Subquadratic Non-adaptive Threshold Group Testing Gianluca De Marco - - PowerPoint PPT Presentation

Subquadratic Non-adaptive Threshold Group Testing

Gianluca De Marco1 Tomasz Jurdzi´ nski2 Michał Ró˙ za´ nski2 Grzegorz Stachowiak2

1Dipartimento di Informatica,

University of Salerno, Italy

2Institute of Computer Science

University of Wrocław, Poland FCT 2017 Bordeaux, France

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Motivation – Group Testing

Introduced by Dorfman in ’43, during World War II, as a part of a large project started by United States Public Health Service, to weed out syphilitic men called up for induction.

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Motivation – Group Testing

Introduced by Dorfman in ’43, during World War II, as a part of a large project started by United States Public Health Service, to weed out syphilitic men called up for induction. Also used in: machine learning, cryptography, multiple access channel communication, DNA sequencing, medical examination, streaming algorithms, and more.

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Motivation – Group Testing

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Motivation – Group Testing

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Motivation – Group Testing

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Motivation – Group Testing

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Motivation – Group Testing

• infected

infected

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Motivation

Group Testing The population is identified with [n]. An individual is a number from the set.

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Motivation

Group Testing The population is identified with [n]. An individual is a number from the set. Distinguished, unknown set of elements P ⊆ [n] of size d, called positives.

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Motivation

Group Testing The population is identified with [n]. An individual is a number from the set. Distinguished, unknown set of elements P ⊆ [n] of size d, called positives. Testing pool – Q ⊆ [n].

FCT 2017 8 / 14

Motivation

Group Testing The population is identified with [n]. An individual is a number from the set. Distinguished, unknown set of elements P ⊆ [n] of size d, called positives. Testing pool – Q ⊆ [n]. Feedback function fP : 2[n] → {0, 1} fP(Q) = 1 if |Q ∩ P| ≥ 1, and 0 otherwise.

FCT 2017 8 / 14

Motivation

Group Testing The population is identified with [n]. An individual is a number from the set. Distinguished, unknown set of elements P ⊆ [n] of size d, called positives. Testing pool – Q ⊆ [n]. Feedback function fP : 2[n] → {0, 1} fP(Q) = 1 if |Q ∩ P| ≥ 1, and 0 otherwise. A sequence Q = (Q1, ..., Qm) is called a pooling strategy of size m.

FCT 2017 8 / 14

Motivation

Group Testing The population is identified with [n]. An individual is a number from the set. Distinguished, unknown set of elements P ⊆ [n] of size d, called positives. Testing pool – Q ⊆ [n]. Feedback function fP : 2[n] → {0, 1} fP(Q) = 1 if |Q ∩ P| ≥ 1, and 0 otherwise. A sequence Q = (Q1, ..., Qm) is called a pooling strategy of size m. A measurement is a vector y = (fP(Q1), ..., fP(Qm)).

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Motivation

Group Testing The population is identified with [n]. An individual is a number from the set. Distinguished, unknown set of elements P ⊆ [n] of size d, called positives. Testing pool – Q ⊆ [n]. Feedback function fP : 2[n] → {0, 1} fP(Q) = 1 if |Q ∩ P| ≥ 1, and 0 otherwise. A sequence Q = (Q1, ..., Qm) is called a pooling strategy of size m. A measurement is a vector y = (fP(Q1), ..., fP(Qm)). Pooling strategy Q is valid if for any set of positives P of size d, it is possible to discover the set P basing only on the measurement vector.

FCT 2017 8 / 14

Motivation

Group Testing The population is identified with [n]. An individual is a number from the set. Distinguished, unknown set of elements P ⊆ [n] of size d, called positives. Testing pool – Q ⊆ [n]. Feedback function fP : 2[n] → {0, 1} fP(Q) = 1 if |Q ∩ P| ≥ 1, and 0 otherwise. A sequence Q = (Q1, ..., Qm) is called a pooling strategy of size m. A measurement is a vector y = (fP(Q1), ..., fP(Qm)). Pooling strategy Q is valid if for any set of positives P of size d, it is possible to discover the set P basing only on the measurement vector. Goal: design a valid pooling strategy with small number of tests.

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Motivation

Example Let Q = (Q1, . . . , Qn), where Qi = {i}.

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Motivation

Example Let Q = (Q1, . . . , Qn), where Qi = {i}. For a measurement y = (1, 0, 0, 1, 0, 1, 0, ...0) we can derive the set of positives P = {1, 3, 7}.

FCT 2017 9 / 14

Motivation

Example Let Q = (Q1, . . . , Qn), where Qi = {i}. For a measurement y = (1, 0, 0, 1, 0, 1, 0, ...0) we can derive the set of positives P = {1, 3, 7}. Q discovers all positives explicitly.

FCT 2017 9 / 14

Motivation

Example Let Q = (Q1, . . . , Qn), where Qi = {i}. For a measurement y = (1, 0, 0, 1, 0, 1, 0, ...0) we can derive the set of positives P = {1, 3, 7}. Q discovers all positives explicitly. How to decode measurement vectors for more complex queries?

FCT 2017 9 / 14

Motivation

Example Let Q = (Q1, . . . , Qn), where Qi = {i}. For a measurement y = (1, 0, 0, 1, 0, 1, 0, ...0) we can derive the set of positives P = {1, 3, 7}. Q discovers all positives explicitly. How to decode measurement vectors for more complex queries? The length of Q is n. Can we do better knowing that there are d ≪ n positives?

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Group Testing

Results Introduction – [Dorfman ’43] Explicit construction O(d2 log2 n), existential result O(d2 log n), connections to coding theory – [Kautz, Singleton ’64] Lower bound Ω(d2 logd n) – [Dyachkov, Rykov ’82; Furedi ’96] Explicit construction with O(d2 log n) tests – [Porat, Rotschild ’08]

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Group Testing

Results Introduction – [Dorfman ’43] Explicit construction O(d2 log2 n), existential result O(d2 log n), connections to coding theory – [Kautz, Singleton ’64] Lower bound Ω(d2 logd n) – [Dyachkov, Rykov ’82; Furedi ’96] Explicit construction with O(d2 log n) tests – [Porat, Rotschild ’08] Research directions Gap between lower and upper bound Efficient construction Decoding – how to obtain a set of positives from a measurement vector effectively? Practical aspects, applications & generalizations

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Threshold Group Testing

Definition The population is identified with [n]. An individual is a number from the set. Distinguished, unknown set of elements P ⊆ [n] of size d, called positives. Testing pool – Q ⊆ [n]. Feedback function fP : 2[n] → {0, 1} fP,t(Q) = 1 if |Q ∩ P| ≥ t, and 0 otherwise. A sequence Q = (Q1, ..., Qm) is called a pooling strategy of size m. A measurement is a vector y = (fP(Q1), ..., fP(Qm)). Group Testing ≡ Threshold Group Testing with threshold 1.

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Threshold Group Testing

Previous Results Introduction of threshold group testing (adaptive)– Damaschke (’05) Non-adaptive result with O(dt+1 log n) tests – Chen, Fu (’09) Improved result for constant threshold, O(d2 log d log(n/d)) (existential), explicit construction with O(d3 log d log(n/d)) tests – Cheraghchi (’13)

FCT 2017 12 / 14

Threshold Group Testing

Previous Results Introduction of threshold group testing (adaptive)– Damaschke (’05) Non-adaptive result with O(dt+1 log n) tests – Chen, Fu (’09) Improved result for constant threshold, O(d2 log d log(n/d)) (existential), explicit construction with O(d3 log d log(n/d)) tests – Cheraghchi (’13) Our result There exists a pooling design of size O(

d2 q(d,t) log n d ) for a threshold group

testing, where q(d, t) = Ω

• dt

d−t

• .

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Threshold Group Testing

Previous Results Introduction of threshold group testing (adaptive)– Damaschke (’05) Non-adaptive result with O(dt+1 log n) tests – Chen, Fu (’09) Improved result for constant threshold, O(d2 log d log(n/d)) (existential), explicit construction with O(d3 log d log(n/d)) tests – Cheraghchi (’13) Our result There exists a pooling design of size O(

d2 q(d,t) log n d ) for a threshold group

testing, where q(d, t) = Ω

• dt

d−t

• .

Corollary: there exists a pooling design of size O(d3/2 log(n/d)) for t = d/2.

FCT 2017 12 / 14

Threshold Group Testing

Previous Results Introduction of threshold group testing (adaptive)– Damaschke (’05) Non-adaptive result with O(dt+1 log n) tests – Chen, Fu (’09) Improved result for constant threshold, O(d2 log d log(n/d)) (existential), explicit construction with O(d3 log d log(n/d)) tests – Cheraghchi (’13) Our result There exists a pooling design of size O(

d2 q(d,t) log n d ) for a threshold group

testing, where q(d, t) = Ω

• dt

d−t

• .

Corollary: there exists a pooling design of size O(d3/2 log(n/d)) for t = d/2. Lower bound: Ω(min (d/t)2, n/t).

FCT 2017 12 / 14

Threshold Group Testing

Previous Results Introduction of threshold group testing (adaptive)– Damaschke (’05) Non-adaptive result with O(dt+1 log n) tests – Chen, Fu (’09) Improved result for constant threshold, O(d2 log d log(n/d)) (existential), explicit construction with O(d3 log d log(n/d)) tests – Cheraghchi (’13) Our result There exists a pooling design of size O(

d2 q(d,t) log n d ) for a threshold group

testing, where q(d, t) = Ω

• dt

d−t

• .

Corollary: there exists a pooling design of size O(d3/2 log(n/d)) for t = d/2. Lower bound: Ω(min (d/t)2, n/t). Multi-threshold group testing: O(d2/t log n) for Θ( √ t) thresholds.

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Further directions

Tight bounds Explicit construction Decoding algorithms

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Thank you!

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