Amir Rothschild

Explicit Nonadaptive Combinatorial Group Testing Schemes

Group testing is a long studied problem in combinatorics: A small set of r ill people should be identified out of the whole (n people) by using only queries (tests) of the form “Does set X contain an ill human?” In this paper we provide an explicit construction of a testing scheme which is better (smaller) than any known explicit construction. This scheme has Θ(min[r2 lnn,n]) tests which is as many as the best nonexplicit schemes have. In our construction, we use a fact that may have a value by its own right: Linear error-correction codes with parameters [m,k,δm]q meeting the Gilbert-Varshamov bound may be constructed quite efficiently, in Θ(qkm) time.

Explicit Non-adaptive Combinatorial Group Testing Schemes

Group testing is a long studied problem in combinatorics: A small set of <i>r</i> ill people should be identified out of the whole (<i>n</i> people) by using only queries (tests) of the form “Does set X contain an ill human?” In this paper we provide an explicit construction of a testing scheme which is better (smaller) than any known explicit construction. This scheme has Θ(min[<i>r</i>² ln<i>n</i>,<i>n</i>]) tests which is as many as the best nonexplicit schemes have. In our construction, we use a fact that may have a value by its own right: Linear error-correction codes with parameters [<i>m</i>,<i>k</i>,δ<i>m</i>]<i>q</i> meeting the Gilbert-Varshamov bound may be constructed quite efficiently, in Θ(<i>qkm</i>) time.

Mismatch Sampling

We consider the well known problem of pattern matching under the Hamming distance. Previous approaches have shown how to count the number of mismatches efficiently, especially when a bound is known for the maximum Hamming distance. Our interest is different in that we wish collect a random sample of mismatches of fixed size at each position in the text. Given a pattern p of length m and a text t of length n , we show how to sample with high probability c mismatches where possible from every alignment of p and t in O ((c + logn )(n + m logm )logm ) time. Further, we guarantee that the mismatches are sampled uniformly and can therefore be seen as representative of the types of mismatches that occur.