Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide

ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012

Stephan Holzer Silvio Frischknecht Roger Wattenhofer

Networks Cannot Compute Their Diameter in Sublinear Time

ETH Zurich – Distributed Computing Group


Distributed network Graph G of n nodes

2

4

5 1

3



2

4

5 1

3



2

4

5 1

3 1 Local infor-

mation only



2

4

5 1

3 2

1 3

Local infor- mation only

Slide inspired by Danupon Nanongkai



2

4

5 1

3 2

1 3


?



2

4

5 1

3

Limited bandwidth

2

1 3


?



2

4

5 1

3

Limited bandwidth

2

1 3


? Time complexity: number of

communication rounds

Synchronized

Internal computations

negligible


Distributed algorithms: a simple example


Count the nodes!


Count the nodes!

1. Compute BFS-Tree


0

Count the nodes!

1. Compute BFS-Tree


1

1

0

1

1

Count the nodes!

1. Compute BFS-Tree


Count the nodes!

1

1

0

2 1

2

1 2

2

2 Runtime: ? Diameter

1. Compute BFS-Tree 2. Count nodes in subtrees


Diameter of a network

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



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



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




Diameter of this network?


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





• Global problems: Ω( D )

2

1 3


?



• Maximal Independent Set • Coloring • Matching



• Maximal Independent Set • Coloring • Matching

• Local problems: runtime independent of / smaller than D

e.g. O(log n)


Fundamental problems



• Diameter appears frequently in distributed

computing




• 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


Networks cannot compute their diameter in sublinear time!



Diameter of this network?














Base graph has diameter 3



has diameter 3 2?



Pair of nodes not connected on both sides?

has diameter 3 2?




has diameter 3 2?



Now: slightly more formal



Pair of nodes not connected on both sides? Label potential edges




1 1




1 1 2 2




1 1 2 2 3 3




1 1 2 2 3 3

4 4




1 1 2 2 3 3

4 4




2 4

2 3 1 1

3 4


Given graph



Θ(n) edges

2 4

2 3 1 1

3 4 4

2 3 4

A B




Θ(n) edges

2 4

2 3 1 1

3 4

2 3 4

Same as “A and B not disjoint?”


A

B

4

2 3

4



Θ(n) edges

2 4

2 3 1 1

3 4

2 3 4

Same as “A and B not disjoint?”


A ⊆ [n2]

B ⊆ [n2]

4

2 3

4


A ⊆ [n2]

B ⊆ [n2]

“A and B not disjoint?”

4

2 3

4




Θ(n) edges

A ⊆ [n2]

B ⊆ [n2]

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

4

2 3

4 Ω(n) time


Same as


Diameter Approximation


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

Extend 2 vs. 3

D = 9 base graph



Extend 2 vs. 3

D = 9 base graph



Extend 2 vs. 3

9 D = 9 base graph



Extend 2 vs. 3

D = 9 base graph



Extend 2 vs. 3

D = 9 base graph

5



Extend 2 vs. 3

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

5



Extend 2 vs. 3

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


Technique is general


G



G



G

cut



cut



cut



cut

f’(Ga,Gb)


Alice Bob

cut

f’(Ga,Gb)


Alice Bob

cut

f’(Ga,Gb)


Alice Bob

cut

f’(Ga,Gb)


Alice Bob

cut

Time(g) Time(f) ≥ ----------- |cut|

f’(Ga,Gb)


Summary

Diameter Ω(n) 7 3/2-eps approximation takes Ω(n1/2)

cut general technique


Thanks!

Documents

Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide