The 5 -graph is a spanner Prosenjit Bose, Pat Morin, Andr e van - - PowerPoint PPT Presentation

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The 5 -graph is a spanner Prosenjit Bose, Pat Morin, Andr e van - - PowerPoint PPT Presentation

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The 5 -graph is a spanner Prosenjit Bose, Pat Morin, Andr e van Renssen and Sander Verdonschot Carleton University June 20, 2013 Sander Verdonschot (Carleton University) The 5 -graph is a spanner June 20, 2013 1 / 19 -graphs

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SLIDE 1

The θ5-graph is a spanner

Prosenjit Bose, Pat Morin, Andr´ e van Renssen and Sander Verdonschot

Carleton University

June 20, 2013

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 1 / 19

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SLIDE 2

θ-graphs

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

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 2 / 19

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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) The θ5-graph is a spanner June 20, 2013 3 / 19

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SLIDE 4

Previous Work

Clarkson 1987 θ-graphs with k > 8 are (1 + ε)-spanners Keil 1988 Ruppert & Seidel 1991 θ-graphs with k > 6 have spanning ratio 1 1 − 2 sin(θ/2)

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 4 / 19

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SLIDE 5

Previous Work

Clarkson 1987 θ-graphs with k > 8 are (1 + ε)-spanners Keil 1988 Ruppert & Seidel 1991 θ-graphs with k > 6 have spanning ratio 1 1 − 2 sin(θ/2) El Molla 2009 θ2 and θ3 are not spanners Bonichon et al. 2010 θ6 is a planar 2-spanner

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 4 / 19

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SLIDE 6

Previous Work

Clarkson 1987 θ-graphs with k > 8 are (1 + ε)-spanners Keil 1988 Ruppert & Seidel 1991 θ-graphs with k > 6 have spanning ratio 1 1 − 2 sin(θ/2) El Molla 2009 θ2 and θ3 are not spanners Bonichon et al. 2010 θ6 is a planar 2-spanner

What about θ4 and θ5?

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 4 / 19

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SLIDE 7

θ5 Challenges

Asymmetric Steps can get further away

u v w

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 5 / 19

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SLIDE 8

θ5 Challenges

Asymmetric Steps can get further away

u w

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 5 / 19

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SLIDE 9

Connectedness

Induction on size of canonical triangle

w u

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 6 / 19

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SLIDE 10

Connectedness

Base case: smallest canonical triangle

w u

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 7 / 19

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SLIDE 11

Connectedness

Base case: smallest canonical triangle IH: There exists a path between every two vertices with a smaller canonical triangle Case1: w lies near the bisector

w u

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 7 / 19

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SLIDE 12

Connectedness

Base case: smallest canonical triangle IH: There exists a path between every two vertices with a smaller canonical triangle Case2: w lies far from the bisector

w u

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 7 / 19

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SLIDE 13

Connectedness

Base case: smallest canonical triangle IH: There exists a path between every two vertices with a smaller canonical triangle Case2: w lies far from the bisector

w u

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 7 / 19

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SLIDE 14

Connectedness

Base case: smallest canonical triangle IH: There exists a path between every two vertices with a smaller canonical triangle Case2: w lies far from the bisector

w u

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 7 / 19

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SLIDE 15

Connectedness

Base case: smallest canonical triangle IH: There exists a path between every two vertices with a smaller canonical triangle Case2: w lies far from the bisector

w u

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 7 / 19

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SLIDE 16

Connectedness

Base case: smallest canonical triangle IH: There exists a path between every two vertices with a smaller canonical triangle Case2: w lies far from the bisector

w vw u

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 7 / 19

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SLIDE 17

Connectedness

Base case: smallest canonical triangle IH: There exists a path between every two vertices with a smaller canonical triangle Case2: w lies far from the bisector

w vw u

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 7 / 19

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SLIDE 18

Connectedness

Base case: smallest canonical triangle IH: There exists a path between every two vertices with a smaller canonical triangle Case2: w lies far from the bisector

w vw u

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 7 / 19

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SLIDE 19

Spanning Ratio - Strategy

Find a vertex v with

A path w v shorter than a · |△uw| w u v

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 8 / 19

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SLIDE 20

Spanning Ratio - Strategy

Find a vertex v with

A path w v shorter than a · |△uw| A canonical triangle smaller than b · |△uw| w u v

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 8 / 19

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SLIDE 21

Spanning Ratio - Strategy

Find a vertex v with

A path w v shorter than a · |△uw| A canonical triangle smaller than b · |△uw|

Then there is a path u w shorter than c · |△uw|

w u v

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 8 / 19

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SLIDE 22

Spanning Ratio - Cases

u w 1 2 3 4

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 9 / 19

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SLIDE 23

Spanning Ratio - Case 1

Case 1

u w

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SLIDE 24

Spanning Ratio - Case 1

Case 1 w v ≤ a · |△uw|

u vw w

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 10 / 19

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SLIDE 25

Spanning Ratio - Case 1

Case 1 w v ≤ a · |△uw| |△uv| ≤ b · |△uw|

u vw w

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 10 / 19

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SLIDE 26

Spanning Ratio - Case 1

Case 1 w v ≤ a · |△uw| |△uv| ≤ b · |△uw| Done!

u vw w

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 10 / 19

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SLIDE 27

Spanning Ratio - Case 2 & 3

Works for Case 2 and 3.

u w 2 3

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 11 / 19

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SLIDE 28

Spanning Ratio - Case 4

Case 4

w v u

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 12 / 19

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SLIDE 29

Spanning Ratio - Case 4

Case 4 Our strategy doesn’t work everywhere

u w v

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 12 / 19

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SLIDE 30

Spanning Ratio - Case 4

Case 4 Our strategy doesn’t work everywhere But it does work in a large part

u w v

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 12 / 19

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SLIDE 31

Spanning Ratio - Case 4

Case 4 Our strategy doesn’t work everywhere But it does work in a large part Left with a small region that we can’t solve

u w

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 12 / 19

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SLIDE 32

Spanning Ratio - Case 4

Case 4 Our strategy doesn’t work everywhere But it does work in a large part Left with a small region that we can’t solve What about vu?

u w

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 12 / 19

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SLIDE 33

Spanning Ratio - Case 4

Case 4 Our strategy doesn’t work everywhere But it does work in a large part Left with a small region that we can’t solve What about vu?

u w b c d e

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 12 / 19

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SLIDE 34

Spanning Ratio - Case 4b

Case 4b w v ≤ a · |△uw| |△uv| ≤ b · |△uw| Done!

u w v

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 13 / 19

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SLIDE 35

Spanning Ratio - Case 4c

Case 4c Convert to worst-case

v u w

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 14 / 19

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SLIDE 36

Spanning Ratio - Case 4c

Case 4c Convert to worst-case

w u v

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 14 / 19

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SLIDE 37

Spanning Ratio - Case 4c

Case 4c Convert to worst-case w v ≈ 0 |△uv| ≈ |△uw| Done!

w ≈ v u

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 14 / 19

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SLIDE 38

Spanning Ratio - Case 4d

Case 4d Convert to worst-case

vw vu w u

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 15 / 19

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SLIDE 39

Spanning Ratio - Case 4d

Case 4d Convert to worst-case

vw vu w u

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 15 / 19

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SLIDE 40

Spanning Ratio - Case 4d

Case 4d Convert to worst-case

vw w vu u

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 15 / 19

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SLIDE 41

Spanning Ratio - Case 4d

Case 4d Convert to worst-case

w vu vw u

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 15 / 19

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SLIDE 42

Spanning Ratio - Case 4d

Case 4d Convert to worst-case

w vu vw u

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 15 / 19

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SLIDE 43

Spanning Ratio - Case 4d

Case 4d Convert to worst-case

w vu vw u

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 15 / 19

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SLIDE 44

Spanning Ratio - Case 4d

Case 4d Convert to worst-case Equivalent to Case 1 Done!

w vw u ≈ vu

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 15 / 19

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SLIDE 45

Spanning Ratio - Case 4e

Case 4e

vw w u vu

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 16 / 19

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SLIDE 46

Spanning Ratio - Case 4e

Case 4e vu is close to w ⇒ Done!

vu w u

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 16 / 19

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SLIDE 47

Spanning Ratio - Case 4e

Case 4e vu is close to w ⇒ Done!

vw w vu u

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 16 / 19

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SLIDE 48

Spanning Ratio - Case 4e

Case 4e vu is close to w ⇒ Done! vu above vw

vw w vu u

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 16 / 19

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SLIDE 49

Spanning Ratio - Case 4e

Case 4e vu is close to w ⇒ Done! vu above vw

Convert to worst-case vw w vu u

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 16 / 19

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SLIDE 50

Spanning Ratio - Case 4e

Case 4e vu is close to w ⇒ Done! vu above vw

Convert to worst-case Done! u vw w vu

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 16 / 19

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SLIDE 51

Spanning Ratio - Case 4e

Case 4e vu is close to w ⇒ Done! vu above vw ⇒ Done! vu right of vw

w vw u vu

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 16 / 19

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SLIDE 52

Spanning Ratio - Case 4e

Case 4e vu is close to w ⇒ Done! vu above vw ⇒ Done! vu right of vw

Convert to worst-case w vw vu u

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 16 / 19

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SLIDE 53

Spanning Ratio - Case 4e

Case 4e vu is close to w ⇒ Done! vu above vw ⇒ Done! vu right of vw

Convert to worst-case w vw vu u

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 16 / 19

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SLIDE 54

Spanning Ratio - Case 4e

Case 4e vu is close to w ⇒ Done! vu above vw ⇒ Done! vu right of vw

Convert to worst-case w vw vu u

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 16 / 19

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SLIDE 55

Spanning Ratio - Case 4e

Case 4e vu is close to w ⇒ Done! vu above vw ⇒ Done! vu right of vw

Convert to worst-case Done! w vw vu u

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 16 / 19

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SLIDE 56

Spanning Ratio - Constant

There is a path between any pair of vertices, of length ≤ c · |△|

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 17 / 19

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SLIDE 57

Spanning Ratio - Constant

There is a path between any pair of vertices, of length ≤ c · |△| = 2(2 + √ 5) · |△| ≈ 8.472 · |△|

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 17 / 19

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SLIDE 58

Spanning Ratio - Constant

There is a path between any pair of vertices, of length ≤ c · |△| = 2(2 + √ 5) · |△| ≈ 8.472 · |△| To compute the spanning ratio, use the smallest of △uw and △wu Worst-case when △uw = △wu

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 17 / 19

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SLIDE 59

Spanning Ratio - Constant

There is a path between any pair of vertices, of length ≤ c · |△| = 2(2 + √ 5) · |△| ≈ 8.472 · |△| To compute the spanning ratio, use the smallest of △uw and △wu Worst-case when △uw = △wu The θ5-graph has spanning ratio at most cos π

10

cos π

5

· c ≈ 9.960

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 17 / 19

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SLIDE 60

Lower bound

u w

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 18 / 19

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SLIDE 61

Lower bound

u w

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 18 / 19

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SLIDE 62

Lower bound

u w v1

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SLIDE 63

Lower bound

u w v1

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SLIDE 64

Lower bound

u w v2 v1

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SLIDE 65

Lower bound

u w v2 v1

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SLIDE 66

Lower bound

u w v3 v2 v1

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 18 / 19

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SLIDE 67

Lower bound

u w v3 v2 v1

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SLIDE 68

Lower bound

u w v3 v2 v1

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SLIDE 69

Lower bound

u w v3 v2 v1

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SLIDE 70

Lower bound

u w v3 v4 v2 v1

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 18 / 19

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SLIDE 71

Lower bound

u w v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17 v18 v19 v20 v21 v22 v23 v24 v25 v26 v27 v28 v29 v30 v31

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 18 / 19

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SLIDE 72

Lower bound

u w v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17 v18 v19 v20 v21 v22 v23 v24 v25 v26 v27 v28 v29 v30 v31

Spanning ratio ≈ 3.798

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 18 / 19

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SLIDE 73

Conclusion

We showed that the θ5-graph is a constant geometric spanner Its spanning ratio lies in 3.798 ≤ ... ≤ 9.960

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 19 / 19

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SLIDE 74

Conclusion

We showed that the θ5-graph is a constant geometric spanner Its spanning ratio lies in 3.798 ≤ ... ≤ 9.960 Open:

Close the gap

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 19 / 19

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SLIDE 75

Conclusion

We showed that the θ5-graph is a constant geometric spanner Its spanning ratio lies in 3.798 ≤ ... ≤ 9.960 Open:

Close the gap Is θ4 a spanner?

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 19 / 19

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SLIDE 76

Conclusion

We showed that the θ5-graph is a constant geometric spanner Its spanning ratio lies in 3.798 ≤ ... ≤ 9.960 Open:

Close the gap Is θ4 a spanner? Yes! (WADS 2013)

Sander Verdonschot (Carleton University) The θ5-graph is a spanner June 20, 2013 19 / 19