a better tomorrow
DESCRIPTION
A Better TOMORROW. ME. fast TOMOgRaphy oveR netwOrks with feW probes. Sheng Cai. Mayank Bakshi. Minghua Chen. Sidharth Jaggi. The Chinese University of Hong Kong. The Institute of Network Coding. FRANTIC. ME. - PowerPoint PPT PresentationTRANSCRIPT
A Better TOMORROWfast TOMOgRaphy oveR netwOrks
with feW probes
ME
Sheng Cai Mayank Bakshi Minghua Chen Sidharth Jaggi
The Chinese University of Hong Kong
The Institute of Network Coding
FRANTICFast Reference-based Algorithm for
Network Tomography vIa Compressive Sensing
ME
Sheng Cai Mayank Bakshi Minghua Chen Sidharth Jaggi
The Institute of Network Coding
The Chinese University of Hong Kong
TomographyComputerized Axial
(CAT scan)
Tomography
Estimate x given y and T
y = Tx
Network Tomography
Measurements y:•End-to-end packet delays
Transform T:•Network connectivity matrix (known a priori)
Infer x:•Link congestion
Hopefully “k-sparse”
Compressive sensing?
Challenge:•Matrix T “fixed”
Idea:•“Mimic” random matrix
1. Better CS [BJCC12] “SHO-FA”
1. Better CS [BJCC12] “SHO-FA”
O(k) measurements,O(k) time
SHO(rt)-FA(st)
O(k) meas., O(k) steps
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1. Better CS [BJCC12] “SHO-FA”
Need “sparse & random” matrix T
SHO-FA
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n ck
Ad=3
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T
1. Better CS [BJCC12] “SHO-FA”
2. Better mimicking of desired T
Node delay estimation
1v3v4v2v
Node delay estimation
4v2v3v
1v
4v2v1v3v
Node delay estimation
€
y = [1 0 1 0] dv
Edge delay estimation
1e 5e6e 3e4e
2e
Idea 1: Cancellation
, ,
Idea 2: “Loopy” measurements
•Fewer measurements•Arbitrary packet injection/
reception•Not just 0/1 matrices (SHO-FA)
,
SHO-FA + Cancellations +
Loopy measurements
• Measurements: O(k log(n)/log(M))• Decoding time: O(k log(n)/log(M))• General graphs, node/edge delay estimation
• n = |V| or |E|• M = “loopiness”• k = sparsity
• Path delay: O(DnM/k) • Path delay: O(D’M/k) (Steiner trees)
• Path delay: O(D’’M/k) (“Average” Steiner trees)
• Path delay: ??? (Graph decompositions)
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1. Graph-Matrix
2. (Most) x-expansion
≥2|S||S|23
Decoding – Leaf Check(1-Passed)
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26
? n
m<n
m
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Compressive sensing
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?
k ≤ m<n
? n
m
k
Robust compressive sensing
Approximate sparsity
Measurement noise
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?
Apps: 1. Compression
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W(x+z)
BW(x+z) = A(x+z)
M.A. Davenport, M.F. Duarte, Y.C. Eldar, and G. Kutyniok, "Introduction to Compressed Sensing,"in Compressed Sensing: Theory and Applications, Cambridge University Press, 2012.
x+z
Apps: 2. Network tomography
Weiyu Xu; Mallada, E.; Ao Tang; , "Compressive sensing over graphs," INFOCOM, 2011M. Cheraghchi, A. Karbasi, S. Mohajer, V.Saligrama: Graph-Constrained Group Testing. IEEE Transactions on Information Theory 58(1): 248-262 (2012)
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Apps: 3. Fast(er) Fourier Transform
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H. Hassanieh, P. Indyk, D. Katabi, and E. Price. Nearly optimal sparse fourier transform. In Proceedings of the 44th symposium on Theory of Computing (STOC '12). ACM, New York, NY, USA, 563-578.
Apps: 4. One-pixel camera
http://dsp.rice.edu/sites/dsp.rice.edu/files/cs/cscam.gif
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y=A(x+z)+e
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y=A(x+z)+e
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y=A(x+z)+e
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y=A(x+z)+e
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y=A(x+z)+e
(Information-theoretically) order-optimal38
(Information-theoretically) order-optimal
• Support Recovery
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SHO(rt)-FA(st)
O(k) meas., O(k) steps
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SHO(rt)-FA(st)
O(k) meas., O(k) steps
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SHO(rt)-FA(st)
O(k) meas., O(k) steps
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1. Graph-Matrix
n ck
d=3
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A
1. Graph-Matrix
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n ck
Ad=3
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1. Graph-Matrix
2. (Most) x-expansion
≥2|S||S|46
3. “Many” leafs
≥2|S||S|L+L’≥2|S|
3|S|≥L+2L’
L≥|S|L+L’≤3|S|
L/(L+L’) ≥1/3L/(L+L’) ≥1/2
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4. Matrix
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Encoding – Recap.
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0
1
0
1
0
Decoding – Initialization
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Decoding – Leaf Check(2-Failed-ID)
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Decoding – Leaf Check (4-Failed-VER)
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Decoding – Leaf Check(1-Passed)
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Decoding – Step 4 (4-Passed/STOP)
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Decoding – Recap.
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0
0
0
0
0
?
?
?0
0
0
1
0
Decoding – Recap.
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0
1
0
1
0
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Noise/approx. sparsity
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Meas/phase error
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Correlated phase meas.
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Correlated phase meas.
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Correlated phase meas.
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