Phoenix: A Weight-based Network Coordinate System Using Matrix Factorization
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Transcript of Phoenix: A Weight-based Network Coordinate System Using Matrix Factorization
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Phoenix: A Weight-Based
Network Coordinate System
Using Matrix Factorization
Yang Chen
Department of Computer Science
Duke University
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Outline
• Background
• System Design
• Evaluation
• Perspective Future Work
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BACKGROUND
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Internet Distance
• Round-trip propagation / transmission delay between two Internet nodes
What?
• Strong indicator of network proximity
• Relatively stable
Why?
• Measurement tool “Ping” is with major operating systems
How?
4
50ms
Alice Bob
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Use Cases
• Knowledge of Internet distance is useful
for…
– P2P content delivery (file sharing/streaming)
– Online/mobile games
– Overlay routing
– Server selection in P2P/Cloud
– Network monitoring
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Scalability
• Huge number of end-to-end paths in large
scale systems
SLOW and COSTLY when the system becomes large!6
N nodes N ´N measurements
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Network Coordinate (NC) Systems
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(5, 10, 2) (-3, 4, -2)
Distance Function
22ms
• Scalable measurement: N2 NK (K << N)
• Every node is assigned with coordinates
• Distance function: compute the distance between
two nodes without explicit measurement
AliceBob
[Ng et al, INFOCOM’02]
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Deployments
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They are all using
Network Coordinate Systems!
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Basic models
• Euclidean Distance-based NC (ENC)
– Modeling the Internet as a Euclidean space
– Systems: Vivaldi [Dabek et al., SIGCOMM’04], GNP [Ng et al,
INFOCOM’02], NPS [Ng et al., USENIX ATC’04], PIC [Costa et al.,
ICDCS’04]…
• Matrix Factorization-based NC (MFNC)
– Factorizing an Internet distance matrix as the
product of two smaller matrices
– Systems: IDES [Mao et al., JSAC’06], Phoenix, …
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Modeling the Internet as
a Euclidean space
• In a d-dimensional
Euclidean space, each
node will be mapped to
a position
• Compute distances
based on coordinates
using Euclidean distance
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d=3
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Triangle Inequality Violation
Czech
Republic
Slovakia
Hungary
5.6 ms
3.6 ms
29.9 ms
A Triangle Inequality Violation (TIV)
example in GEANT network
29.9 > 5.6+3.6
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Lots of TIVs in the Internet
due sub-optimal routing!!
Predicted distances in
Euclidean space must
satisfy triangle
inequality
[Zheng et al, PAM’05]
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Correlation in Internet Distance Matrices
Duke UNC Yale Aachen Oxford Toronto THU NUS
Duke - 3 24 107 122 37 219 252
UNC 3 - 24 106 109 38 219 253
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Internet paths with nearby
end nodes are often overlap!!
Rows in different Internet distance matrices are large correlated (low
effective rank)
[Tang et al, IMC’03], [Lim et al, ToN’05], [Liao et al, CoNEXT’11]
Distance measurement using PlanetLab nodes
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Factorization of an Internet Distance Matrix
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» ´{N rowsN columns
d columns
Mij » Xi
×Yj
X7
= [ 1 0 3 ],Y2
= [ 2 0 5 ]
M72 » X7
×Y2
=1´2 + 0 +3´ 5 =17
M X Y T
[Mao et al., JSAC’06]
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Matrix Factorization-Based NC
• Each node i has an outgoing vector Xi and an
incoming vector Yi
• Distance function is the dot product.14
» ´{N rowsN columns
d columns
M X Y T
X2
Y2
No triangle inequality constrain in this model!
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SYSTEM DESIGN
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Goals
• Substantial improvement in prediction
accuracy
• Decentralized and scalable
• Robust to dynamic Internet
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Workflow of Phoenix
System Initialization
Peer Discovery
Scalable Measurement
Coordinates Calculation
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System Initialization
Peer Discovery
Scalable Measurement
Coordinates Calculation
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System Initialization
• Early nodes (N<K): Full-mesh measurement
• Compute coordinates of early nodes by minimizing the overall discrepancy
between predicted distances and measured distances
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Measured Distance
Predicted Distance
H1
H2
H3
H4
H1
H2
H3
H4
(X1,Y1)(X2,Y2)
(X3,Y3)(X4,Y4)
Nonnegative matrix factorization: [D. D. Lee and H. S. Seung, Nature, 401(6755):788–791,
1999.]
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Dynamic Peer Discovery
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Tracker
H2 H3 H5 H3 H4 H6
H2 H3 H4 H5 H6 H1 H3 H4 H5 H6
H1H2
Gossip among nodes
• N>K, all nodes become ordinary nodes
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Reference Node Selection
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• Every new node randomly selects K existing nodes as
reference nodes
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Measurement and
Bootstrap Coordinates Calculation
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Measured Distance
Predicted Distance
R1R2 RK
• Node Hnew computes its own coordinates by
minimizing the overall discrepancy between predicted
distances and measured distances (Non-negative
least squares)
Hnew
(X1,Y1)(XK,YK)(X2,Y2)
(Xnew,Ynew)
R1R2 RK
Hnew
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Accuracy of Reference Coordinates
0 50 100 150
Node 1
Node 2
Node 3
…
Node N
Predicted Distance
Measured distance
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(XA,YA)
Distance between Node A and every other node
Node A
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Accuracy of Reference Coordinates (cont.)
0 20 40 60 80 100 120
Node 1
Node 2
Node 3
…
Node N
Predicted Distance
Measured Distance
23Distance between Node B and every other node
(XB,YB)
Misleading the nodes
referring to Node B!!
Node B
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Referring to Inaccurate
Coordinates
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(X1,Y1)(XK,YK)(X2,Y2)
(Xnew,Ynew)
R1R2 RK
Hnew
Error Propagation:
Hnew may mislead
nodes refer to it
Minimize
the impact
of RK
Give preference to
accurate reference
coordinates
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Heuristic Weight Assignment
0 50 100 150 200
R1
R2
R3
…
RK Predicted Distance
Measured distance
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Bootstrap Coordinates
Distance between Hnew and every reference node
Enhanced Coordinates
Updating coordinates
regularly
Hnew
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EVALUATION
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Evaluation Setup
• Data sets
– PL: 169 PlanetLab nodes
– King: 1740 Internet DNS servers
• Metric
– Relative Error (RE)
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RE =MeasuredDist -PredictedDist
min(MeasuredDist,PredictedDist)
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Evaluation: Relative Error
28
90th Percentile
Relative Error
Phoenix Phoenix
(Simple)
Vivaldi IDES
0.63 0.91 0.83 0.89
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Evaluation (cont.)
• Other findings through evaluation
– Robust to node churn
– Fast convergence
– Robust to measurement anomalies
– Robust to distance variation
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FUTURE WORK
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Perspective Topics
• NC systems in mobile-centric environment
– Access latency, host mobility, host churn
• Scalable Prediction of other important
network parameters
– Available bandwidth, shortest-path distance in
social graph
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Software
• NCSim
– Simulator of Decentralized Network
Coordinate Algorithms
– http://code.google.com/p/ncsim/
• Phoenix
– Original Phoenix simulator in IEEE TNSM
paper
– http://www.cs.duke.edu/~ychen/Phoenix_TNS
M_2011.zip
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