Noisy Group Testing (Quick and Efficient) Sheng Cai, Mayank Bakshi, Sidharth Jaggi The Chinese University of Hong Kong Mohammad Jahangoshahi Sharif University of Technology

q Group Testing For Pr(error)< ε, Lower bound of number of tests: What’s known [CCJS11] 2 q Chun Lam Chan; Pak Hou Che; Jaggi, S.; Saligrama, V.;, "Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms," 49th Annual Allerton Conference on Communication, Control, and Computing, pp , Sept [CCJS11] Adaptive vs. Non-adaptive

Decoding complexity # Tests  Lower bound Lower bound      Adaptive Non-Adaptive Two-Stage Adaptive This work   O(poly(D)log(N)),O(D 2 log(N)) O(DN),O(Dlog(N)) [NPR12] O(Dlog(N)) 3

Hammer: GROTESQUE Testing 4

4

4

4

Testing Matrix IN OUT Negative 0 Positive 1 5

Multiplicity ? 6

Multiplicity (d = 0) 7

7

7

7 d = 0 No positive tests

Multiplicity (d = 1) 8

8

8

8 d = 1 50% positive tests

Multiplicity (d = 2) 9

9

9

9 d = 2 75% positive tests Statistical Difference!

Multiplicity ? 10

Localization 11 ?

Localization BSC (q) Channel Expander Codes Decoder Signature Test Outcome Particular Signature 12

Hammer: GROTESQUE Testing 4

Nail: “Good” Partioning N items D defectives 13

Adaptive Group Testing Groups 14

Adaptive Group Testing 14 Decaying geometrically Tests Groups

Adaptive Group Testing 14 The number of unidentified defectives <

Adaptive Group Testing 14 Tests of size Coupon Collection

Non-Adaptive Group Testing Groupsconstant fraction of “Good” groups Tests 15

Non-Adaptive Group Testing 15

Non-Adaptive Group Testing 15 Independent partitions Coupon Collection Tests

2-Stage Adaptive Group Testing 16 Groups (Birthdays)

2-Stage Adaptive Group Testing Non-adaptive Group Testing 16 + Tests

   Summary of this work Decoding complexity # Tests O(Dlog(N)) 3

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