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Networks Cannot Compute Their Diameter in Sublinear Time. Stephan Holzer. Silvio Frischknecht Roger Wattenhofer. ETH Zurich – Distributed Computing Group . Distributed network G raph G of n nodes. 2. 1. 5. 3. 4. - PowerPoint PPT Presentation
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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!