Periods, Partial Words, and a Result of Guibas and Odlyzko

Abstract

A well known and unexpected result of Guibas and Odlyzko states that the set of all periods of a word is independent of the alphabet size (alphabets with one symbol are excluded here). More specifically, for every word u, there exists a word v over the alphabet {0,1} such that u and v have the same length and the same set of periods. Recently, Blanchet-Sadri and Chriscoe extended this fundamental result to words with one “do not know” symbol also called partial words with one hole. They showed that for every partial word u with one hole, there exists a partial word v with at most one hole over the alphabet {0,1} such that u and v have the same length, the same set of periods, and the same set of weak periods. In this talk, we (Blanchet-Sadri and Shirey) extend this result further to all partial words. Given such a partial word u, our proof provides an algorithm to compute a partial word v over {0,1} sharing the same length and the same sets of periods and weak periods as u. A World Wide Web server interface at http://www.uncg.edu/mat/bintwo/ has been established for automated use of the program.