A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max...
-
Upload
madison-harrington -
Category
Documents
-
view
227 -
download
0
description
Transcript of A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max...
![Page 1: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/1.jpg)
A randomized linear time algorithm A randomized linear time algorithm forfor
graph spannersgraph spanners
Surender Baswana
Postdoctoral Researcher Max Planck Institute for Computer Science
Saarbruecken, Germany
![Page 2: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/2.jpg)
Graph SpannersGraph SpannersDefinition :
Given a graph G=(V,E), a spanner is a sub-graph G=(V,Es) which has the following two crucial properties
![Page 3: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/3.jpg)
Graph SpannersGraph SpannersDefinition :
Given a graph G=(V,E), a spanner is a sub-graph G=(V,Es) which has the following two crucial properties
1. sparse
![Page 4: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/4.jpg)
Graph SpannersGraph SpannersDefinition :
Given a graph G=(V,E), a spanner is a sub-graph G=(V,Es) which has the following two crucial properties
1. sparse
2. preserves approximate distances pair-wise.
![Page 5: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/5.jpg)
Graph SpannersGraph SpannersDefinition :
Given a graph G=(V,E), a spanner is a sub-graph G=(V,Es) which has the following two crucial properties
1. sparse
2. preserves approximate distances pair-wise.
δ(u,v) ≤ δs(u,v) ≤ t δ(u,v) for some constant t ≥ 1
![Page 6: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/6.jpg)
Graph SpannersGraph SpannersDefinition :
Given a graph G=(V,E), a spanner is a sub-graph G=(V,Es) which has the following two crucial properties
1. sparse
2. preserves approximate distances pair-wise.
δ(u,v) ≤ δs(u,v) ≤ t δ(u,v) for some constant t ≥ 1
t : stretch of the spanner
![Page 7: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/7.jpg)
Communication network :Communication network :Motivation for spannersMotivation for spanners
![Page 8: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/8.jpg)
Communication network :Communication network :Motivation for spannersMotivation for spanners
Each edge has• cost• weight (length)
![Page 9: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/9.jpg)
Communication network :Communication network :Motivation for spannersMotivation for spanners
Minimizing the total cost : sparseness is desirable
![Page 10: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/10.jpg)
Communication network :Communication network :Motivation for spannersMotivation for spanners
Minimizing the total cost : sparseness is desirable
u
v
![Page 11: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/11.jpg)
Communication network :Communication network :Motivation for spannersMotivation for spanners
Minimizing the pair-wise distances : small stretch is desirable
![Page 12: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/12.jpg)
Communication network :Communication network :Motivation for spannersMotivation for spanners
Minimizing the pair-wise distances : small stretch is desirable
![Page 13: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/13.jpg)
Graph spannersGraph spanners A trade off between sparseness and stretch
![Page 14: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/14.jpg)
Graph spannersGraph spanners A trade off between sparseness and stretch
Sparse
δ(u,v) ≤ δs(u,v) ≤ t δ(u,v)
![Page 15: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/15.jpg)
Graph spannersGraph spanners A trade off between sparseness and stretch
Sparse
δ(u,v) ≤ δs(u,v) ≤ t δ(u,v)
t-Spanner
![Page 16: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/16.jpg)
Aim :
To compute the sparsest spanner of a weighted graph with stretch t.
![Page 17: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/17.jpg)
Applications of Graph SpannerApplications of Graph Spanner Distributed Computing
Design of Synchronizers Compact routing tables
Computational Biology Reconstruction of Phylogenetic trees
All-pairs Approximate Shortest Paths
![Page 18: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/18.jpg)
Organization of the talkOrganization of the talk
Optimal size of a t-spanner
Earlier algorithms
A algorithm
![Page 19: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/19.jpg)
Optimal size of a t-spanner
![Page 20: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/20.jpg)
Optimal size of aOptimal size of a tt--spannerspanner
u v
![Page 21: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/21.jpg)
Optimal size of aOptimal size of a tt--spannerspanner
u v
![Page 22: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/22.jpg)
Optimal size of aOptimal size of a tt--spannerspanner
u v
???Length of Smallest cycle= t
![Page 23: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/23.jpg)
Optimal size of aOptimal size of a tt--spannerspanner
u v
stretch ≥ t-1Length of Smallest cycle= t
![Page 24: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/24.jpg)
Optimal size of aOptimal size of a tt--spannerspanner
u v
stretch ≥ t-1Length of Smallest cycle= t
How dense can a graph with shortest cycle length ≥ t be ?
![Page 25: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/25.jpg)
Optimal size of aOptimal size of a tt--spannerspanner
u v
stretch ≥ t-1
Girth Conjecture [Erdös[1960], Bondy & Simonovits [1974], Bollobas [1978]]There are graph with shortest cycle length > 2k and Ω(n1+1/k) edges
Length of Smallest cycle= t
![Page 26: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/26.jpg)
Optimal size of aOptimal size of a tt--spannerspannerLet k be any positive integer
There are graphs whose (2k-1)-spanner (a 2k-spanner) must have Ω(n1+1/k) edges
![Page 27: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/27.jpg)
Optimal size of aOptimal size of a tt--spannerspannerLet k be any positive integer
There are graphs whose (2k-1)-spanner (a 2k-spanner) must have Ω(n1+1/k) edges
4- spanner and 3-spanner : Ω(n3/2)
6-spanner and 5-spanner : Ω(n5/4)
8-spanner and 7-spanner : Ω(n7/6)
![Page 28: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/28.jpg)
Aim of an AlgorithmistAim of an Algorithmist To design an algorithm A A such that
AAG=(V, E) G=(V, Es)
|ES| = O (minimum (m , n1+1/k))
Input Graph (2k-1)-Spanner
![Page 29: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/29.jpg)
Earlier algorithms for (2k-1)-spanner
![Page 30: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/30.jpg)
Earlier algorithms for graph Earlier algorithms for graph spannersspanners
Althofer, Das, Dobkin,Joseph, Soares
DCG 1993 2k-1 O(mn1+1/k) O(n1+1/k)
Cohen SIAM J. Computing1998
(2k-1)(1+ε)
O(mn1/k) O(n1+1/k)
Thorup, Zwick
JACM 2005
2k-1 O(mn1/k) (randomized)
O(n1+1/k)
Stretch ConstructionTime
Size
![Page 31: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/31.jpg)
Althofer, Das, Dobkin,Joseph, Soares
DCG 1993 2k-1 O(mn1+1/k) O(n1+1/k)
Cohen SIAM J. Computing 1998
(2k-1)(1+ε)
O(mn1/k) O(n1+1/k)
Thorup, Zwick
JACM 2005
2k-1 O(mn1/k) (randomized)
O(n1+1/k)
Stretch ConstructionTime
Size
Can we Compute Can we Compute (2k-1)-(2k-1)-spanners in linear time ?spanners in linear time ?
![Page 32: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/32.jpg)
A algorithm
Computing a Computing a (2k-1)-(2k-1)-spanner in expectedspanner in expected O(m)O(m) timetime
![Page 33: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/33.jpg)
Local approachLocal approachLet G=(V,ES) be a spanner of G=(V,E)
Edge in Spanner
Edge not in Spanner
![Page 34: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/34.jpg)
Local approachLocal approachLet G=(V,ES) be a spanner of G=(V,E)
Edge in Spanner
Edge not in Spanner
![Page 35: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/35.jpg)
Local approachLocal approachLet G=(V,ES) be a spanner of G=(V,E)
2
1 t-1≤w
≤w ≤w
≤w
w
Pt : For each edge not in the spanner , there is a path in the spanner connecting its endpoints
• with at-most t edges• none heavier than the edge
Edge not in Spanner
Edge in Spanner
![Page 36: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/36.jpg)
Local approachLocal approachLet G=(V,ES) be a spanner of G=(V,E)
u v
2
1 t-1≤w
≤w ≤w
≤w
w
Pt : For each edge not in the spanner , there is a path in the spanner connecting its endpoints
• with at-most t edges• none heavier than the edge
Edge not in Spanner
Edge in Spanner
![Page 37: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/37.jpg)
Local approachLocal approachLet G=(V,ES) be a spanner of G=(V,E)
u v
2
1 t-1≤w
≤w ≤w
≤w
w
Pt : For each edge not in the spanner , there is a path in the spanner connecting its endpoints
• with at-most t edges• none heavier than the edge
Edge not in Spanner
Edge in Spanner
t-spanner
![Page 38: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/38.jpg)
NewNew Algorithms for ( Algorithms for (2k-12k-1)-)-spannerspanner
An External-memory algorithm for (2k-1)-spanner
Time complexity : Integer sorting
![Page 39: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/39.jpg)
NewNew Algorithms for ( Algorithms for (2k-12k-1)-)-spannerspanner
A distributed algorithm for (2k-1)-spanner :
Number of Rounds : O(1) , Communication complexity : O(m) (linear)
![Page 40: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/40.jpg)
NewNew Algorithms for ( Algorithms for (2k-12k-1)-)-spannerspanner
A streaming algorithm for (2k-1)-spanner Number of passes : O(1) Processing time per edge : O(1)
![Page 41: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/41.jpg)
Algorithm for 3-spanner
![Page 42: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/42.jpg)
Algorithm for 3-spannerAlgorithm for 3-spannerEasy caseEasy case : fewer than : fewer than n½ edges edges
![Page 43: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/43.jpg)
Algorithm for 3-spannerAlgorithm for 3-spannerDifficult caseDifficult case : much more than : much more than n½ edges edges
![Page 44: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/44.jpg)
Algorithm for 3-spannerAlgorithm for 3-spannerDifficult caseDifficult case : much more than : much more than n½ edges edges
Which n½ edges to select ?
![Page 45: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/45.jpg)
Algorithm for 3-spannerAlgorithm for 3-spanner Phase 1 : Clustering
Phase 2 : Adding edges between vertices and clusters
Initially all edges are Red
![Page 46: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/46.jpg)
Algorithm for 3-spannerAlgorithm for 3-spanner Phase 1 : Clustering
Phase 2 : Adding edges between vertices and clusters
center
Initially all edges are Red
![Page 47: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/47.jpg)
Algorithm for 3-spannerAlgorithm for 3-spannerPhase 1 : Phase 1 : ClusteringClustering
1. S : select each vertex independently with probability p.
![Page 48: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/48.jpg)
Algorithm for 3-spannerAlgorithm for 3-spannerPhase 1 : Phase 1 : ClusteringClustering
1. S : select each vertex independently with probability p.
2. Process each v Є V \S as follows
![Page 49: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/49.jpg)
Algorithm for 3-spannerAlgorithm for 3-spannerPhase 1 : Phase 1 : ClusteringClustering
1. S : select each vertex independently with probability p.
2. Process each v Є V \S as follows1. If v is not adjacent to any sampled vertex
![Page 50: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/50.jpg)
Algorithm for 3-spannerAlgorithm for 3-spannerPhase 1 : Phase 1 : ClusteringClustering
1. S : select each vertex independently with probability p .
2. Process each v Є V \S as follows1. If v is not adjacent to any sampled vertex.
v
S
V \S
![Page 51: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/51.jpg)
Algorithm for 3-spannerAlgorithm for 3-spannerPhase 1 : Phase 1 : ClusteringClustering
1. S : select each vertex independently with probability p .
2. Process each v Є V \S as follows1. If v is not adjacent to any sampled vertex. add all its
edges.
v
S
V \S
![Page 52: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/52.jpg)
Algorithm for 3-spannerAlgorithm for 3-spannerPhase 1 : Phase 1 : ClusteringClustering
1. S : select each vertex independently with probability p.
2. Process each v Є V \S as follows1. If v is not adjacent to any sampled vertex. add all its
edges.2. If v is adjacent to some sampled vertex
S
V \S
![Page 53: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/53.jpg)
Algorithm for 3-spannerAlgorithm for 3-spannerPhase 1 : Phase 1 : ClusteringClustering
1. S : select each vertex independently with probability p .
2. Process each v Є V \S as follows1. If v is not adjacent to any sampled vertex. add all its
edges.2. If v is adjacent to some sampled vertex.
weights
v
x
S
V \S
![Page 54: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/54.jpg)
Algorithm for 3-spannerAlgorithm for 3-spannerPhase 1 : Phase 1 : ClusteringClustering
1. S : select each vertex independently with probability p.
2. Process each v Є V \S as follows1. If v is not adjacent to any sampled vertex. add all its
edges.2. If v is adjacent to some sampled vertex.
weights
v
x
S
V \S
![Page 55: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/55.jpg)
Algorithm for 3-spannerAlgorithm for 3-spannerPhase 1 : Phase 1 : ClusteringClustering
1. S : select each vertex independently with probability p2. Process each v Є V \S as follows
1. If v is not adjacent to any sampled vertex. add all its edges.
2. If v is adjacent to some sampled vertex.
weights
v
x
S
V \S
![Page 56: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/56.jpg)
Algorithm for 3-spannerAlgorithm for 3-spannerPhase 1 : Phase 1 : ClusteringClustering
1. S : select each vertex independently with probability p2. Process each v Є V \S as follows
1. If v is not adjacent to any sampled vertex. add all its edges.
2. If v is adjacent to some sampled vertex.
weights
x
v
S
V \S
![Page 57: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/57.jpg)
Algorithm for 3-spannerAlgorithm for 3-spannerPhase 1 : Phase 1 : ClusteringClustering
G=(V,E) G=(V1,E1)Phase 1
Spanner (partial)
RemainingRed edges
Red edges
![Page 58: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/58.jpg)
Algorithm for 3-spannerAlgorithm for 3-spannerPhase 1 : Phase 1 : ClusteringClustering
G=(V,E) G=(V1,E1)
Every v Є V1 is clustered
v o
Phase 1
Spanner (partial)
RemainingRed edges
Red edges
![Page 59: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/59.jpg)
Algorithm for 3-spannerAlgorithm for 3-spannerPhase 1 : Phase 1 : ClusteringClustering
G=(V,E) G=(V1,E1)
Every v Є V1 is clustered Every red edge (w-v) Є E1 is ......
v o
Phase 1
Spanner (partial)
RemainingRed edges
Red edges
![Page 60: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/60.jpg)
Algorithm for 3-spannerAlgorithm for 3-spannerPhase 1 : Phase 1 : ClusteringClustering
G=(V,E) G=(V1,E1)
Every v Є V1 is clustered Every red edge (w-v) Є E1 is at-least as heavy as (v-o)
v o
Phase 1
Spanner (partial)
RemainingRed edges
Red edges
![Page 61: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/61.jpg)
Algorithm for 3-spannerAlgorithm for 3-spannerPhase 1 : Phase 1 : ClusteringClustering
G=(V,E) G=(V1,E1)
Every v Є V1 is clustered Every red edge (w-v) Є E1 is at-least as heavy as (v-o)
v o
Phase 1
Spanner (partial)
Observation I
Red edges RemainingRed edges
![Page 62: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/62.jpg)
Algorithm for 3-spannerAlgorithm for 3-spannerDifficult caseDifficult case : much more than : much more than n½ edges edges
Which n½ edges to select ?
![Page 63: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/63.jpg)
Algorithm for 3-spannerAlgorithm for 3-spannerDifficult caseDifficult case : much more than : much more than n½ edges edges
v
![Page 64: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/64.jpg)
Algorithm for 3-spannerAlgorithm for 3-spannerPhase 2 : Phase 2 : adding edges between vertices and adding edges between vertices and
clustersclusters
v
![Page 65: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/65.jpg)
Analysis of the algorithmAnalysis of the algorithm
Size of the spanner
Edges added during Phase 1 + Edges added during Phase 2 n/p + n2p = n3/2 , for p = 1/√n
Correctness ??
![Page 66: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/66.jpg)
Spanner has stretch 3Spanner has stretch 3PropertyProperty P3 holdsholds
x y
![Page 67: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/67.jpg)
Spanner has stretch 3Spanner has stretch 3PropertyProperty P3 holdsholds
x y
Both x and y are clustered
![Page 68: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/68.jpg)
Spanner has stretch 3Spanner has stretch 3PropertyProperty P3 holdsholds
x y
yx
![Page 69: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/69.jpg)
Spanner has stretch 3Spanner has stretch 3PropertyProperty P3 holdsholds
x y
yxObservation I
![Page 70: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/70.jpg)
Spanner has stretch 3Spanner has stretch 3PropertyProperty P3 holdsholds
x y
yx yxo
![Page 71: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/71.jpg)
Spanner has stretch 3Spanner has stretch 3PropertyProperty P3 holdsholds
x y
yx yxo
ß
α
z
![Page 72: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/72.jpg)
Spanner has stretch 3Spanner has stretch 3PropertyProperty P3 holdsholds
x y
yx yxo
ß
α
z
Observation I
![Page 73: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/73.jpg)
Algorithm for (Algorithm for (2k-12k-1)-spanner)-spanner
![Page 74: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/74.jpg)
Algorithm for (Algorithm for (2k-12k-1)-spanner)-spanner
n
n1-1/k
n1/k
V0
V1
Vk-1
#Vertices #Clusters
![Page 75: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/75.jpg)
Algorithm for (Algorithm for (2k-12k-1)-spanner)-spanner Invariant : At level i, we have graph G=(Vi,Ei)
Every vertex in Vi is clustered For every edge e Є Ei
i edges
e
![Page 76: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/76.jpg)
An openAn open problemproblem
Fully Dynamic algorithm for (2k-1)-spanner ?
![Page 77: A randomized linear time algorithm for graph spanners Surender Baswana Postdoctoral Researcher Max Planck Institute for Computer Science Saarbruecken,](https://reader035.fdocuments.us/reader035/viewer/2022062413/5a4d1b457f8b9ab0599a32c5/html5/thumbnails/77.jpg)
Thank you Thank you