networks cannot compute their diameter in sublinear time soda.pdfsilvio frischknecht roger...
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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
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Distributed network Graph G of n nodes
2
4
5 1
3
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Distributed network Graph G of n nodes
2
4
5 1
3
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Distributed network Graph G of n nodes
2
4
5 1
3 1 Local infor-
mation only
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Distributed network Graph G of n nodes
2
4
5 1
3 2
1 3
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
5 1
3 2
1 3
Local infor- mation only
?
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Distributed network Graph G of n nodes
2
4
5 1
3
Limited bandwidth
2
1 3
Local infor- mation only
?
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Distributed network Graph G of n nodes
2
4
5 1
3
Limited bandwidth
2
1 3
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
2 1
2
1 2
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
1 3
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 1 2 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 1 2 2 3 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 1 2 2 3 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 1 2 2 3 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?
2 4
2 3 1 1
3 4
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
2 4
2 3 1 1
3 4 4
2 3 4
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
2 4
2 3 1 1
3 4
2 3 4
Same as “A and B not disjoint?”
Networks cannot compute their diameter in sublinear time!
A
B
4
2 3
4
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Pair of nodes not connected on both sides?
Θ(n) edges
2 4
2 3 1 1
3 4
2 3 4
Same as “A and B not disjoint?”
Networks cannot compute their diameter in sublinear time!
A ⊆ [n2]
B ⊆ [n2]
4
2 3
4
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
A ⊆ [n2]
B ⊆ [n2]
“A and B not disjoint?”
4
2 3
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 ⊆ [n2]
B ⊆ [n2]
“A and B not disjoint?” Communication Complexity randomized: Ω(n 2) bits
4
2 3
4 Ω(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 Ω(n1/2)
Extend 2 vs. 3
D = 9 base graph
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
3/2-ε approximating the diameter takes Ω(n1/2)
Extend 2 vs. 3
D = 9 base graph
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
3/2-ε approximating the diameter takes Ω(n1/2)
Extend 2 vs. 3
9 D = 9 base graph
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
3/2-ε approximating the diameter takes Ω(n1/2)
Extend 2 vs. 3
D = 9 base graph
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
3/2-ε approximating the diameter takes Ω(n1/2)
Extend 2 vs. 3
D = 9 base graph
5
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
3/2-ε approximating the diameter takes Ω(n1/2)
Extend 2 vs. 3
D = 9 base graph D = 7 good graph ------------- Factor: 3/2 7
5
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
3/2-ε approximating the diameter takes Ω(n1/2)
Extend 2 vs. 3
D = 9 base graph D = 7 good graph ------------- Factor: 3/2 -ε 7
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) 7 3/2-eps approximation takes Ω(n1/2)
cut general technique
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Thanks!
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