Geometric Spanner Networks -...

103
Geometric Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ-Graph Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics Geometric Spanner Networks Spring 2014

Transcript of Geometric Spanner Networks -...

Page 1: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Geometric Spanner Networks

Spring 2014

Page 2: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Textbook:

Giri Narasimhan, Michiel Smid, Geometric SpannerNetworks, CAMBRIDGE UNIVERSITY PRESS, 2007.

Page 3: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Example of networks

London Underground Network

Page 4: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Example of networks

Ad hoc Network

Page 5: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Example of networks

Yeast Protein Interaction Network

Page 6: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Example of networks

MyrielMlle. Baptistine

Mme. Magloire

Countess de Lo

Geborand

Champtercier

Cravatte

Count

Old ManNapoleon

Valjean

Labarre

Marguerite

Mme. de R

IsabeauGervais

Fantine

Thenardier

Cosette

JavertFaucheleventBamatabois

Simplice

ScaufflaireWoman 1

Judge

Champmathieu

BrevetChenildieu

CochepailleMother Innocent

Mlle. Gillenormand

Marius

Enjolras

Bossuet

Gueulemer

Babet

Claquesous

Montparnasse

Toussaint

Tholomyes

Listolier

Fameuil

Blacheville

Favourite

DahliaZephine

PerpetuePontmercy

Eponine

Boulatruelle

Brujon

Lt. GillenormandGillenormand

Gribier

Mme. Pontmercy

Mabeuf

Jondrette

Mme. Burgon

Combeferre

Prouvaire

Feuilly

Bahorel

Joly

Grantaire

Child 1

Child 2

Mme. Hucheloup

Baroness T

Mlle. Vaubois

Mother Plutarch

Anzelma

Mme. Thenardier

Woman 2

CourfeyracGavroche

Magnon

A Social Network (Les Miserables characters)

Page 7: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Geometric Network

Page 8: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Geometric Network

Page 9: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Geometric Network

Page 10: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Geometric Network

Page 11: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Geometric Network

1

1.4

2.2

2.2

2.8

3

3.1

3.6

3.6

3.6

3.6

4.1

4.4

4.4

4.4

5

5

6.3(5,5)

(9,2)

(4,1)

(1,0)

(6,4)

(0,7)

(3,6)

(7,8)

(1,3)

(3,1)

Geometric NetworkWeighted undirected graphG(V,E) s.t.

V ⊂ Rd.∀e = (u, v) ∈ E, wt(e) = |uv|.

Page 12: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Network Quality

Driving distance: 256 km. Actual distance: 198 km.Driving distanceActual distance =1.27.

Page 13: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Network Quality

Driving distance: 180 km. Actual distance: 136 km.Driving distanceActual distance =1.32.

Page 14: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Network Quality

Driving distance: 143 km. Actual distance: 100 km.Driving distanceActual distance =1.43.

Page 15: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Network Quality

1

1.4

2.2

2.2

2.8

3

3.1

3.6

3.6

3.6

3.6

4.1

4.4

4.4

4.4

5

5

6.3(5,5)

(9,2)

(4,1)

(1,0)

(6,4)

(0,7)

(3,6)

(7,8)

(1,3)

(3,1)

Dilation (stretch factor)between a pair of vertices=

Distance in the graphEuclidean distance

of a network= maximumdilation between all pairs.

t-spannerA network with dilation atmost t, or∀u, v ∈ V , there is a pathbetween u and v of length≤ t× |uv|. (t-path)

Page 16: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Network Quality

1

1.4

2.2

2.2

2.8

3

3.1

3.6

3.6

3.6

3.6

4.1

4.4

4.4

4.4

5

5

6.3(5,5)

(9,2)

(4,1)

(1,0)

(6,4)

(0,7)

(3,6)

(7,8)

(1,3)

(3,1)

Dilation (stretch factor)between a pair of vertices=

Distance in the graphEuclidean distance

of a network= maximumdilation between all pairs.

t-spannerA network with dilation atmost t, or∀u, v ∈ V , there is a pathbetween u and v of length≤ t× |uv|. (t-path)

Page 17: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Network Quality

1

1.4

2.2

2.2

2.8

3

3.1

3.6

3.6

3.6

3.6

4.1

4.4

4.4

4.4

5

5

6.3(5,5)

(9,2)

(4,1)

(1,0)

(6,4)

(0,7)

(3,6)

(7,8)

(1,3)

(3,1)

Dilation (stretch factor)between a pair of vertices=

Distance in the graphEuclidean distance

of a network= maximumdilation between all pairs.

t-spannerA network with dilation atmost t, or∀u, v ∈ V , there is a pathbetween u and v of length≤ t× |uv|. (t-path)

Page 18: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Network Quality

1

1.4

2.2

2.2

2.8

3

3.1

3.6

3.6

3.6

3.6

4.1

4.4

4.4

4.4

5

5

6.3(5,5)

(9,2)

(4,1)

(1,0)

(6,4)

(0,7)

(3,6)

(7,8)

(1,3)

(3,1)

Dilation (stretch factor)between a pair of vertices=

Distance in the graphEuclidean distance

of a network= maximumdilation between all pairs.

t-spannerA network with dilation atmost t, or∀u, v ∈ V , there is a pathbetween u and v of length≤ t× |uv|. (t-path)

Page 19: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Network Quality

1

1.4

2.2

2.2

2.8

3

3.1

3.6

3.6

3.6

3.6

4.1

4.4

4.4

4.4

5

5

6.3(5,5)

(9,2)

(4,1)

(1,0)

(6,4)

(0,7)

(3,6)

(7,8)

(1,3)

(3,1)

Dilation (stretch factor)between a pair of vertices=

Distance in the graphEuclidean distance

of a network= maximumdilation between all pairs.

t-spannerA network with dilation atmost t, or∀u, v ∈ V , there is a pathbetween u and v of length≤ t× |uv|. (t-path)

Page 20: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Network Quality

(1 + ε)-Spanners approximate the complete graphs witherror ε.

Page 21: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Example

10-spanner for 532 US-cities

Page 22: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Example

5-spanner for 532 US-cities

Page 23: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Example

3-spanner for 532 US-cities

Page 24: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Example

2-spanner for 532 US-cities

Page 25: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Example

1.5-spanner for 532 US-cities

Page 26: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

How to compute a good spanner?

Given a set V and t > 1 Sparse t-Spanner

Quality measurement:Number of edges (size)Weight (compared with MST)Maximum degreeDiameter

Page 27: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

How to compute a good spanner?

Given a set V and t > 1 Sparse t-Spanner

Quality measurement:Number of edges (size)Weight (compared with MST)Maximum degreeDiameter

Page 28: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

How to compute a good spanner?

Constructing sparse t-spanners:Greedy (Bern (1989) and Althöfer et al. (1993)).Θ-graph (Clarkson (1987) and Keil (1988)).Ordered Θ-graph (Bose et. al. (2004)).Well-Separated Pair Decomposition (Arya et. al.(1995)).

Page 29: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

(Org.) Greedy Algorithm

(6, 1)

(8, 3)(4, 3)

(1, 5)

(5, 7)

(5, 8)(2, 8)

Page 30: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

(Org.) Greedy Algorithm

(6, 1)

(8, 3)(4, 3)

(1, 5)

(5, 7)

(5, 8)(2, 8)

Page 31: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

(Org.) Greedy Algorithm

(6, 1)

(8, 3)(4, 3)

(1, 5)

(5, 7)

(5, 8)(2, 8)

Page 32: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

(Org.) Greedy Algorithm

(6, 1)

(8, 3)(4, 3)

(1, 5)

(5, 7)

(5, 8)(2, 8)

Page 33: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

(Org.) Greedy Algorithm

(6, 1)

(8, 3)(4, 3)

(1, 5)

(5, 7)

(5, 8)(2, 8)

Page 34: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

(Org.) Greedy Algorithm

(6, 1)

(8, 3)(4, 3)

(1, 5)

(5, 7)

(5, 8)(2, 8)

Page 35: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

(Org.) Greedy Algorithm

(6, 1)

(8, 3)(4, 3)

(1, 5)

(5, 7)

(5, 8)(2, 8)

Page 36: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

(Org.) Greedy Algorithm

(6, 1)

(8, 3)(4, 3)

(1, 5)

(5, 7)

(5, 8)(2, 8)

Page 37: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

(Org.) Greedy Algorithm

(6, 1)

(8, 3)(4, 3)

(1, 5)

(5, 7)

(5, 8)(2, 8)

Page 38: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

(Org.) Greedy Algorithm

(6, 1)

(8, 3)(4, 3)

(1, 5)

(5, 7)

(5, 8)(2, 8)

Page 39: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

(Org.) Greedy Algorithm

(6, 1)

(8, 3)(4, 3)

(1, 5)

(5, 7)

(5, 8)(2, 8)

Page 40: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

(Org.) Greedy Algorithm

(6, 1)

(8, 3)(4, 3)

(1, 5)

(5, 7)

(5, 8)(2, 8)

Page 41: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

(Org.) Greedy Algorithm

ORG. GREEDY

Input: V and t > 1Output: t-spanner G(V,E)Sort pairs of points by non-decreasing order of distance;E := ∅;G := (V,E) ;for each pair (u, v) of points (in sorted order) do

if SHORTESTPATH(G, u, v) > t · |uv| thenAdd (u, v) to E;

endendreturn G(V,E);

Time Complexity: O(n3 log n).

Storage Complexity: O(n2).

Page 42: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

(Org.) Greedy Algorithm

ORG. GREEDY

Input: V and t > 1Output: t-spanner G(V,E)Sort pairs of points by non-decreasing order of distance;E := ∅;G := (V,E) ;for each pair (u, v) of points (in sorted order) do

if SHORTESTPATH(G, u, v) > t · |uv| thenAdd (u, v) to E;

endendreturn G(V,E);

Time Complexity: O(n3 log n).

Storage Complexity: O(n2).

Page 43: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Imp. Greedy AlgorithmORG. GREEDY

Input: V and t > 1Output: t-spanner G(V,E)Sort pairs of points by non-decreasing order of distance;E := ∅; G := (V,E) ;for each pair (u, v) of points (in sorted order) do

if SHORTESTPATH(G, u, v) > t · |uv| thenAdd (u, v) to E;

endendreturn G(V,E);

Number of shortest path queries: Θ(n2).

Page 44: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Imp. Greedy AlgorithmORG. GREEDY

Input: V and t > 1Output: t-spanner G(V,E)Sort pairs of points by non-decreasing order of distance;E := ∅; G := (V,E) ;for each pair (u, v) of points (in sorted order) do

if SHORTESTPATH(G, u, v) > t · |uv| thenAdd (u, v) to E;

endendreturn G(V,E);

Number of shortest path queries: Θ(n2).Observations:

We only want to know if there is a t-path between uand v.The graph is only updated O(n) times.

Page 45: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Imp. Greedy Algorithm

IMP. GREEDY

Input: V and t > 1Output: t-spanner G(V,E)for each pair (u, v) ∈ V 2 do Set Weight(u, v) :=∞;Sort pairs of points by non-decreasing order of distance;E := ∅; G := (V,E) ;for each pair (u, v) of points (in sorted order) do

if Weight(u, v) ≤ t · |uv| thenSkip (u, v);

elseCompute single source shortest path with source u;for each w do update Weight(u,w) and Weight(w, u);if Weight(u, v) ≤ t · |uv| then Skip (u, v);else Add (u, v) to E;

endendreturn G(V,E);

Page 46: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Imp. Greedy Algorithm

Conjecture:The running time of IMP. GREEDY is O(n2 log n).

Bose, Carmi, Farshi, Maheshvari and Smid (2008)The conjecture is wrong!They presented an algorithm which computes thegreedy spanner in O(n2 log n) time (even for pointsfrom some metric spaces).

Page 47: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Apx. Greedy Algorithm

t-spanner AlgorithmPoint set

t

t-spanner

Constant degree

Page 48: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Apx. Greedy Algorithm

t-spanner Algorithm Pruning Algorithm

Point set

t

t-spanner

t′

(t · t′)-spannerApproximate

Constant degree

Page 49: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Apx. Greedy Algorithm

t-spanner Algorithm Pruning Algorithm

Point set

t

t-spanner

t′

(t · t′)-spannerApproximate

Constant degree

Sink Spanner

O(n) edges

Page 50: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Apx. Greedy Algorithm

t-spanner Algorithm Pruning Algorithm

Point set

t

t-spanner

t′

(t · t′)-spannerApproximate

Constant degree

Sink Spanner

O(n) edges

Time Complexity: O(n log2 n)

Storage Complexity: O(n).

Page 51: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Θ-Graph Algorithmt = 3,Θ = π/6

Page 52: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Θ-Graph Algorithmt = 3,Θ = π/6

Page 53: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Θ-Graph Algorithmt = 3,Θ = π/6

Page 54: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Θ-Graph Algorithmt = 3,Θ = π/6

Page 55: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Θ-Graph Algorithmt = 3,Θ = π/6

Page 56: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Θ-Graph Algorithmt = 3,Θ = π/6

Page 57: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Θ-Graph Algorithmt = 3,Θ = π/6

Page 58: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Θ-Graph Algorithmt = 3,Θ = π/6

Page 59: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Θ-Graph Algorithmt = 3,Θ = π/6

Page 60: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Θ-Graph Algorithmt = 3,Θ = π/6

Page 61: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Θ-Graph Algorithm

Θ-GRAPH

Input: V and t > 1Output: t-spanner G(V,E)Set k:= the smallest integer such that t = 1

cos θ−sin θ forθ = 2π/k;E := ∅;for each point u ∈ V do

C1, . . . , Ck := non-overlapping cones with angle θand with apex at u;for each cone Ci do

Connect u to the closest point in Ci;end

endreturn G(V,E);

Time Complexity: O(n log n).Storage Complexity: O(n).

Page 62: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Θ-Graph Algorithm

Θ-GRAPH

Input: V and t > 1Output: t-spanner G(V,E)Set k:= the smallest integer such that t = 1

cos θ−sin θ forθ = 2π/k;E := ∅;for each point u ∈ V do

C1, . . . , Ck := non-overlapping cones with angle θand with apex at u;for each cone Ci do

Connect u to the closest point in Ci;end

endreturn G(V,E);

Time Complexity: O(n log n).Storage Complexity: O(n).

Page 63: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Variants of Θ-Graph Algorithm

Ordered Θ-Graph– O(log n) maximum degreeSame as the Θ-graph algorithm, except we add pointsone by one in a special order.

Random Ordered Θ-Graph– O(log n) spannerdiameterWe add points one by one in a random order.

Sink Spanner– bounded degreeDecrease the degree of nodes by replacing some edgesby paths within other nodes.

Skip-List Spanner– O(log n) spanner diameterDecrease the diameter of Θ-graph by adding some extraedges.

Page 64: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Variants of Θ-Graph Algorithm

Ordered Θ-Graph– O(log n) maximum degreeSame as the Θ-graph algorithm, except we add pointsone by one in a special order.

Random Ordered Θ-Graph– O(log n) spannerdiameterWe add points one by one in a random order.

Sink Spanner– bounded degreeDecrease the degree of nodes by replacing some edgesby paths within other nodes.

Skip-List Spanner– O(log n) spanner diameterDecrease the diameter of Θ-graph by adding some extraedges.

Page 65: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Variants of Θ-Graph Algorithm

Ordered Θ-Graph– O(log n) maximum degreeSame as the Θ-graph algorithm, except we add pointsone by one in a special order.

Random Ordered Θ-Graph– O(log n) spannerdiameterWe add points one by one in a random order.

Sink Spanner– bounded degreeDecrease the degree of nodes by replacing some edgesby paths within other nodes.

Skip-List Spanner– O(log n) spanner diameterDecrease the diameter of Θ-graph by adding some extraedges.

Page 66: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Variants of Θ-Graph Algorithm

Ordered Θ-Graph– O(log n) maximum degreeSame as the Θ-graph algorithm, except we add pointsone by one in a special order.

Random Ordered Θ-Graph– O(log n) spannerdiameterWe add points one by one in a random order.

Sink Spanner– bounded degreeDecrease the degree of nodes by replacing some edgesby paths within other nodes.

Skip-List Spanner– O(log n) spanner diameterDecrease the diameter of Θ-graph by adding some extraedges.

Page 67: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Sink SpannerA variant of Θ-graph with bounded degree

Input: V and t > 1Output: t-spanner G(V,E)

Construct a directed√t-spanner

−→G with bounded

out-degree;for each point q ∈ V do

Replace the “star” pointing to q by a√t-q-sink

spannerendreturn G(V,E);

Time Complexity: O(n log n) Storage Complexity: O(n).

Page 68: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Sink SpannerA variant of Θ-graph with bounded degree

Input: V and t > 1Output: t-spanner G(V,E)

Construct a directed√t-spanner

−→G with bounded

out-degree;for each point q ∈ V do

Replace the “star” pointing to q by a√t-q-sink

spannerendreturn G(V,E);

Time Complexity: O(n log n) Storage Complexity: O(n).

Page 69: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Skip-List SpannerA variant of Θ-graph with O(logn) spanner diameter

Input: V and t > 1Output: t-spanner G(V,E)Set V0 := V ; i := 1;while Vi−1 6= ∅ do

Vi contains each points of Vi−1 with probability 1/2;endfor each i do

Construct a t-spanner Gi(Vi, Ei) using the Θ-graphalgorithm;

endE = ∪iEi;return G(V,E);

Time Complexity: O(n log n) Storage Complexity: O(n).

Page 70: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Skip-List SpannerA variant of Θ-graph with O(logn) spanner diameter

Input: V and t > 1Output: t-spanner G(V,E)Set V0 := V ; i := 1;while Vi−1 6= ∅ do

Vi contains each points of Vi−1 with probability 1/2;endfor each i do

Construct a t-spanner Gi(Vi, Ei) using the Θ-graphalgorithm;

endE = ∪iEi;return G(V,E);

Time Complexity: O(n log n) Storage Complexity: O(n).

Page 71: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Well Separated Pair Decomposition (WSPD)

Well Separated Pair:A,B ⊂ Rd are s-well separated (s > 0), if ∃ disjoint balls,DA and DB such that

Page 72: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Well Separated Pair Decomposition (WSPD)

Well Separated Pair:A,B ⊂ Rd are s-well separated (s > 0), if ∃ disjoint balls,DA and DB such that

A

B

Page 73: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Well Separated Pair Decomposition (WSPD)

Well Separated Pair:A,B ⊂ Rd are s-well separated (s > 0), if ∃ disjoint balls,DA and DB such that

A ⊆ DA and B ⊆ DB.

A

B

DB

DA

rA

rB

Page 74: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Well Separated Pair Decomposition (WSPD)

Well Separated Pair:A,B ⊂ Rd are s-well separated (s > 0), if ∃ disjoint balls,DA and DB such that

A ⊆ DA and B ⊆ DB.d(DA, DB) ≥ s×max(radius(DA), radius(DB)).

A

B

DB

DA

≥ s×max(rA, rB)

rA

rB

Page 75: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Well Separated Pair Decomposition (WSPD)

Well Separated Pair Decomposition:Let V ⊂ Rd and s > 0. A WSPD for V with respect to s isa set {(Ai, Bi)}mi=1 of pairs of non-empty subsets of Vsuch that

∀i, Ai and Bi are s-well separated,∀p, q ∈ V , there is exactly one index i s. t.

p ∈ Ai and q ∈ Bi orq ∈ Ai and p ∈ Bi.

m : Size of WSPD.

Callahan & Kosaraju (1995)For each set of n points, we can construct a WSPD ofsize O(sd · n) in O(n log n) time using O(sd · n) space.

Page 76: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Well Separated Pair Decomposition (WSPD)

Well Separated Pair Decomposition:Let V ⊂ Rd and s > 0. A WSPD for V with respect to s isa set {(Ai, Bi)}mi=1 of pairs of non-empty subsets of Vsuch that

∀i, Ai and Bi are s-well separated,∀p, q ∈ V , there is exactly one index i s. t.

p ∈ Ai and q ∈ Bi orq ∈ Ai and p ∈ Bi.

m : Size of WSPD.

Callahan & Kosaraju (1995)For each set of n points, we can construct a WSPD ofsize O(sd · n) in O(n log n) time using O(sd · n) space.

Page 77: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Well Separated Pair Decomposition (WSPD)

Well Separated Pair Decomposition:Let V ⊂ Rd and s > 0. A WSPD for V with respect to s isa set {(Ai, Bi)}mi=1 of pairs of non-empty subsets of Vsuch that

∀i, Ai and Bi are s-well separated,∀p, q ∈ V , there is exactly one index i s. t.

p ∈ Ai and q ∈ Bi orq ∈ Ai and p ∈ Bi.

m : Size of WSPD.

Callahan & Kosaraju (1995)For each set of n points, we can construct a WSPD ofsize O(sd · n) in O(n log n) time using O(sd · n) space.

Page 78: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Well Separated Pair Decomposition (WSPD)

Well Separated Pair Decomposition:Let V ⊂ Rd and s > 0. A WSPD for V with respect to s isa set {(Ai, Bi)}mi=1 of pairs of non-empty subsets of Vsuch that

∀i, Ai and Bi are s-well separated,∀p, q ∈ V , there is exactly one index i s. t.

p ∈ Ai and q ∈ Bi orq ∈ Ai and p ∈ Bi.

m : Size of WSPD.

Callahan & Kosaraju (1995)For each set of n points, we can construct a WSPD ofsize O(sd · n) in O(n log n) time using O(sd · n) space.

Page 79: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Well Separated Pair Decomposition (WSPD)

Well Separated Pair Decomposition:Let V ⊂ Rd and s > 0. A WSPD for V with respect to s isa set {(Ai, Bi)}mi=1 of pairs of non-empty subsets of Vsuch that

∀i, Ai and Bi are s-well separated,∀p, q ∈ V , there is exactly one index i s. t.

p ∈ Ai and q ∈ Bi orq ∈ Ai and p ∈ Bi.

m : Size of WSPD.

Callahan & Kosaraju (1995)For each set of n points, we can construct a WSPD ofsize O(sd · n) in O(n log n) time using O(sd · n) space.

Page 80: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

WSPD-based Algorithm

WSPD Algorithm

Input: V and t > 1Output: t-spanner G(V,E)

SetW := WSPD of V w.r.t. s := 4(t+1)t−1 ;

Set E = ∅;for each (Ai, Bi) ∈ W do

Select an arbitrary node u ∈ Ai and an arbitrary nodev ∈ Bi;Add edge (u, v) to E.

endreturn G(V,E).

Time Complexity: O(n log n).Storage Complexity: O(n).

Page 81: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

WSPD-based Algorithm

WSPD Algorithm

Input: V and t > 1Output: t-spanner G(V,E)

SetW := WSPD of V w.r.t. s := 4(t+1)t−1 ;

Set E = ∅;for each (Ai, Bi) ∈ W do

Select an arbitrary node u ∈ Ai and an arbitrary nodev ∈ Bi;Add edge (u, v) to E.

endreturn G(V,E).

Time Complexity: O(n log n).Storage Complexity: O(n).

Page 82: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Theoretical bounds

- Size Weight Degree Time

Greedy spanner O(n) O(wt(MST)) O(1) O(n2 logn)

Apx. greedy spanner O(n) O(wt(MST)) O(1) O(n logn)

Θ-graph O(n) Θ(n · wt(MST)) Θ(n) O(n logn)

O. Θ-graph O(n) O(n · wt(MST)) O(logn) O(n logn)

WSPD spanner O(n) O(logn · wt(MST)) Θ(n) O(n logn)

Sink-spanner O(n) O(n · wt(MST)) O(1) O(n logn)

Skip-list spanner O(n)∗ Θ(n · wt(MST))∗ Θ(n) O(n logn)∗

(*): Expected with high probability

Page 83: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Applications

Page 84: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

ApplicationsDesigning approximation algorithms with spanners

Traveling Salesperson Problem (TSP)Find the shortest tour that visits each point exactly onceand return to the starting point.

Known results:The problem is NP-hard even in Rd.A 2-approximation algorithm for metric spaces byRosenkrantz et al. (1977).A 1.5-approximation algorithm by Christofides et al.(1976).A PTAS ((1 + ε)-approx. Alg.) for geometric case byArora (1998) and Mitchell (1999).A PTAS for geometric case using spanners by Raoand Smith (1998).

Page 85: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

ApplicationsDesigning approximation algorithms with spanners

Traveling Salesperson Problem (TSP)Find the shortest tour that visits each point exactly onceand return to the starting point.

Known results:The problem is NP-hard even in Rd.A 2-approximation algorithm for metric spaces byRosenkrantz et al. (1977).A 1.5-approximation algorithm by Christofides et al.(1976).A PTAS ((1 + ε)-approx. Alg.) for geometric case byArora (1998) and Mitchell (1999).A PTAS for geometric case using spanners by Raoand Smith (1998).

Page 86: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

ApplicationsDesigning approximation algorithms with spanners

Definition:If G is a graph with vertex set P , then a tour of P in G is a(possibly non-simple) cycle in G that visits each point ofP at least once.

Observation:For any t-spanner G for P , there is a tour of P in G,whose weight is at most t · wt(TSP(P )).

Theorem (Rao and Smith, 1998)Given a (1 + ε)-spanner of a set of n points with O(n)size and O(wt(MST)) weight, we can compute a(1 + ε)-approximation of TSP(P ) in O(n log n) time.

Page 87: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

ApplicationsDesigning approximation algorithms with spanners

Definition:If G is a graph with vertex set P , then a tour of P in G is a(possibly non-simple) cycle in G that visits each point ofP at least once.

Observation:For any t-spanner G for P , there is a tour of P in G,whose weight is at most t · wt(TSP(P )).

Theorem (Rao and Smith, 1998)Given a (1 + ε)-spanner of a set of n points with O(n)size and O(wt(MST)) weight, we can compute a(1 + ε)-approximation of TSP(P ) in O(n log n) time.

Page 88: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

ApplicationsDesigning approximation algorithms with spanners

Definition:If G is a graph with vertex set P , then a tour of P in G is a(possibly non-simple) cycle in G that visits each point ofP at least once.

Observation:For any t-spanner G for P , there is a tour of P in G,whose weight is at most t · wt(TSP(P )).

Theorem (Rao and Smith, 1998)Given a (1 + ε)-spanner of a set of n points with O(n)size and O(wt(MST)) weight, we can compute a(1 + ε)-approximation of TSP(P ) in O(n log n) time.S. B. Rao and W. D. Smith, Approximating Geometrical Graphs via“Spanners” and “Banyans”, STOC’98, pp. 540–550, 1998.

Page 89: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

ApplicationsMetric space searching

Approximate proximity searching:Multimedia information retrieval,Data mining,Pattern recognition,Machine learning,Computer vision andBiomedical databases.

Page 90: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

ApplicationsMetric space searching

Approximate proximity searching:Multimedia information retrieval,Data mining,Pattern recognition,Machine learning,Computer vision andBiomedical databases.

Page 91: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

ApplicationsMetric space searching

What is the role of spanners?A meter show the similarity between any two objects.But evaluating the distances are expensive.One way to speedup is computing the distancebetween any two objects and save them, but it needO(n2) space (AESA).A t-spanner can be used as a sparse data structureto reduce the space.

Page 92: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

ApplicationsMetric space searching

What is the role of spanners?A meter show the similarity between any two objects.But evaluating the distances are expensive.One way to speedup is computing the distancebetween any two objects and save them, but it needO(n2) space (AESA).A t-spanner can be used as a sparse data structureto reduce the space.

Page 93: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

ApplicationsMetric space searching

What is the role of spanners?A meter show the similarity between any two objects.But evaluating the distances are expensive.One way to speedup is computing the distancebetween any two objects and save them, but it needO(n2) space (AESA).A t-spanner can be used as a sparse data structureto reduce the space.

Page 94: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

ApplicationsMetric space searching

What is the role of spanners?A meter show the similarity between any two objects.But evaluating the distances are expensive.One way to speedup is computing the distancebetween any two objects and save them, but it needO(n2) space (AESA).A t-spanner can be used as a sparse data structureto reduce the space.

G. Navarro, R. Paredes, and E. Chávez, t-Spanners for metric spacesearching, Data & Knowledge Engineering, pp. 820-854, 2007.

Page 95: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

ApplicationsProtein Visualization

D. Russel and L. Guibas, Exploring Protein Folding TrajectoriesUsing Geometric Spanners, Pacific Symposium on Biocomputing,pp. 40-51, 2005.

Page 96: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Research Topics

Current and Future Works:Dynamic spanners (insert and remove nodes).Kinetic spanners (when points move and we want tomaintain an spanner all the time).Fault-tolerant spanners (vertex/edge fault tolerant orregion fault tolerant).Spanners among obstacles.Optimization problems.External memory (I/O efficient) algorithms forgenerating spanners.Experimental works on spanner algorithms.

Page 97: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Research Topics

Current and Future Works:Dynamic spanners (insert and remove nodes).Kinetic spanners (when points move and we want tomaintain an spanner all the time).Fault-tolerant spanners (vertex/edge fault tolerant orregion fault tolerant).Spanners among obstacles.Optimization problems.External memory (I/O efficient) algorithms forgenerating spanners.Experimental works on spanner algorithms.

Page 98: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Research Topics

Current and Future Works:Dynamic spanners (insert and remove nodes).Kinetic spanners (when points move and we want tomaintain an spanner all the time).Fault-tolerant spanners (vertex/edge fault tolerant orregion fault tolerant).Spanners among obstacles.Optimization problems.External memory (I/O efficient) algorithms forgenerating spanners.Experimental works on spanner algorithms.

Page 99: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Research Topics

Current and Future Works:Dynamic spanners (insert and remove nodes).Kinetic spanners (when points move and we want tomaintain an spanner all the time).Fault-tolerant spanners (vertex/edge fault tolerant orregion fault tolerant).Spanners among obstacles.Optimization problems.External memory (I/O efficient) algorithms forgenerating spanners.Experimental works on spanner algorithms.

Page 100: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Research Topics

Current and Future Works:Dynamic spanners (insert and remove nodes).Kinetic spanners (when points move and we want tomaintain an spanner all the time).Fault-tolerant spanners (vertex/edge fault tolerant orregion fault tolerant).Spanners among obstacles.Optimization problems.External memory (I/O efficient) algorithms forgenerating spanners.Experimental works on spanner algorithms.

Page 101: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Research Topics

Current and Future Works:Dynamic spanners (insert and remove nodes).Kinetic spanners (when points move and we want tomaintain an spanner all the time).Fault-tolerant spanners (vertex/edge fault tolerant orregion fault tolerant).Spanners among obstacles.Optimization problems.External memory (I/O efficient) algorithms forgenerating spanners.Experimental works on spanner algorithms.

Page 102: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics

Research Topics

Current and Future Works:Dynamic spanners (insert and remove nodes).Kinetic spanners (when points move and we want tomaintain an spanner all the time).Fault-tolerant spanners (vertex/edge fault tolerant orregion fault tolerant).Spanners among obstacles.Optimization problems.External memory (I/O efficient) algorithms forgenerating spanners.Experimental works on spanner algorithms.

Page 103: Geometric Spanner Networks - Yazdcs.yazd.ac.ir/farshi/Teaching/Spanner-3922/Slides/GSN-Course.pdf · Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy

GeometricSpanner Networks

Course OutlineTextbook

Introduction

Algorithms ReviewGreedy Algorithm (Org. andImp.)

Apx. Greedy Algorithm

(Ordered) Θ-GraphAlgorithm (Sink andSkip-list spanner)

Sink Spanner

WSPD-based Algorithm

Theoreticalbounds

ApplicationsDesigning approximationalgorithms with spanners

Metric space searching

Protein Visualization

Research Topics