The Ehrenfeucht-Silberger Problem

We consider repetitions in words and solve a longstanding open problem about the relation between the period and the length of its longest unbordered factor. A word u is called bordered if there exists a proper prefix that is also a suffix of u, otherwise it is called unbordered. In 1979 Ehrenfeucht and Silberger raised the following problem: What is the maximum length of a word w, w.r.t. the length of its longest unbordered factor still allowing that is shorter than the period of w. We show that if w is longer than 7( - 1)/3 then = which gives the optimal asymtotic bound.

• Type: Conference
• Year: 2009
• Where: ICALP
• Authors: Stepan Holub, Dirk Nowotka
• Keywords: ICALP 2009, Longest Unbordered Factor, Longstanding Open Problem, Maximum Length, Theoretical Computer Science