Graph spanners : static, dynamic and fault tolerant

1
Graph spanners static, dynamic and fault
tolerant

• Surender Baswana
• Department of CSE
• IIT Kanpur

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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

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
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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

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• Aim
• The sparsest spanner of a weighted graph with
  stretch t.

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Organization of the talk

• Optimal size of a t-spanner
• Spanners
• Static Spanner
• Dynamic spanners
• Fault Tolerant Spanners
• A simple and linear time algorithm for static
  spanner (how problem itself guides to its
  solution)
• A simple algorithm for fault-tolerant spanner

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Optimal size of a t-spanner
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Optimal size of a t-spanner
v
u
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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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Static spanners
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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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Dynamic spanners
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A dynamic spanner

• an initial graph G(V,E) followed by an online
  sequence of updates
• i,i,d,i,d,i,i,d,d,i,i,d, ....

Deletion of an edge
Insertion of an edge
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A dynamic spanner

• an initial graph G(V,E) followed by an online
  sequence of updates
• i,i,d,i,d,i,i,d,d,i,i,d, ....
• Aim
• To maintain a t-spanner in online manner
  efficiently

Deletion of an edge
Insertion of an edge
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Dynamic algorithms for graph spanners
33
Fault Tolerant spanners
34
Why do we need a fault tolerant spanner ?

• For networks where
• Failures occur rarely
• There is simultaneous repair going on
• At any time, there are at most f edges which may
  be dead.

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What is a fault tolerant spanner ?

• A subgraph Es is called f-fault tolerant spanner
  if for any set F
• of f edges from E,
• Es -F is a (2k-1)-spanner
  of E-F

36
Results on size of fault tolerant spanners
37
A simple and linear time algorithm for static
spanner
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Earlier algorithms for graph spanners
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Can we compute a (2k-1)-spanner in O(m) time ?
Question
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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

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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
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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External memory algorithms for (2k-1)-spanner

• Time complexity Integer sorting

46
Distributed Algorithms for (2k-1)-spanner

• Number of Rounds O(k) ,
• Communication complexity O(km) (linear)

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Streaming Algorithm for (2k-1)-spanner

• For Unweighted graphs
• Number of passes 1
• Processing time per edge O(1)

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Algorithm for 3-spanner
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Aims

• Given a graph G(V,E)
• The number of edges O(n3/2 )
• Stretch 3
• Computation time O(m)

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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
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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
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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
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Algorithm for 3-spannerPhase 2 adding edges
between vertices and clusters
v
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Analysis of the algorithm

• Size of the spanner
• Edges added during Phase 1 Edges added during
  Phase 2
• Correctness ??

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Analysis of the algorithm

• Size of the spanner
• Edges added during Phase 1 Edges added during
  Phase 2
• Correctness ??

How many coin tosses to get a HEAD
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Analysis of the algorithm

• Size of the spanner
• Edges added during Phase 1 Edges added during
  Phase 2
• n/p
• Correctness ??

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Analysis of the algorithm

• Size of the spanner
• Edges added during Phase 1 Edges added during
  Phase 2
• n/p n2p
• Correctness ??

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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
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Algorithm for fault tolerant spanner
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Algorithm for f-edge fault tolerant spanners

• Given graph (V,E)
• Es F
• For i1 to f1 do
• Ei spanner(E-Es ,2k-1)
• Es Es U Ei
• return Es

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Algorithm for f-edge fault tolerant spanners

• Given graph G(V,E)
• Es F
• For i1 to f1 do
• Ei spanner(E-Es ,2k-1)
• Es Es U Ei
• return Es
• Observation Ei and Ej are edge disjoint for
  iltgtj.

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Algorithm for f-edge fault tolerant spanners

• Let F be any set of f edges.
• Let (u,v) ? E-F
• Aim
• There has to be a path of length at most 2k-1
  between u and v in E-Es

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Algorithm for f-edge fault tolerant spanners

• Let F be any set of i edges.
• Let (u,v) ? E-F
• Aim
• There is a path of length 2k-1 between u and v in
  E-Es
• (follows from observation)

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Summary

• Simple and efficient algorithms exist for graph
  spanners
• Static spanners
• Dynamic spanners
• Fault tolerant spanners
• Open problems (exist for additive spanners)

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Thank you