COVERING A CIRCULAR STRING WITH SUBSTRINGS OF FIXED LENGTH

Abstract

A nonempty circular string C(x) of length n is said to be covered by a set Uk of strings each of fixed length k≤n iff every position in C(x) lies within an occurrence of some string u∈Uk. In this paper we consider the problem of determining the minimum cardinality of a set Uk 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 Uk so that, for sufficiently large k, uk≈σk–m, where uk=|Uk| 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.