Information- theory and Information theoretic and algorithmic aspects algorithms for GT and QGT of binary and quantitative group testing in the Oliver Gebhard sublinear regime Basic setup Binary group testing Oliver Gebhard Quantitative group testing Goethe-University Frankfurt gebhard@math.uni-frankfurt.de Joint work with A. Coja-Oghlan, M.Hahn-Klimroth, P. Loick and M.Hahn-Klimroth, D. Kaaser, P. Loick
Overview Information- theory and algorithms for GT and QGT Oliver Gebhard 1 Basic setup Basic setup Binary group testing Quantitative 2 Binary group testing group testing Quantitative group testing 3
Basic setup Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing
Basic setup Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing
Basic setup Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing Goal :Are we able to identify the sick individuals?
Basic setup Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing �� ����� ����� �
Basic setup Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing �� ����� ����� � Goal :Are we able to reduce the number of tests?
Basic setup Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing
Basic setup Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing Binary: Is a sick individual contained? Quantitative: Number of sick individuals?
Basic setup Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Options to choose the underlying testing procedure Binary group testing 1 Number of stages Quantitative 2 Pooling procedure group testing
Basic setup Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Options to choose the underlying testing procedure Binary group testing 1 Number of stages Quantitative 2 Pooling procedure group testing To Do: Rigorous Analysis of the choice
Basic setup Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing 0 n Number of tests
Basic setup Information- theory and algorithms for GT and QGT Oliver Information Gebhard Theoretic Bound Basic setup Binary group testing Quantitative group testing 0 n Number of tests
Basic setup Information- theory and algorithms for GT and QGT Oliver Information Algorithmic Gebhard Theoretic Bound Bound Basic setup Binary group testing Quantitative group testing 0 n Number of tests
Binary group testing Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing
Binary group testing Information- theory and algorithms for x 7 a 5 GT and QGT Oliver Gebhard x 6 Basic setup a 4 x 5 Binary group testing Quantitative x 4 a 3 group testing x 3 a 2 x 2 x 1 a 1
Binary group testing: Information Theory Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Counting Bound implies m > k ⋅ log 2 ( n / k ) . Binary group testing Quantitative group testing
Binary group testing: Information Theory Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Counting Bound implies m > k ⋅ log 2 ( n / k ) . Binary group Baldassini et al: Adaptive testing strategies achieve this testing bound. Quantitative group testing
Binary group testing: Information Theory Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Counting Bound implies m > k ⋅ log 2 ( n / k ) . Binary group Baldassini et al: Adaptive testing strategies achieve this testing bound. Quantitative group testing Question: Is non-adaptive group testing able to achieve the bound as well?
Binary group testing: Information Theory Information- theory and algorithms for GT and QGT Assume a constant weight testing scheme Oliver Gebhard Basic setup Binary group testing Quantitative group testing
Binary group testing: Information Theory Information- theory and algorithms for GT and QGT Assume a constant weight testing scheme Oliver The underlying graph structure: Gebhard Basic setup Binary group testing Quantitative group testing
Binary group testing: Information Theory Information- theory and algorithms for GT and QGT Assume a constant weight testing scheme Oliver The underlying graph structure: Gebhard Bipartite Factor Graph Basic setup Fixed variable nopde degree Binary group testing Quantitative group testing
Binary group testing: Information Theory Information- theory and algorithms for GT and QGT Assume a constant weight testing scheme Oliver The underlying graph structure: Gebhard Bipartite Factor Graph Basic setup Fixed variable nopde degree Binary group testing Theorem 1 [CGHL18] Quantitative group testing n θ log ( n ) Let m inf = min { 1 , 1 − θ θ log ( 2 )} log ( 2 ) 1 m < ( 1 − ǫ ) m inf : No algorithm exists to output the right configuration for the constant weight pooling 2 m > ( 1 + ǫ ) m inf : There exist an algorithm, which outputs the right configuration w.h.p.
Binary group testing: Information Theory Information- theory and algorithms for GT and QGT Lower Bound: Derive the value m ∗ , below which Oliver Gebhard infected/uninfected individuals occur that may swap status Basic setup without harming the test result Binary group testing Quantitative group testing
Binary group testing: Information Theory Information- theory and algorithms for GT and QGT Lower Bound: Derive the value m ∗ , below which Oliver Gebhard infected/uninfected individuals occur that may swap status Basic setup without harming the test result Binary group Upper Bound: Derive the value m ∗∗ above which no testing satisfying assignment beside the original one exists Quantitative group testing
Binary group testing: Information Theory Information- theory and algorithms for GT and QGT Lower Bound: Derive the value m ∗ , below which Oliver Gebhard infected/uninfected individuals occur that may swap status Basic setup without harming the test result Binary group Upper Bound: Derive the value m ∗∗ above which no testing satisfying assignment beside the original one exists Quantitative group testing 1 Small overlap argument: Analyse probability that configuration with certain overlap fulfills the test result 2 Large overlap argument: Analyse probability that positive (negative) tests stay positive (negative)
Binary group testing: Information Theory Information- theory and algorithms for GT and QGT Lower Bound: Derive the value m ∗ , below which Oliver Gebhard infected/uninfected individuals occur that may swap status Basic setup without harming the test result Binary group Upper Bound: Derive the value m ∗∗ above which no testing satisfying assignment beside the original one exists Quantitative group testing 1 Small overlap argument: Analyse probability that configuration with certain overlap fulfills the test result 2 Large overlap argument: Analyse probability that positive (negative) tests stay positive (negative) m ∗ , m ∗∗ set conditions to derive the m inf ( n ,θ ) as stated
Binary group testing: Algorithmic Aspect Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Allemann: multi-stage algorithm at the predicted lower Binary group bound testing Quantitative group testing
Binary group testing: Algorithmic Aspect Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Allemann: multi-stage algorithm at the predicted lower Binary group bound testing Aldridge, Scarlett et. al.: Sub-optimal non-adaptive Quantitative group testing strategies available
Binary group testing: Algorithmic Aspect Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Allemann: multi-stage algorithm at the predicted lower Binary group bound testing Aldridge, Scarlett et. al.: Sub-optimal non-adaptive Quantitative group testing strategies available Most promising algorithms: SCOMP, Definite Defective
Binary group testing: Algorithmic Aspect Information- theory and Truth DD-Start algorithms for GT and QGT x 7 x 7 a 5 Oliver Gebhard Basic setup x 6 x 6 a 4 Binary group testing x 5 x 5 Quantitative group testing x 4 x 4 a 3 x 3 x 3 a 2 x 2 x 2 x 1 x 1 a 1
Binary group testing: Algorithmic Aspect Information- theory and algorithms for x 7 a 5 GT and QGT Oliver Gebhard x 6 Basic setup a 4 x 5 Binary group testing Quantitative x 4 a 3 group testing x 3 a 2 x 2 x 1 a 1
Binary group testing: Algorithmic Aspect Information- theory and algorithms for x 7 a 5 GT and QGT Oliver Gebhard x 6 Basic setup a 4 x 5 Binary group testing Quantitative x 4 a 3 group testing x 3 a 2 x 2 x 1 a 1
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