Tracing a Single User Joint work with Noga Alon

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.

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.

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.

Molecular Biology (cont.) In some applications, we are interested in finding one sequence that contains the short subsequence, rather than all of them.

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.

Non-Adaptive Tests a1a1 a2a2....amam T1T T2T TnTn

r-SUT Definition Definition: Let F be a family of subsets of [n] = {1,…,n}. F is called r-single-user-tracing superimposed (r-SUT) if  F 1,…, F k  F with | F i |  r, In other words, given the union of up to r sets from F, one can identify at least one of those sets.

Communication Suppose that m users share a common channel. Each user is associated with a vector in {0,1} n. 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.

Maximal r-SUT Families Let g(n,r) denote the maximum size of an r-SUT family of subsets of [n]. Let R g (r) = lim sup n  log g(n,r) / n. Csűrös and Ruszinkó: There exist constants c 1,c 2 >0 s.t.. Our result: R g (r) =  (1/r) (and hence  (1/r)).

Lower Bound Let m = 2 n/(20r). We construct a family F ={F 1,…,F m } of subsets of [n] at random as follows:  1 ≤ i ≤ m and 1 ≤ j ≤ n independently, put j in F i with probability 1/r.

Lower Bound (cont.) We show that F is r-SUT with positive probability. We say a configuration of F 1,…, F k  F with | F i |  r and is bad if all the unions are equal. We show that with positive probability there are no bad configurations.

Lower Bound (cont.) We show that with probability > ½ no small configuration is bad, and that with probability > ½ no large configuration is bad. Therefore, with positive probability there is no bad configuration.

Small Configurations Proposition: With probability > ½ the following holds:  s<2r and distinct A 1,…,A s  F,  j  [n] that belongs to exactly one of the sets A 1,…,A s. Corollary: With probability > ½ no small configuration is bad.

Small Configurations (cont.) A1A1 A3A3 A4A4 A6A6 A8A8 A7A7 A2A2 A5A5 A9A9

Large Configurations Proposition: With probability > ½ the following holds. For all distinct A 1,…,A r,B 1,…,B r  F, Corollary: With probability > ½ no large configuration is bad.

Large Configurations (cont.) B1B1 B3B3 A2A2 A1A1 B2B2 A3A3 B1B1 B2B2 B3B3 AiAi

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 (MUT k (r)) if given the union of any ℓ  r sets from F, one can identify at least min(k, ℓ ) of them.

Tracing Multiple Users (cont.) Let h(n,r,k) denote the maximum size of a MUT k (r) family of subsets of [n]. Let R h (r,k) = lim sup n  log h(n,r,k) / n. We have shown that there are constants c 1,c 2,c 3,c 4 >0 s.t..

Open Problems We have shown that R g (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  R g (2)  1/2+o(1). Follows from a result of Coppersmith and Shearer By a careful analysis of the random construction

Open Problems (cont.) We show how to construct an r-SUT family in time m O(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