Combinatorial Group Testing Problems on Various Models Speaker: - - PowerPoint PPT Presentation

Combinatorial Group Testing Problems on Various Models

Speaker: Huilan Chang Advisor: Hung-Lin Fu

National Chiao Tung University

Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models

Group Testing Problem

Classical group testing: Origination: blood testing in World War II

Venkatesh Saligrama at Boston University

Rosenblatt and Dorfman (1940s) suggested to pool the blood samples.

Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models 1 / 2 6

Group Testing Problem

A set N of n clones, each of which is either positive or negative, is given. Goal: identify all positive ones by as few number of group tests as possible. A group test is an experiment on a subset of clones.

3 2 1 4 5 6 T2 positive T1 negative positive T3 positive clone negative clone

Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models 2 / 2 6

Group Testing Problem

In combinatorial group testing, the number of positive items is usually assumed at most d (<< n).

Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models 3 / 2 6

Preliminaries on Algorithms

A nonadaptive algorithm specifies all tests simultaneously. Matrix representation:

1 2 3 4 5 6 1 1 1 0 0 0 1 0 0 1 1 0 0 1 0 1 0 1 0 0 1 0 1 1

• utcomes

1 1 clones tests (pools)

Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models 4 / 2 6

Preliminaries on Algorithms

1 2 3 4 5 6 1 1 1 0 0 0 1 0 0 1 1 0 0 1 0 1 0 1 0 0 1 0 1 1

• utcomes

1 1 {1,2}\{1,4}={2}

Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models 5 / 2 6

Preliminaries on Algorithms

1 2 3 4 5 6 1 1 1 0 0 0 1 0 0 1 1 0 0 1 0 1 0 1 0 0 1 0 1 1

• utcomes

1 1 {1,2}\{1,4}={2}

A binary matrix is d-disjunct if for any d + 1 columns C0, C1, · · · , Cd,

• C0 \

d

• i=1

Ci

• ≥ 1.

Then

• τ0(C) > 0

for a negative clone C, τ0(P) = 0 for a positive clone P.

Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models 6 / 2 6

Various Group Testing Models

Inhibitor Model (Farach et al., 1997) Complex Model (Torney, 1999) Inhibitor Complex Model (Chang et al., 2010a) Graph Reconstruction

Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models 7 / 2 6

Inhibitor Model

An inhibitor is a third type of clones whose existence may cancel the effect of positive clones (Farach et al., 1997). Application:

• In drug discovery, certain compound can block the

detection of potent compounds (Xie et al. 2001).

• Enzyme inhibitors are molecules that interact in some way

with the enzyme to prevent it from working normally.

Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models 8 / 2 6

Complex Model

A subset of clones, called a complex, can induce a positive or negative effect (Torney, 1999). General setting: no complex contains another. Application: Protein-to-protein interaction (Lappe and Holm, 2004):

http://www.mdc-berlin.de Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models 9 / 2 6

Inhibitor Complex Model

Inhibitor Complex Model (Chang et al., 2010a): besides the positive complexes and negative complexes, an inhibitor is a third type of complexes.

1 5 4 3

inhibitor positive complex negative complex positive outcome

3 4 2

negative outcome

4 5 3 2 Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models 1 / 2 6

Inhibitor Complex Model

Inhibitor Complex Model (Chang et al., 2010a): besides the positive complexes and negative complexes, an inhibitor is a third type of complexes.

1 5 4 3

inhibitor positive complex negative complex positive outcome

3 4 2

negative outcome

4 5 3 2

Consider the setting: a group test is positive only when it contains a positive complex and no inhibitor. Combinatorial assumption: there are at most d positive complexes and at most h inhibitors, and each complex contains at most r clones.

Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models 1 1 / 2 6

Inhibitor Complex Model

A binary matrix is (d, r; z]-disjunct if for any r + d columns C1, · · · , Cr+d,

• r
• i=1

Ci \

r+d

• i=r+1

Ci

• ≥ z.

Such matrics are closely related to MDS codes (D’yachkov et al., 2002) and separating hash families (Stinson et al. 2004).

1 2 3 4 5 6 1 1 0 0 0 0 tests (pools)

• utcomes

positive negative X * * z

For a negative or inhibitory complex X, τ0(X) ≥ z.

Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models 1 2 / 2 6

negative or inhibitor

Inhibitor Complex Model

A matrix is (h, r; y]-inclusive if for any r + h columns C1, · · · , Cr+h,

• (

r

• i=1

Ci)

• (

r+h

• i=r+1

Ci)

• ≤ y.

1 2 3 4 5 6 1 1 1 0 0 0 1 1 1 1 0 0 ests

• ols)
• utcomes

positive inhibitor P * * y h

For a positive complex P with |P| = r, τ0(P) ≤ y. (P appears in at most y negative pools.)

Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models 1 3 / 2 6

Inhibitor Complex Model

Theorem (Chang et al., 2010b) A (d, r; z]-disjunct and (h, r; y]-inclusive matrix with z > y can solve the general inhibitor complex model when each positive complex contains exactly r clones. Decoding procedure: {X : τ0(X) ≤ y} is the set of positive complexes.

Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models 1 4 / 2 6

Inhibitor Complex Model

Our results on the inhibitor complex model (Chang et al, 2010a, 2010b): (A) (H, d + h; 2e + 1)-disjunct, H is the set of complexes. (B) (d + h, r; 2e + 1]-disjunct. (C) (d, r; z]-disjunct and (h, r; y]-inclusive with z − e > y + e. (A)⇐(B)⇐(C).

Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models 1 5 / 2 6

Inhibitor Complex Model

Our results on the inhibitor complex model (Chang et al, 2010a, 2010b): (A) (H, d + h; 2e + 1)-disjunct, H is the set of complexes. (B) (d + h, r; 2e + 1]-disjunct. (C) (d, r; z]-disjunct and (h, r; y]-inclusive with z − e > y + e. (A)⇐(B)⇐(C). Design Decoding complexity Testing performance (A) O( |H|−1

h

• (hr)t|H|)

O(( 2(d + h)r lg n lg((d + h)r lg n))r+1) (Gao et al.,2006) (B) O( n−r

h

• (h + r)t|H|)

O(c(d, h, r) lg n) (C) O(rt|H|) (Chang et al. 2010b) r = 1: O((d + h)2 lg n) (Hwang and S´

• s, 1987)

r > 1: T-design, direct constructions

Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models 1 6 / 2 6

Inhibitor Complex Model

Classification problem on inhibitor complex model: classify all complexes by group tests. Motivation: Enzyme inhibitors are molecules that bind to enzymes and decrease their activity. Since blocking an enzyme’s activity can kill a pathogen or correct a metabolic imbalance, many drugs are enzyme inhibitors.

Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models 1 7 / 2 6

Inhibitor Complex Model

Theorem (Chang et al., 2010b) A (d + h, 2r; 1]-disjunct matrix can classify all complexes under the inhibitor complex model. Decoding procedure: Identify inhibitors first.

• {X : τ1(X) = 0} ≡ I is the set of inhibitors.

Then distinguish positive from negative.

• {X ∈ H \ I : τ IX

0 (X) = 0} is the set of positive complexes.

Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models 1 8 / 2 6

Graph Reconstruction

Given a family G of labeled graphs on [n], a hidden graph G is known belonging to G.

• Example:G = {Hamiltonian cycles on [4]} =

1 2 3 4 1 3 2 4 1 2 4 3

The main task is to reconstruct G by asking queries as few as possible. A query Q(S) for S ⊆ [n] is “Does S induce at least one edge of G?”.

• Example: Q({1, 3}) = 0.

Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models 1 9 / 2 6

Graph Reconstruction

Given a family G of labeled graphs on [n], a hidden graph G is known belonging to G.

• Example:G = {Hamiltonian cycles on [4]} =

1 2 3 4 1 3 2 4 1 2 4 3

The main task is to reconstruct G by asking queries as few as possible. A query Q(S) for S ⊆ [n] is “Does S induce at least one edge of G?”.

• Example: Q({1, 3}) = 0. Then G = 1234.

Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models 2 / 2 6

Graph Reconstruction

Application: Whole-genome shotgun sequencing

Contigs Primers ea/b Circular genome 1 2 3

Multiplex PCR: Q({1, 2, 3}) = 1 and Q({1, 3}) = 0.

Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models 2 1 / 2 6

Graph Reconstruction

2 4 2A 2B 1A 1B 4B 4A 3A 3B

contig consecutive primers

2A 2B 1A 1B 4B 4A 3A 3B

gap

1 3 (b) (a)

(a) Primers are vertices: G contains all perfect matchings on {1A, 1B, 2A, 2B, 3A, 3B, 4A, 4B}. (b) Contigs are vertices: G contains all Hamiltonian cycles on {1, 2, 3, 4}.

Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models 2 2 / 2 6

Graph Reconstruction

We consider sequential algorithms and improve results on G = {Hamiltonian cycles} or { matchings} or {stars} or {cliques}. Hamiltonian cycles matchings stars cliques Information lower bound n lg n (1 + o(1))( n

2 lg n)

(1 + o(1))n n Grebinski and Kucherov, 1998 Bouvel et al., 2005 2n lg n (1 + o(1))(n lg n) 2n 2n n lg n + cn (1 + o(1))( n

2 lg n)

n + 2 lg n n + lg n Chang et al., 2010c

Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models 2 3 / 2 6

Graph Reconstruction

Our strategies:

• Affine plane method (Tettelin et al., 1996).

Example: An affine plane of order 3 contains 9 points and 12 blocks of size 3. Each pair of points appear together in exactly

• ne block.

1 3 2 4 6 5 7 9 8

3 1 2 4 5 6 7 6 2 4 3 4 7 5 3 1 2 6 8 4 1 7 positive blocks

For n > 242, there exists a prime p such that n ≤ p2 ≤ 1.44n < 2n. (Nagura, 1952)

• Reconstruct a maximal matching first.

Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models 2 4 / 2 6

References

• F. H. Chang, H. Chang, and F. K. Hwang, Pooling designs for

clone library screening in the inhibitor complex model, J. Comb.

• Optim. (2010a) DOI 10.1007/s10878-009-9279-9.
• H. Chang, H. B. Chen, and H. L. Fu, Identification and

classification problems on pooling designs for inhibitor model, J.

• Comput. Biol. (2010b) to appear.
• H. Chang, H. B. Chen, H. L. Fu, and C. H. Shi, Reconstruction of

hidden graphs and threshold group testing, J. Comb. Optim. (2010c) DOI 10.1007/s10878-010-9291-0.

Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models 2 5 / 2 6

Thank you for your attention!

Speaker: Huilan Chang Advisor: Hung-Lin Fu Combinatorial Group Testing Problems on Various Models 2 6 / 2 6