Rotation of Periodic Strings and Short Superstrings

Abstract

This paper presents two simple approximation algorithms for the shortest superstring problem, with approximation ratios 2.67 and 2.596, improving the best previously published 2.75
approximation. The framework of our improved algorithms is similar to that of previous algorithms in the sense that they construct a superstring by computing some optimal cycle covers on the distance graph of the given strings, and
then break and merge the cycles to finally obtain a Hamiltonian path, but we make use of new bounds on the overlap between two strings. We prove that for each periodic semi-infinite string alpha = a1 a2 ... of period q, there exists an integer k, such that for any (finite) string s of period p which is inequivalent to alpha, the overlap between s and the rotation alpha[k] = ak ak+1 ... is at most p+ q/2. Moreover, if p<=q, then the overlap between s and alpha[k]
is not larger than 2/3 (p+q). In the previous shortest superstring algorithms p+q was used as the standard bound on overlap between two strings with periods p and q.