a randomized linear time algorithm for graph spanners surender baswana postdoctoral researcher max...
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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
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
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
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.
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
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
Communication network :Communication network :Motivation for spannersMotivation for spanners
Communication network :Communication network :Motivation for spannersMotivation for spanners
Each edge has• cost• weight (length)
Communication network :Communication network :Motivation for spannersMotivation for spanners
Minimizing the total cost : sparseness is desirable
Communication network :Communication network :Motivation for spannersMotivation for spanners
Minimizing the total cost : sparseness is desirable
u
v
Communication network :Communication network :Motivation for spannersMotivation for spanners
Minimizing the pair-wise distances : small stretch is desirable
Communication network :Communication network :Motivation for spannersMotivation for spanners
Minimizing the pair-wise distances : small stretch is desirable
Graph spannersGraph spanners A trade off between sparseness and stretch
Graph spannersGraph spanners A trade off between sparseness and stretch
Sparse
δ(u,v) ≤ δs(u,v) ≤ t δ(u,v)
Graph spannersGraph spanners A trade off between sparseness and stretch
Sparse
δ(u,v) ≤ δs(u,v) ≤ t δ(u,v)
t-Spanner
Aim :
To compute the sparsest spanner of a weighted graph with stretch t.
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
Organization of the talkOrganization of the talk
Optimal size of a t-spanner
Earlier algorithms
A algorithm
Optimal size of a t-spanner
Optimal size of aOptimal size of a tt--spannerspanner
u v
Optimal size of aOptimal size of a tt--spannerspanner
u v
Optimal size of aOptimal size of a tt--spannerspanner
u v
???Length of Smallest cycle= t
Optimal size of aOptimal size of a tt--spannerspanner
u v
stretch ≥ t-1Length of Smallest cycle= t
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 ?
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
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
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)
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
Earlier algorithms for (2k-1)-spanner
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
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 ?
A algorithm
Computing a Computing a (2k-1)-(2k-1)-spanner in expectedspanner in expected O(m)O(m) timetime
Local approachLocal approachLet G=(V,ES) be a spanner of G=(V,E)
Edge in Spanner
Edge not in Spanner
Local approachLocal approachLet G=(V,ES) be a spanner of G=(V,E)
Edge in Spanner
Edge not in Spanner
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
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
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
NewNew Algorithms for ( Algorithms for (2k-12k-1)-)-spannerspanner
An External-memory algorithm for (2k-1)-spanner
Time complexity : Integer sorting
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)
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)
Algorithm for 3-spanner
Algorithm for 3-spannerAlgorithm for 3-spannerEasy caseEasy case : fewer than : fewer than n½ edges edges
Algorithm for 3-spannerAlgorithm for 3-spannerDifficult caseDifficult case : much more than : much more than n½ edges edges
Algorithm for 3-spannerAlgorithm for 3-spannerDifficult caseDifficult case : much more than : much more than n½ edges edges
Which n½ edges to select ?
Algorithm for 3-spannerAlgorithm for 3-spanner Phase 1 : Clustering
Phase 2 : Adding edges between vertices and clusters
Initially all edges are Red
Algorithm for 3-spannerAlgorithm for 3-spanner Phase 1 : Clustering
Phase 2 : Adding edges between vertices and clusters
center
Initially all edges are Red
Algorithm for 3-spannerAlgorithm for 3-spannerPhase 1 : Phase 1 : ClusteringClustering
1. S : select each vertex independently with probability p.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Algorithm for 3-spannerAlgorithm for 3-spannerDifficult caseDifficult case : much more than : much more than n½ edges edges
Which n½ edges to select ?
Algorithm for 3-spannerAlgorithm for 3-spannerDifficult caseDifficult case : much more than : much more than n½ edges edges
v
Algorithm for 3-spannerAlgorithm for 3-spannerPhase 2 : Phase 2 : adding edges between vertices and adding edges between vertices and
clustersclusters
v
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 ??
Spanner has stretch 3Spanner has stretch 3PropertyProperty P3 holdsholds
x y
Spanner has stretch 3Spanner has stretch 3PropertyProperty P3 holdsholds
x y
Both x and y are clustered
Spanner has stretch 3Spanner has stretch 3PropertyProperty P3 holdsholds
x y
yx
Spanner has stretch 3Spanner has stretch 3PropertyProperty P3 holdsholds
x y
yxObservation I
Spanner has stretch 3Spanner has stretch 3PropertyProperty P3 holdsholds
x y
yx yxo
Spanner has stretch 3Spanner has stretch 3PropertyProperty P3 holdsholds
x y
yx yxo
ß
α
z
Spanner has stretch 3Spanner has stretch 3PropertyProperty P3 holdsholds
x y
yx yxo
ß
α
z
Observation I
Algorithm for (Algorithm for (2k-12k-1)-spanner)-spanner
Algorithm for (Algorithm for (2k-12k-1)-spanner)-spanner
n
n1-1/k
n1/k
V0
V1
Vk-1
#Vertices #Clusters
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
An openAn open problemproblem
Fully Dynamic algorithm for (2k-1)-spanner ?
Thank you Thank you
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