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Non-Adaptive Fault Diagnosis for All-Optical Networks via Combinatorial Group Testing on Graphs

Non-Adaptive Fault Diagnosis for All-Optical Networks via Combinatorial Group Testing on Graphs. Nick Harvey, Mihai Pătraşcu, Yonggang Wen, Sergey Yekhanin Vincent W.S. Chan. Group Testing. n drugs: n-1 good, 1 bad. Hi! I’m a probe. Group Testing. n drugs: n-1 good, 1 bad

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Non-Adaptive Fault Diagnosis for All-Optical Networks via Combinatorial Group Testing on Graphs

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  1. Non-Adaptive Fault Diagnosis for All-Optical Networks via Combinatorial Group Testing on Graphs Nick Harvey, Mihai Pătraşcu, Yonggang Wen, Sergey Yekhanin Vincent W.S. Chan

  2. Group Testing • n drugs: n-1 good, 1 bad Hi! I’m a probe.

  3. Group Testing • n drugs: n-1 good, 1 bad • better idea: Hi! I’m a Walsh character.

  4. Very Useful in Practice To: nickh, eewyg, chan Subject: is any of you willing to give the talk?

  5. Combinatorial Group Testing • adaptive / nonadaptive • m = # elements • s = # bad elements • T* = # necessary probes adaptive:

  6. Group Testing on Graphs • elements = edges • bad elements = failed edges • probes = connectedsubgraphs  1 element = $1 Assumptions Graph Testing: 1 connected subset = $1 CGT: 1 subset = $1

  7. Cost Model: All-Optical Networks Practical assumption: undirected graph an edge = 2 parallel optical fibers • => testing entire connected subgraph = 1 lightpath

  8. Cost Model: All-Optical Networks “1 lightpath = $1” Optical routing => real cost: transmission / reception => real delay: adaptivity routing with negligible attenuation speed of light = fast

  9. Assumptions Failure model: • links fail (cable cuts…) • permanent failure model (link = up/down) • adversarial failures (up to s) Network model: • undirected • separate control network [alternative: want to find the connected component of a central node]

  10. Some Special Cases adaptive: Nonadaptive group testing on graphs: • line, ring • complete graph • grid, torus

  11. Lesson 1: Small Connectivity Hurts Lower bound for line: • probes = interval • if one vertex has no [ or ] => can’t distinguish adjacent edges ?

  12. Lesson 2: Just Need a Trusted Subnet Solution for complete graph: • test the star • assuming 1. says “fail”: apply CGT on star edges • assuming 1. says “not fail”: apply CGT on all other edges Cost:

  13. General Results: Well Connected min-cut 2(s+1) If G containtss+1 edge-disjoint spanning trees: Proof: • test each subtree => at least one is ok • use this subtree to do CGT on the other edges Corollaries: • 2D torus has min-cut 4 => 2D grid similar, more complicated • complete graph has min-cut n-1 => can handle

  14. General Results: Trees Tree T of depth D: Proof: • for all d≤D, assume failure is at depth d use tree to depth d-1 to do CGT at depth d centroid decomposition

  15. General Results: Low Diameter G connected, diameter D: Proof: • let T = shortest path tree (depth D) Cost • test T 1 • assuming T is ok, do CGT on G\T O(log n) • assuming T in not ok, apply tree algorithm on T O(D+log2n)

  16. Summary • trees * line, ring: • diameter D: • s+1 edge-disjoint spanning trees * complete graph * torus, grid

  17. The End I have a question about slides {1,2,5,6,9}

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