Covering a Circular String with Substrings of Fixed Length

A nonempty circular string C(x) of length n is said to be covered by a set U k of strings each of xed length k n ii every position in C(x) lies within an occurrence of some string u 2 U k. In this paper we consider the problem of determining the minimum cardinality of a set U k which guarantees that every circular string C(x) of length n k can be covered. In particular, we show how, for any positive integer m, to choose the elements of U k so that, for suuciently large k, u k k?m , where u k = jU k j and is the size of the alphabet on which the strings are deened. The problem has application to DNA sequencing by hybridization using oligonucleotide probes.