Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger...
Transcript of 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
![Page 2: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/2.jpg)
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Distributed network Graph G of n nodes
2
4
5 1
3
![Page 3: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/3.jpg)
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Distributed network Graph G of n nodes
2
4
5 1
3
![Page 4: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/4.jpg)
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
![Page 5: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/5.jpg)
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
![Page 6: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/6.jpg)
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
?
![Page 7: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/7.jpg)
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
?
![Page 8: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/8.jpg)
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
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ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Distributed algorithms: a simple example
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ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Count the nodes!
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ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Count the nodes!
1. Compute BFS-Tree
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ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
0
Count the nodes!
1. Compute BFS-Tree
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ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
1
1
0
1
1
Count the nodes!
1. Compute BFS-Tree
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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
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ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Diameter of a network
• Distance between two nodes = Number of hops of shortest path
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ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Diameter of a network
• Distance between two nodes = Number of hops of shortest path
![Page 17: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/17.jpg)
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
![Page 18: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/18.jpg)
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?
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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
![Page 20: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/20.jpg)
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
?
![Page 21: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/21.jpg)
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Fundamental graph-problems
• Maximal Independent Set • Coloring • Matching
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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)
![Page 23: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/23.jpg)
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
![Page 24: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/24.jpg)
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
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ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Networks cannot compute their diameter in sublinear time!
![Page 26: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/26.jpg)
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
![Page 27: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/27.jpg)
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Networks cannot compute their diameter in sublinear time!
![Page 28: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/28.jpg)
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Networks cannot compute their diameter in sublinear time!
![Page 29: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/29.jpg)
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Networks cannot compute their diameter in sublinear time!
![Page 30: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/30.jpg)
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Networks cannot compute their diameter in sublinear time!
![Page 31: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/31.jpg)
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Networks cannot compute their diameter in sublinear time!
![Page 32: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/32.jpg)
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Networks cannot compute their diameter in sublinear time!
Base graph has diameter 3
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ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Networks cannot compute their diameter in sublinear time!
has diameter 3 2?
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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?
![Page 35: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/35.jpg)
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?
![Page 36: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/36.jpg)
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!
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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!
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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!
![Page 39: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/39.jpg)
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!
![Page 40: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/40.jpg)
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!
![Page 41: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/41.jpg)
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!
![Page 42: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/42.jpg)
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!
![Page 43: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/43.jpg)
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
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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!
![Page 45: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/45.jpg)
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
![Page 46: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/46.jpg)
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
![Page 47: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/47.jpg)
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!
![Page 48: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/48.jpg)
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
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ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Diameter Approximation
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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
![Page 51: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/51.jpg)
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
![Page 52: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/52.jpg)
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
![Page 53: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/53.jpg)
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
![Page 54: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/54.jpg)
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
![Page 55: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/55.jpg)
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
![Page 56: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/56.jpg)
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
![Page 57: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/57.jpg)
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Technique is general
![Page 58: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/58.jpg)
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
G
Technique is general
![Page 59: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/59.jpg)
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
G
Technique is general
![Page 60: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/60.jpg)
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
G
cut
Technique is general
![Page 61: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/61.jpg)
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
cut
Technique is general
![Page 62: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/62.jpg)
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
cut
Technique is general
![Page 63: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/63.jpg)
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
cut
f’(Ga,Gb)
![Page 64: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/64.jpg)
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Alice Bob
cut
f’(Ga,Gb)
![Page 65: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/65.jpg)
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Alice Bob
cut
f’(Ga,Gb)
![Page 66: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/66.jpg)
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Alice Bob
cut
f’(Ga,Gb)
![Page 67: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/67.jpg)
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Alice Bob
cut
Time(g) Time(f) ≥ ----------- |cut|
f’(Ga,Gb)
![Page 68: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/68.jpg)
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
![Page 69: Networks Cannot Compute Their Diameter in Sublinear Time SODA.pdfSilvio Frischknecht Roger Wattenhofer . Networks Cannot Compute Their Diameter in Sublinear Time ... Thanks for slide](https://reader033.fdocuments.us/reader033/viewer/2022042016/5e748bfd4b2d6b7f8e1399d4/html5/thumbnails/69.jpg)
ETH Zurich – Distributed Computing Group Stephan Holzer SODA 2012
Thanks!