Affiliation:
1. University of Turku, Turku, Finland
2. University of Stuttgart, Stuttgart, Germany
Abstract
The relationship between the length of a word and the maximum length of its unbordered factors is investigated in this article. Consider a finite word
w
of length
n
. We call a word
bordered
if it has a proper prefix, which is also a suffix of that word. Let μ(
w
) denote the maximum length of all unbordered factors of
w
, and let ∂(
w
) denote the period of
w
. Clearly, μ(
w
) ≤ ∂(
w
).
We establish that μ(
w
) = ∂(
w
), if
w
has an unbordered prefix of length μ(
w
) and
n
≥ 2μ(
w
) − 1. This bound is tight and solves the stronger version of an old conjecture by Duval [1983]. It follows from this result that, in general,
n
≥ 3μ(
w
) − 3 implies μ(
w
) = ∂(
w
), which gives an improved bound for the question raised by Ehrenfeucht and Silberger in 1979.
Publisher
Association for Computing Machinery (ACM)
Subject
Artificial Intelligence,Hardware and Architecture,Information Systems,Control and Systems Engineering,Software
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