaining cases

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 9 / 15

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Improvements for Y5

What if we only apply the lemma for small angles?

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 10 / 15

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Improvements for Y5

If the edge is very short, we’re still okay

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 10 / 15

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Improvements for Y5

Case 1: Both edges are long and they cross

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 10 / 15

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Improvements for Y5

Case 1: Both edges are long and they cross

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 10 / 15

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Improvements for Y5

Case 1: Both edges are long and they cross

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 10 / 15

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Improvements for Y5

Case 2: Both edges are long and do not cross

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 10 / 15

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Improvements for Y5

Case 2: Both edges are long and do not cross

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 10 / 15

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Improvements for Y5

Case 2: Both edges are long and do not cross

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 10 / 15

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Improvements for Y5

Case 2: Both edges are long and do not cross

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 10 / 15

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Improvements for Y5

Case 2: Both edges are long and do not cross

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 10 / 15

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Improvements for Y5

Case 2: Both edges are long and do not cross

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 10 / 15

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Improvements for Y5

Case 2: Both edges are long and do not cross

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 10 / 15

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Our Results

k > 6 1/(1 − 2 sin(θ/2)) (Bose et al., 2010) k > 3 and odd 1/(1 − 2 sin(3θ/8)) Y6 17.7 (Damian & Raudonis, 2012) Y5 10.9 3.74 Y4 663 (Bose et al., 2012) Y3

×

(El Molla, 2009) Y2

×

(El Molla, 2009)

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 11 / 15

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Spanning ratio of Y6

Same general idea: Strategically apply basic lemma Handle remaining cases

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 12 / 15

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Spanning ratio of Y6

Split cone into center and margins

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 13 / 15

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Spanning ratio of Y6

Destination in center → Apply basic lemma

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 13 / 15

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Spanning ratio of Y6

Closest in center → Apply basic lemma

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 13 / 15

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Spanning ratio of Y6

Closest in center → Apply basic lemma

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 13 / 15

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Spanning ratio of Y6

Look from the other side

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 13 / 15

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Spanning ratio of Y6

Closest in center → Apply (variation of) basic lemma

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 13 / 15

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Spanning ratio of Y6

If closest lies in both margins,

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 13 / 15

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Spanning ratio of Y6

If closest lies in both margins, consider the other closest

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 13 / 15

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Spanning ratio of Y6

If closest lies in both margins, consider the other closest

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 13 / 15

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Spanning ratio of Y6

If closest lies in both margins, consider the other closest

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 13 / 15

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Spanning ratio of Y6

A few more cases...

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 13 / 15

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Our Results

k > 6 1/(1 − 2 sin(θ/2)) (Bose et al., 2010) k > 3 and odd 1/(1 − 2 sin(3θ/8)) Y6 17.7 5.8 (Damian & Raudonis, 2012) Y5 10.9 3.74 Y4 663 (Bose et al., 2012) Y3

×

(El Molla, 2009) Y2

×

(El Molla, 2009)

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 14 / 15

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Future work

Improved lower bounds Competitive routing

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 15 / 15

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Future work

Improved lower bounds Competitive routing

Questions?

Sander Verdonschot (Carleton University) Spanning Ratios for Yao Graphs SoCG 2014 15 / 15