Tracing a Single User

1
Tracing a Single User

• Joint work with Noga Alon

2
Group Testing

• Dorfman raised the following problem in 1941
• All American inductees gave blood samples, that
  were tested for the presence of a syphilitic
  antigen.
• We assume that the number of infected blood
  samples r is much smaller than the total number
  m.
• Testing each sample separately requires m tests.

3
Group Testing (cont.)

• Instead, one can test pools that contain blood
  from a set of samples.
• If the outcome is negative none of the samples
  in the pool is infected.
• Otherwise, the pool contains at least one
  infected sample, which can be determined by
  further tests.
• This way, less than m tests are needed.

4
Molecular Biology

• In recent years this problem has gained
  popularity again in the field of molecular
  biology.
• For example, when we are given a large set of DNA
  sequences, and we look for all those that contain
  a specific short subsequence.
• We can use a method similar to that of the blood
  testing problem.

5
Molecular Biology (cont.)

• In some applications, we are interested in
  finding one sequence that contains the short
  subsequence, rather than all of them.

6
Parallelization

• Often, we would prefer to conduct all experiments
  simultaneously, even at the cost of increasing
  the number of experiments.
• Thus, we need our tests to be non-adaptive, i.e.
  the pool tested in each experiment is independent
  of the outcomes of other experiments.

7
Non-Adaptive Tests
a1 a2 . . . . am
T1 0 1 1 1 1 0 0
T2 1 1 1 1 0 0 1
. 0 1 0 0 1 1 0
. 1 0 1 0 0 0 0
. 1 0 0 1 1 0 1
Tn 0 0 1 0 0 1 1
1
1
1
0
1
0
8
r-SUT Definition

• Definition Let F be a family of subsets ofn
  1,,n. F is called r-single-user-tracing
  superimposed (r-SUT) if ?F1,,Fk?F with Fi?r,
• In other words, given the union of up to r sets
  from F, one can identify at least one of those
  sets.

9
Communication

• Suppose that m users share a common channel.
• Each user is associated with a vector in 0,1n.
• All active users transmit their vectors, and a
  single receiver gets the OR of all transmitted
  vectors.
• Given that at most r users are active
  simultaneously, we would like the receiver to be
  able to identify at least one of them.

10
Maximal r-SUT Families

• Let g(n,r) denote the maximum size of anr-SUT
  family of subsets of n.
• Let Rg(r) lim sup n?? log g(n,r) / n.
• Csurös and Ruszinkó There exist constants
  c1,c2gt0 s.t.
• .
• Our result Rg(r) ??(1/r) (and hence ?(1/r)).

11
Lower Bound

• Let m 2n/(20r).
• We construct a family FF1,,Fm of subsets of
  n at random as follows
• ? 1 i m and 1 j n independently, put j in
  Fi with probability 1/r.

12
Lower Bound (cont.)

• We show that F is r-SUT with positive
  probability.
• We say a configuration of F1,,Fk?F with Fi?r
  and is bad if all the unions
  are equal.
• We show that with positive probability there are
  no bad configurations.

13
Lower Bound (cont.)

• We show that with probability gt ½ no small
  configuration is bad, and that with probability gt
  ½ no large configuration is bad.
• Therefore, with positive probability there is no
  bad configuration.

14
Small Configurations

• Proposition With probability gt ½ the following
  holds?slt2r and distinct A1,,As?F, ?j?n that
  belongs to exactly one of the sets A1,,As.
• Corollary With probability gt ½ no small
  configuration is bad.

15
Small Configurations (cont.)
A5
A7
A2
A1
A8
A4
A9
A3
A6
16
Large Configurations

• Proposition With probability gt ½ the following
  holds. For all distinct A1,,Ar,B1,,Br?F,
• Corollary With probability gt ½ no large
  configuration is bad.

17
Large Configurations (cont.)
B3
B2
B1
B1
Ai
B3
A1
A3
B2
A2
18
Tracing Multiple Users

• Recently, Laczay and Ruszinkó have introduced the
  following generalization of r-SUT families.
• For integers n, r?2, and 1?k?r, a family F of
  subsets of n is called k-out-of-r
  multiple-user-tracing superimposed (MUTk(r)) if
  given the union of any l?r sets from F, one can
  identify at least min(k,l) of them.

19
Tracing Multiple Users (cont.)

• Let h(n,r,k) denote the maximum size of aMUTk(r)
  family of subsets of n.
• Let Rh(r,k) lim sup n?? log h(n,r,k) / n.
• We have shown that there are constants
  c1,c2,c3,c4gt0 s.t.
  .

20
Open Problems

• We have shown that Rg(r) ?(1/r), but the
  question of finding the exact constant is still
  open.
• This problem is open even for the case of r 2.
• 1/3 ? Rg(2) ? 1/2o(1).

Follows from a result of Coppersmith and Shearer
By a careful analysis of the random construction
21
Open Problems (cont.)

• We show how to construct an r-SUT family in time
  mO(r), where m is the size of the family.It
  would be interesting to find explicit
  constructions for all r.
• There are other related problems for which there
  are still gaps between lower and upper bounds
• Multiple-user tracing families
• r-superimposed families
• Disjointly r-superimposed families
• Graph identifying codes