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

9

1. Better CS [BJCC12] “SHO-FA”

Need “sparse & random” matrix T

SHO-FA

11

n ck

Ad=3

12

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)

22

1. Graph-Matrix

2. (Most) x-expansion

≥2|S||S|23

Decoding – Leaf Check(1-Passed)

24

26

? n

m<n

m

27

Compressive sensing

28

?

k ≤ m<n

? n

m

k

Robust compressive sensing

Approximate sparsity

Measurement noise

29

?

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

32

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

33

y=A(x+z)+e

34

y=A(x+z)+e

35

y=A(x+z)+e

36

y=A(x+z)+e

37

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

44

n ck

Ad=3

45

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)

51

Decoding – Leaf Check (4-Failed-VER)

52

Decoding – Leaf Check(1-Passed)

53

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.

28

0

1

0

1

0

57

Noise/approx. sparsity

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Meas/phase error

59

Correlated phase meas.

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Correlated phase meas.

61

Correlated phase meas.

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