Stephan Holzer

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

1

Stephan HolzerSilvio Frischknecht Roger Wattenhofer

Networks Cannot Compute Their Diameter in Sublinear Time

ETH Zurich – Distributed Computing Group

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Distributed network Graph G of n nodes

2

4

51

3

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Distributed network Graph G of n nodes

2

4

51

3

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Distributed network Graph G of n nodes

2

4

51

31 Local infor-

mation only

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Distributed network Graph G of n nodes

2

4

51

32

13

Local infor-mation only

Slide inspired by Danupon Nanongkai

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Distributed network Graph G of n nodes

2

4

51

32

13

Local infor-mation only

?

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Distributed network Graph G of n nodes

2

4

51

3

Limitedbandwidth

log n

log

nlog n log n

log n

log n

2

13

Local infor-mation only

?

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Distributed network Graph G of n nodes

2

4

51

3

Limitedbandwidth

log n

log

nlog n log n

log n

log n

2

13

Local infor-mation only

?Time complexity:

number of communication rounds

Synchronized

Internal computations

negligible

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Distributed algorithms:a simple example

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Count the nodes!

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Count the nodes!

1. Compute BFS-Tree

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

0

Count the nodes!

1. Compute BFS-Tree

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

1

1

0

1

1

Count the nodes!

1. Compute BFS-Tree

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Count the nodes!

1

1

0

21

2

12

2

2

Runtime: ?Diameter

1. Compute BFS-Tree

2. Count nodes in subtrees

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Diameter of a network

• Distance between two nodes = Number of hops of shortest path

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Diameter of a network

• Distance between two nodes = Number of hops of shortest path

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Diameter of a network

• Distance between two nodes = Number of hops of shortest path• Diameter of network = Maximum distance, between any two nodes

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Diameter of a network

• Distance between two nodes = Number of hops of shortest path• Diameter of network = Maximum distance, between any two nodes

Diameter of this network?

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Fundamental graph-problems

• Spanning Tree – Broadcasting, Aggregation, etc• Minimum Spanning Tree – Efficient

broadcasting, etc. • Shortest path – Routing, etc.• Steiner tree – Multicasting, etc. • Many other graph problems.

Thanks for slide to Danupon Nanongkai

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Fundamental graph-problems

• Spanning Tree – Broadcasting, Aggregation, etc• Minimum Spanning Tree – Efficient

broadcasting, etc. • Shortest path – Routing, etc.• Steiner tree – Multicasting, etc. • Many other graph problems.

• Global problems: Ω( D )2

13

Local infor-mation only

?

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Fundamental graph-problems

• Maximal Independent Set• Coloring• Matching

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Fundamental graph-problems

• Maximal Independent Set• Coloring• Matching

• Local problems: runtime independent of / smaller than D

e.g. O(log n)

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Fundamental problems

• Spanning Tree – Broadcasting, Aggregation, etc• Minimum Spanning Tree – Efficient

broadcasting, etc. • Shortest path – Routing, etc.• Steiner tree – Multicasting, etc. • Many other graph problems.

• Diameter appears frequently in distributed computing

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

• Spanning Tree – Broadcasting, Aggregation, etc• Minimum Spanning Tree – Efficient

broadcasting, etc. • Shortest path – Routing, etc.• Steiner tree – Multicasting, etc. • Many other graph problems.

• Diameter appears frequently in distributed computing

Fundamental problems 1. Formal definition?

Complexity of computing D? .

Known: Ω (D)

Ω(n)

≈Ω(1)

This talk:Even if D = 3

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Networks cannot compute their diameter in sublinear time!

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Diameter of a network

Diameter of this network?

• Distance between two nodes = Number of hops of shortest path• Diameter of network = Maximum distance, between any two nodes

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Networks cannot compute their diameter in sublinear time!

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Networks cannot compute their diameter in sublinear time!

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Networks cannot compute their diameter in sublinear time!

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Networks cannot compute their diameter in sublinear time!

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Networks cannot compute their diameter in sublinear time!

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Networks cannot compute their diameter in sublinear time!

Base graph has diameter 3

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Networks cannot compute their diameter in sublinear time!

has diameter 3 2?

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Networks cannot compute their diameter in sublinear time!

Pair of nodes not connected on both sides?

has diameter 3 2?

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Networks cannot compute their diameter in sublinear time!

Pair of nodes not connected on both sides?

has diameter 3 2?

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Pair of nodes not connected on both sides?

Now: slightly more formal

Networks cannot compute their diameter in sublinear time!

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Pair of nodes not connected on both sides?

Label potential edges

Networks cannot compute their diameter in sublinear time!

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Pair of nodes not connected on both sides?

Label potential edges

1 1

Networks cannot compute their diameter in sublinear time!

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Pair of nodes not connected on both sides?

Label potential edges

1 12 2

Networks cannot compute their diameter in sublinear time!

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Pair of nodes not connected on both sides?

Label potential edges

1 12 23 3

Networks cannot compute their diameter in sublinear time!

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Pair of nodes not connected on both sides?

Label potential edges

1 12 23 3

4 4

Networks cannot compute their diameter in sublinear time!

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Pair of nodes not connected on both sides?

1 12 23 3

4 4

Networks cannot compute their diameter in sublinear time!

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Pair of nodes not connected on both sides?

24

2 31 1

34

Networks cannot compute their diameter in sublinear time!

Given graph

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Pair of nodes not connected on both sides?

Θ(n) edges

24

2 31 1

344

2 34

A B

Networks cannot compute their diameter in sublinear time!

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Pair of nodes not connected on both sides?

Θ(n) edges

24

2 31 1

34

2 34

Same as “A and B not disjoint?”

Networks cannot compute their diameter in sublinear time!

A

B

4

23

4

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Pair of nodes not connected on both sides?

Θ(n) edges

24

2 31 1

34

2 34

Same as “A and B not disjoint?”

Networks cannot compute their diameter in sublinear time!

A [n⊆ 2]

B [n⊆ 2]

4

23

4

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

A [n⊆ 2]

B [n⊆ 2]

“A and B not disjoint?”

4

23

4

Networks cannot compute their diameter in sublinear time!

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Pair of nodes not connected on both sides?

Θ(n) edges

A [n⊆ 2]

B [n⊆ 2]

“A and B not disjoint?”Communication Complexity randomized: Ω(n 2) bits

4

23

4Ω(n) time

Networks cannot compute their diameter in sublinear time!

Same as

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Diameter Approximation

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

3/2-ε approximating the diameter takes W(n1/2)

Extend2 vs. 3

D = 9 base graph

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

3/2-ε approximating the diameter takes W(n1/2)

Extend2 vs. 3

D = 9 base graph

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

3/2-ε approximating the diameter takes W(n1/2)

Extend2 vs. 3

9

D = 9 base graph

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

3/2-ε approximating the diameter takes W(n1/2)

Extend2 vs. 3

D = 9 base graph

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

3/2-ε approximating the diameter takes W(n1/2)

Extend2 vs. 3

D = 9 base graph

5

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

3/2-ε approximating the diameter takes W(n1/2)

Extend2 vs. 3

D = 9 base graphD = 7 good graph-------------Factor: 3/27

5

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

3/2-ε approximating the diameter takes W(n1/2)

Extend2 vs. 3

D = 9 base graphD = 7 good graph-------------Factor: 3/2 -ε7

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Technique is general

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

G

Technique is general

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

G

Technique is general

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

G

cut

Technique is general

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

cut

Technique is general

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

cut

Technique is general

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

cut

f’(Ga,Gb)

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Alice Bob

cut

f’(Ga,Gb)

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Alice Bob

cut

f’(Ga,Gb)

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Alice Bob

cut

f’(Ga,Gb)

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Alice Bob

cut

Time(g) Time(f) ≥ -----------

|cut|

f’(Ga,Gb)

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Summary

Diameter Ω(n) 73/2-eps approximation takes Ω(n1/2)

cutgeneral technique

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Thanks!