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 fixed length k≤n iff every position in C(x) lies within an occurrence of some string u∈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 sufficiently large k, u k ≈σ k–m , where u k =|U k | and σ is the size of the alphabet on which the strings are defined. The problem has application to DNA sequencing by hybridization using oligonucleotide probes.