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A Better Tomorrow: Fast Reference-based Network Tomography Using Compressive Sensing

This paper presents an innovative approach to network tomography using a fast reference-based algorithm called FRANTIC ME. Implementing compressive sensing techniques, the method infers network link congestion and delays from end-to-end packet measurements with reduced measurement requirements. By leveraging a "k-sparse" framework, the algorithm achieves efficient node and edge delay estimations while producing accurate results. Key techniques include "loopy" measurements and cancellations, resulting in a robust and effective solution for network diagnostics.

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A Better Tomorrow: Fast Reference-based Network Tomography Using Compressive Sensing

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  1. 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

  2. FRANTIC ME Fast Reference-based Algorithm for Network Tomography vIa Compressive Sensing Sheng Cai Mayank Bakshi Minghua Chen Sidharth Jaggi The Chinese University of Hong Kong The Institute of Network Coding

  3. Computerized Axial Tomography (CAT scan)

  4. Tomography y = Tx Estimate x given y and T

  5. Network Tomography • Transform T: • Network connectivity matrix (known a priori) • Measurements y: • End-to-end packet delays • Infer x: • Link congestion Hopefully “k-sparse” Compressive sensing? • Challenge: • Matrix T “fixed” • Idea: • “Mimic” random matrix

  6. 1. Better CS [BJCC12] “SHO-FA”

  7. 1. Better CS [BJCC12] “SHO-FA” O(k) measurements, O(k) time

  8. SHO(rt)-FA(st)O(k) meas., O(k) steps

  9. 1. Better CS [BJCC12] “SHO-FA” Need “sparse & random” matrix T

  10. SHO-FA A d=3 ck n

  11. 1. Better CS [BJCC12] “SHO-FA” T

  12. 2. Better mimicking of desired T

  13. Node delay estimation

  14. Node delay estimation

  15. Node delay estimation

  16. Edge delay estimation

  17. Idea 1: Cancellation , ,

  18. Idea 2: “Loopy” measurements , • Fewer measurements • Arbitrary packet injection/ • reception • Not just 0/1 matrices (SHO-FA)

  19. SHO-FA + Cancellations + Loopy measurements • n = |V| or |E| • M = “loopiness” • k = sparsity • Measurements: O(k log(n)/log(M)) • Decoding time: O(k log(n)/log(M)) • General graphs, node/edge delay estimation • 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)

  20. 1. Graph-Matrix

  21. 2. (Most) x-expansion ≥2|S| |S|

  22. Decoding – Leaf Check(1-Passed)

  23. m ? n m<n

  24. Compressive sensing ? m ? n k k ≤ m<n

  25. Robust compressive sensing ? e z y=A(x+z)+e Approximate sparsity Measurement noise

  26. Apps: 1. Compression W(x+z) 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. 

  27. Apps: 2. Network tomography Weiyu Xu; Mallada, E.; Ao Tang; , "Compressive sensing over graphs," INFOCOM, 2011 M. Cheraghchi, A. Karbasi, S. Mohajer, V.Saligrama: Graph-Constrained Group Testing. IEEE Transactions on Information Theory 58(1): 248-262 (2012)

  28. Apps: 3. Fast(er) Fourier Transform 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.

  29. Apps: 4. One-pixel camera http://dsp.rice.edu/sites/dsp.rice.edu/files/cs/cscam.gif

  30. y=A(x+z)+e

  31. y=A(x+z)+e

  32. y=A(x+z)+e

  33. y=A(x+z)+e

  34. y=A(x+z)+e (Information-theoretically) order-optimal

  35. (Information-theoretically) order-optimal • Support Recovery

  36. SHO(rt)-FA(st)O(k) meas., O(k) steps

  37. SHO(rt)-FA(st)O(k) meas., O(k) steps

  38. SHO(rt)-FA(st)O(k) meas., O(k) steps

  39. 1. Graph-Matrix A d=3 ck n

  40. 1. Graph-Matrix A d=3 ck n

  41. 1. Graph-Matrix

  42. 2. (Most) x-expansion ≥2|S| |S|

  43. 3. “Many” leafs L+L’≥2|S| ≥2|S| |S| 3|S|≥L+2L’ L≥|S| L+L’≤3|S| L/(L+L’) ≥1/2 L/(L+L’) ≥1/3

  44. 4. Matrix

  45. Encoding – Recap. 0 1 0 1 0

  46. Decoding – Initialization

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