distributed slicing in dynamic systems a. fernández, v. gramoli, e. jiménez, a-m. kermarrec, m....
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![Page 1: Distributed Slicing in Dynamic Systems A. Fernández, V. Gramoli, E. Jiménez, A-M. Kermarrec, M. Raynal](https://reader036.vdocument.in/reader036/viewer/2022062714/56649d545503460f94a307e2/html5/thumbnails/1.jpg)
Distributed Slicing in Dynamic Systems
A. Fernández, V. Gramoli, E. Jiménez, A-M. Kermarrec, M. Raynal
![Page 2: Distributed Slicing in Dynamic Systems A. Fernández, V. Gramoli, E. Jiménez, A-M. Kermarrec, M. Raynal](https://reader036.vdocument.in/reader036/viewer/2022062714/56649d545503460f94a307e2/html5/thumbnails/2.jpg)
ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Context
Distributed System
Interconnected set of nodes
Heterogeneous environment
Resources are heterogeneously spread• Bandwidth
• Processing Power
• Storage Space
• Uptime
• …
Large-scale dynamic environment
Nodes leave and join at any time
Large amount of nodes
No global information
1,50,3
1
10098
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ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Problem
What means rich/poor in this context?
![Page 4: Distributed Slicing in Dynamic Systems A. Fernández, V. Gramoli, E. Jiménez, A-M. Kermarrec, M. Raynal](https://reader036.vdocument.in/reader036/viewer/2022062714/56649d545503460f94a307e2/html5/thumbnails/4.jpg)
ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Problem
What means rich/poor in this context?
20
Am I rich or poor?
![Page 5: Distributed Slicing in Dynamic Systems A. Fernández, V. Gramoli, E. Jiménez, A-M. Kermarrec, M. Raynal](https://reader036.vdocument.in/reader036/viewer/2022062714/56649d545503460f94a307e2/html5/thumbnails/5.jpg)
ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Problem
What means rich/poor in this context?
202
6
15
rich?
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ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Problem
What means rich/poor in this context?
20300
150
815220
poor?
![Page 7: Distributed Slicing in Dynamic Systems A. Fernández, V. Gramoli, E. Jiménez, A-M. Kermarrec, M. Raynal](https://reader036.vdocument.in/reader036/viewer/2022062714/56649d545503460f94a307e2/html5/thumbnails/7.jpg)
ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Applications
Remedy the Gnutella problem in Peer-to-peer Gnutella performance was limited by poor nodes
Kazaa/Skype sollicitate rich nodes rather than poor nodes
Allocating resources/nodes to services Streaming service needs nodes with highest bandwidth Non-critical service can run on unstable nodes File-sharing service requires nodes with many files …
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ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Objectives
Classifying nodes into categories, slices Based on individual characteristics: attributes A slice corresponds to a portion of the system
Typically, answering the question:
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ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Objectives
HOW RICH AM I COMPARED TO OTHERS?
Classifying nodes into categories, slices Based on individual characteristics: attributes A slice corresponds to a portion of the system
Typically, answering the question:
![Page 10: Distributed Slicing in Dynamic Systems A. Fernández, V. Gramoli, E. Jiménez, A-M. Kermarrec, M. Raynal](https://reader036.vdocument.in/reader036/viewer/2022062714/56649d545503460f94a307e2/html5/thumbnails/10.jpg)
ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Classifying the system nodes
![Page 11: Distributed Slicing in Dynamic Systems A. Fernández, V. Gramoli, E. Jiménez, A-M. Kermarrec, M. Raynal](https://reader036.vdocument.in/reader036/viewer/2022062714/56649d545503460f94a307e2/html5/thumbnails/11.jpg)
ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Classifying the system nodes
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…using their attribute values (assume a single attribute for simplicity reason)
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ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Classifying the system nodes
68
70
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0 100
Attribute values ai
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ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Classifying the system nodes
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7565
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0 100
0 1
Attribute values ai
NormalizedIndices pi
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ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
#4#3#2#1
Classifying the system nodes
68
70
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7262
7565
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0 100
0 1
NormalizedIndices pi
0 1Slices si
Attribute values ai
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ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
ModelDynamic system System of n nodes Nodes join and leave the system at any time Nodes may crash too
Each node i knows its attribute value ai, its position estimate pi’, the slices (ex: 10 equally sized slices, each containing
10% of the nodes), a communication view Vi:
• constant number of neighbors j,• their position estimate pj’, their attribute aj, (and their age)
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ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Underlying Gossip-based Overlay
Each node i periodically: Exchanges information with its neighbors (nodes of its
view)• Views, values…
Computes its new view and its new state• View, value…
Dynamic overlay: Failed nodes are naturally removed from views Joining nodes are naturally added into views
Scalable overlay: Limited amount of information stored Limited amount of information exchanged
![Page 17: Distributed Slicing in Dynamic Systems A. Fernández, V. Gramoli, E. Jiménez, A-M. Kermarrec, M. Raynal](https://reader036.vdocument.in/reader036/viewer/2022062714/56649d545503460f94a307e2/html5/thumbnails/17.jpg)
ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Jelasity, Kermarrec 2006 (JK)
Each node i sets pi’ as a uniform random value
Each node i looks for a misplaced neighbor j A neighbor j is misplaced with i iff (ai – aj)(pi’ – pj’) < 0
Then, nodes i and j exchange pi’ and pj’
The sequence of position estimates matches the sequence of attribute values
For any node couple i and j, ai<aj <=> pi’<pj’
Convergence to this ordering is exponentially fast
(in the number of execution)
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ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Algorithm 1: Ordering
Similar to JK
Uses Local Disorder Measure LDM Let lpj’ (resp. lpj) be the normalized index of j random value
(resp. attribute value) among all j of Vi i
LDM(i) = ∑ j in Vi i (lpj’ – lpj)²
ProtocolDo periodically {
Update view Vi using an underlying protocol.Choose the neighbor j that minimizes the LDM(i).Exchange random values pi’ and pj’ Update the random value pi’ w/ pj’ if necessary.Update slice assignment si’ := s : pi’ in s.
}
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ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Algorithm 1: Ordering
Similar to JK
Uses Local Disorder Measure LDM Let lpj’ (resp. lpj) be the normalized index of j random value (resp.
attribute value) among all j of Vi i
LDM(i) = ∑ j in Vi i (lpj’ – lpj)²
ProtocolDo periodically {
Update view Vi using an underlying protocol.Choose the neighbor j that minimizes the LDM(i).Exchange random values pi’ and pj’. Update the random value pi’ w/ pj’ if necessary.Update slice assignment si’ := s : pi’ in s.
}
Difference with JK
![Page 20: Distributed Slicing in Dynamic Systems A. Fernández, V. Gramoli, E. Jiménez, A-M. Kermarrec, M. Raynal](https://reader036.vdocument.in/reader036/viewer/2022062714/56649d545503460f94a307e2/html5/thumbnails/20.jpg)
ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Algorithm 1: Ordering
68
70
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7565
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4/11 9/11
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ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Algorithm 1: Ordering
68
70
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7565
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9/11
7/11
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ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Algorithm 1: Ordering
68
70
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7262
7565
20
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5989
27
![Page 23: Distributed Slicing in Dynamic Systems A. Fernández, V. Gramoli, E. Jiménez, A-M. Kermarrec, M. Raynal](https://reader036.vdocument.in/reader036/viewer/2022062714/56649d545503460f94a307e2/html5/thumbnails/23.jpg)
ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Algorithm 1: Ordering
68
70
8
7262
7565
20
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7/11
2/11
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ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Algorithm 1: Ordering
68
70
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7262
7565
20
71
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![Page 25: Distributed Slicing in Dynamic Systems A. Fernández, V. Gramoli, E. Jiménez, A-M. Kermarrec, M. Raynal](https://reader036.vdocument.in/reader036/viewer/2022062714/56649d545503460f94a307e2/html5/thumbnails/25.jpg)
ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Algorithm 1: Ordering
68
70
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7262
7565
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2/11
4/11
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ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Algorithm 1: Ordering
68
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ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Result: Slight convergence speed up
n = 104
#slices = 10|V| = 20
![Page 28: Distributed Slicing in Dynamic Systems A. Fernández, V. Gramoli, E. Jiménez, A-M. Kermarrec, M. Raynal](https://reader036.vdocument.in/reader036/viewer/2022062714/56649d545503460f94a307e2/html5/thumbnails/28.jpg)
ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
If random values are not perfectly uniformly distributed
…some nodes might never find their slice e.g. the 3 nodes of S2 in the example above
Problem: Wrong Slice Assignment
#4#3#2#1
0 1Slices
Normalizedindices
![Page 29: Distributed Slicing in Dynamic Systems A. Fernández, V. Gramoli, E. Jiménez, A-M. Kermarrec, M. Raynal](https://reader036.vdocument.in/reader036/viewer/2022062714/56649d545503460f94a307e2/html5/thumbnails/29.jpg)
ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Algorithm 2: RankingNo random values!
ProtocolDo periodically {
Update/Shuffle view Vi using an underlying protocol.l += #neighbors with lower attribute value.g += #neighborsSends ai to a randomly chosen neighborSends ai to the neighbor that is the closest to a slice boundary
Update slice assignment si’ = s such that l/g in s.}Upon reception {
Receive aj from jif (aj < ai) l += 1;
g+=1 ;Update slice assignment si’ = s such that l/g in s.
}
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ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Algorithm 2: RankingNo random values!
ProtocolDo periodically {
Update/Shuffle view Vi using an underlying protocol.l += #neighbors with lower attribute value.g += #neighborsSends ai to a randomly chosen neighborSends ai to the neighbor that is the closest to a slice boundary
Update slice assignment si’ = s such that l/g in s.}Upon reception {
Receive aj from jif (aj < ai) l += 1; g+=1 ;Update slice assignment si’ = s such that l/g in s.
}
Same number of messages
![Page 31: Distributed Slicing in Dynamic Systems A. Fernández, V. Gramoli, E. Jiménez, A-M. Kermarrec, M. Raynal](https://reader036.vdocument.in/reader036/viewer/2022062714/56649d545503460f94a307e2/html5/thumbnails/31.jpg)
ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Algorithm 2: Ranking
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4/11
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ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Algorithm 2: Ranking
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ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Algorithm 2: Ranking
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0/2
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ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Algorithm 2: Ranking
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7220
![Page 35: Distributed Slicing in Dynamic Systems A. Fernández, V. Gramoli, E. Jiménez, A-M. Kermarrec, M. Raynal](https://reader036.vdocument.in/reader036/viewer/2022062714/56649d545503460f94a307e2/html5/thumbnails/35.jpg)
ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Algorithm 2: Ranking
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1/4
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ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Algorithm 2: Ranking
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ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Algorithm 2: Ranking
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1/3
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ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Result 1: Unlimited convergence
n = 104
#slices = 100|V| = 20
![Page 39: Distributed Slicing in Dynamic Systems A. Fernández, V. Gramoli, E. Jiménez, A-M. Kermarrec, M. Raynal](https://reader036.vdocument.in/reader036/viewer/2022062714/56649d545503460f94a307e2/html5/thumbnails/39.jpg)
ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Result 1: Unlimited convergence
Ranking precision keeps improving
n = 104
#slices = 100|V| = 20
![Page 40: Distributed Slicing in Dynamic Systems A. Fernández, V. Gramoli, E. Jiménez, A-M. Kermarrec, M. Raynal](https://reader036.vdocument.in/reader036/viewer/2022062714/56649d545503460f94a307e2/html5/thumbnails/40.jpg)
ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Result 2: Tolerating Dynamism
Churn is correlated with attribute values!
e.g. the attribute is the remaining batery lifetime or available storage space.
Churn
![Page 41: Distributed Slicing in Dynamic Systems A. Fernández, V. Gramoli, E. Jiménez, A-M. Kermarrec, M. Raynal](https://reader036.vdocument.in/reader036/viewer/2022062714/56649d545503460f94a307e2/html5/thumbnails/41.jpg)
ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Performance Analysis
d, is the distance from pi’ to the closest slice boundary.For confidence coefficient of 99,99%, the required number of attribute value drawn is
mi ≥ z pi’ (1 – pi’) / d2,
with z <16, a constant.
![Page 42: Distributed Slicing in Dynamic Systems A. Fernández, V. Gramoli, E. Jiménez, A-M. Kermarrec, M. Raynal](https://reader036.vdocument.in/reader036/viewer/2022062714/56649d545503460f94a307e2/html5/thumbnails/42.jpg)
ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Conclusion
Churn-tolerant algorithmGossip-based mechanisms.Slice belongingness re-approximation.
Scalable algorithmLimited number of neighbors.Size of the system is unknown.
Applications of the distributed slicingResource allocationSupernodes / Ultrapeers election
Future workCan we obtain convergence speed of the first algorithm with the
accuracy of the second algorithm?
![Page 43: Distributed Slicing in Dynamic Systems A. Fernández, V. Gramoli, E. Jiménez, A-M. Kermarrec, M. Raynal](https://reader036.vdocument.in/reader036/viewer/2022062714/56649d545503460f94a307e2/html5/thumbnails/43.jpg)
ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
References
Ordered Slicing of Very Large-Scale Overlay NetworksM. Jelasity and A.-M. Kermarrec In Proc. of the 6th IEEE Conference on P2P Computing, 2006.
Randomized AlgorithmsR. Motwani, P. RaghavanCambridge University Press, 1995
Time Bounds for SelectionM. Blum, R. Floyd, V. Pratt, R. Rivest, and R. TarjanJournal Computer and System Sciences 7:448-461, 1972
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ICDCS 2007June
Fernandez, Gramoli, Jimenez, Kermarrec, Raynal
Result 3: Feasibility