A randomized linear time algorithm for graph spanners

1
A randomized linear time algorithm forgraph
spanners

• Surender Baswana
• Postdoctoral Researcher
• Max Planck Institute for Computer Science
• Saarbruecken, Germany

2
Graph Spanners

• Definition
• Given a graph G(V,E), a spanner is a sub-graph
  G(V,Es) which has the following two crucial
  properties

3
Graph Spanners

• Definition
• Given a graph G(V,E), a spanner is a sub-graph
  G(V,Es) which has the following two crucial
  properties
• sparse

4
Graph Spanners

• Definition
• Given a graph G(V,E), a spanner is a sub-graph
  G(V,Es) which has the following two crucial
  properties
• sparse
• preserves approximate distances pair-wise.

5
Graph Spanners

• Definition
• Given a graph G(V,E), a spanner is a sub-graph
  G(V,Es) which has the following two crucial
  properties
• sparse
• preserves approximate distances pair-wise.
• d(u,v) ds(u,v) t d(u,v) for some
  constant t 1

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Graph Spanners

• Definition
• Given a graph G(V,E), a spanner is a sub-graph
  G(V,Es) which has the following two crucial
  properties
• sparse
• preserves approximate distances pair-wise.
• d(u,v) ds(u,v) t d(u,v) for some
  constant t 1

t stretch of the spanner
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Communication network Motivation for spanners
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Communication network Motivation for spanners

• Each edge has
• cost
• weight (length)

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Communication network Motivation for spanners

• Minimizing the total cost sparseness is
  desirable

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Communication network Motivation for spanners

• Minimizing the total cost sparseness is
  desirable

u
v
11
Communication network Motivation for spanners

• Minimizing the pair-wise distances small
  stretch is desirable

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Communication network Motivation for spanners

• Minimizing the pair-wise distances small
  stretch is desirable

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Graph spanners

• A trade off between sparseness and stretch

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Graph spanners

• A trade off between sparseness and stretch
• Sparse
• d(u,v) ds(u,v) t d(u,v)

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Graph spanners

• A trade off between sparseness and stretch
• Sparse
• d(u,v) ds(u,v) t d(u,v)
• t-Spanner

16

• Aim
• To compute the sparsest spanner of a weighted
  graph with stretch t.

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Applications of Graph Spanner

• Distributed Computing
• Design of Synchronizers
• Compact routing tables
• Computational Biology
• Reconstruction of Phylogenetic trees
• All-pairs Approximate Shortest Paths

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Organization of the talk

• Optimal size of a t-spanner
• Earlier algorithms
• A algorithm

New
19
Optimal size of a t-spanner
20
Optimal size of a t-spanner
v
u
21
Optimal size of a t-spanner
v
u
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Optimal size of a t-spanner
v
u
???
Length of Smallest cycle t
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Optimal size of a t-spanner
v
u
stretch t-1
Length of Smallest cycle t
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Optimal size of a t-spanner
v
u
stretch t-1
Length of Smallest cycle t
How dense can a graph with shortest cycle length
t be ?
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Optimal size of a t-spanner
v
u
stretch t-1
Length of Smallest cycle t
Girth Conjecture Erdös1960, Bondy
Simonovits 1974, Bollobas 1978
There are graph with shortest cycle length gt 2k
and O(n11/k) edges
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Optimal size of a t-spanner

• Let k be any positive integer
• There are graphs whose (2k-1)-spanner (a
  2k-spanner) must have O(n11/k) edges

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Optimal size of a t-spanner

• Let k be any positive integer
• There are graphs whose (2k-1)-spanner (a
  2k-spanner) must have O(n11/k) edges

4- spanner and 3-spanner O(n3/2) 6-spanner and
5-spanner O(n5/4) 8-spanner and 7-spanner
O(n7/6)
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Aim of an Algorithmist

• To design an algorithm A such that

A
G(V, Es)
G(V, E)
(2k-1)-Spanner
Input Graph
ES O (minimum (m , n11/k))
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Earlier algorithms for (2k-1)-spanner
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Earlier algorithms for graph spanners
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Can we Compute (2k-1)-spanners in linear time ?
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A algorithm Computing a (2k-1)-spanner
in expected O(m) time
New
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Local approach

• Let G(V,ES) be a spanner of G(V,E)

Edge in Spanner
Edge not in Spanner
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Local approach

• Let G(V,ES) be a spanner of G(V,E)

Edge in Spanner
Edge not in Spanner
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Local approach

• Let G(V,ES) be a spanner of G(V,E)

Edge in Spanner
2
Edge not in Spanner
w
w
1
t-1
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

36
Local approach

• Let G(V,ES) be a spanner of G(V,E)

Edge in Spanner
2
Edge not in Spanner
w
w
1
t-1
w
w
w
u
v

• 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

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Local approach

• Let G(V,ES) be a spanner of G(V,E)

Edge in Spanner
2
Edge not in Spanner
w
w
1
t-1
t-spanner
w
w
w
u
v

• 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

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New Algorithms for (2k-1)-spanner

• An External-memory algorithm for (2k-1)-spanner
  
• Time complexity Integer sorting

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New Algorithms for (2k-1)-spanner

• A distributed algorithm for (2k-1)-spanner
• Number of Rounds O(1) ,
• Communication complexity O(m) (linear)

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New Algorithms for (2k-1)-spanner

• A streaming algorithm for (2k-1)-spanner
  
• Number of passes O(1)
• Processing time per edge O(1)

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Algorithm for 3-spanner
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Algorithm for 3-spannerEasy case fewer than n½
edges
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Algorithm for 3-spannerDifficult case much
more than n½ edges
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Algorithm for 3-spannerDifficult case much
more than n½ edges
Which n½ edges to select ?
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Algorithm for 3-spanner
Initially all edges are Red

• Phase 1 Clustering
• Phase 2 Adding edges between vertices and
  clusters

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Algorithm for 3-spanner
Initially all edges are Red

• Phase 1 Clustering
• Phase 2 Adding edges between vertices and
  clusters

center
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Algorithm for 3-spannerPhase 1 Clustering

• S select each vertex independently with
  probability p.

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Algorithm for 3-spannerPhase 1 Clustering

• S select each vertex independently with
  probability p.
• Process each v ? V \S as follows

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Algorithm for 3-spannerPhase 1 Clustering

• S select each vertex independently with
  probability p.
• Process each v ? V \S as follows
• If v is not adjacent to any sampled vertex

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Algorithm for 3-spannerPhase 1 Clustering

• S select each vertex independently with
  probability p .
• Process each v ? V \S as follows
• If v is not adjacent to any sampled vertex.

v
S
V \S
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Algorithm for 3-spannerPhase 1 Clustering

• S select each vertex independently with
  probability p .
• Process each v ? V \S as follows
• If v is not adjacent to any sampled vertex. add
  all its edges.

v
S
V \S
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Algorithm for 3-spannerPhase 1 Clustering

• S select each vertex independently with
  probability p.
• Process each v ? V \S as follows
• If v is not adjacent to any sampled vertex. add
  all its edges.
• If v is adjacent to some sampled vertex

S
V \S
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Algorithm for 3-spannerPhase 1 Clustering

• S select each vertex independently with
  probability p .
• Process each v ? V \S as follows
• If v is not adjacent to any sampled vertex. add
  all its edges.
• If v is adjacent to some sampled vertex.

v
S
V \S
x
weights
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Algorithm for 3-spannerPhase 1 Clustering

• S select each vertex independently with
  probability p.
• Process each v ? V \S as follows
• If v is not adjacent to any sampled vertex. add
  all its edges.
• If v is adjacent to some sampled vertex.

v
S
V \S
x
weights
55
Algorithm for 3-spannerPhase 1 Clustering

• S select each vertex independently with
  probability p
• Process each v ? V \S as follows
• If v is not adjacent to any sampled vertex. add
  all its edges.
• If v is adjacent to some sampled vertex.

v
S
V \S
x
weights
56
Algorithm for 3-spannerPhase 1 Clustering

• S select each vertex independently with
  probability p
• Process each v ? V \S as follows
• If v is not adjacent to any sampled vertex. add
  all its edges.
• If v is adjacent to some sampled vertex.

v
S
V \S
x
weights
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Algorithm for 3-spannerPhase 1 Clustering

• G(V,E) G(V1,E1)

Remaining Red edges
Red edges
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Algorithm for 3-spannerPhase 1 Clustering

• G(V,E) G(V1,E1)
• Every v ? V1 is clustered

Remaining Red edges
Red edges
v
o
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Algorithm for 3-spannerPhase 1 Clustering

• G(V,E) G(V1,E1)
• Every v ? V1 is clustered
• Every red edge (w-v) ? E1 is ......

Remaining Red edges
Red edges
v
o
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Algorithm for 3-spannerPhase 1 Clustering

• G(V,E) G(V1,E1)
• Every v ? V1 is clustered
• Every red edge (w-v) ? E1 is at-least as heavy as
  (v-o)

Remaining Red edges
Red edges
v
o
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Algorithm for 3-spannerPhase 1 Clustering

• G(V,E) G(V1,E1)
• Every v ? V1 is clustered
• Every red edge (w-v) ? E1 is at-least as heavy as
  (v-o)

Red edges
Remaining Red edges
v
o
Observation I
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Algorithm for 3-spannerDifficult case much
more than n½ edges
Which n½ edges to select ?
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Algorithm for 3-spannerDifficult case much
more than n½ edges
v
64
Algorithm for 3-spannerPhase 2 adding edges
between vertices and clusters
v
65
Analysis 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/vn
• Correctness ??

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Spanner has stretch 3Property P3 holds
x
y
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Spanner has stretch 3Property P3 holds
x
y
Both x and y are clustered
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Spanner has stretch 3Property P3 holds
x
y
y
x
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Spanner has stretch 3Property P3 holds
x
y
y
x
Observation I
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Spanner has stretch 3Property P3 holds
x
y
o
y
x
y
x
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Spanner has stretch 3Property P3 holds
x
y
z
a
o
y
x
ß
y
x
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Spanner has stretch 3Property P3 holds
x
y
z
a
o
y
x
ß
y
x
Observation I
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Algorithm for (2k-1)-spanner
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Algorithm for (2k-1)-spanner
Vertices
Clusters
n1/k
n1-1/k
n
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Algorithm for (2k-1)-spanner

• Invariant At level i, we have graph G(Vi,Ei)
• Every vertex in Vi is clustered
• For every edge e ? Ei

e
76
An open problem
Fully Dynamic algorithm for (2k-1)-spanner ?
77
Thank you