4,233 research outputs found
On the maximal sum of exponents of runs in a string
A run is an inclusion maximal occurrence in a string (as a subinterval) of a
repetition with a period such that . The exponent of a run
is defined as and is . We show new bounds on the maximal sum of
exponents of runs in a string of length . Our upper bound of is
better than the best previously known proven bound of by Crochemore &
Ilie (2008). The lower bound of , obtained using a family of binary
words, contradicts the conjecture of Kolpakov & Kucherov (1999) that the
maximal sum of exponents of runs in a string of length is smaller than Comment: 7 pages, 1 figur
DFAs and PFAs with Long Shortest Synchronizing Word Length
It was conjectured by \v{C}ern\'y in 1964, that a synchronizing DFA on
states always has a shortest synchronizing word of length at most ,
and he gave a sequence of DFAs for which this bound is reached. Until now a
full analysis of all DFAs reaching this bound was only given for ,
and with bounds on the number of symbols for . Here we give the full
analysis for , without bounds on the number of symbols.
For PFAs the bound is much higher. For we do a similar analysis as
for DFAs and find the maximal shortest synchronizing word lengths, exceeding
for . For arbitrary n we give a construction of a PFA on
three symbols with exponential shortest synchronizing word length, giving
significantly better bounds than earlier exponential constructions. We give a
transformation of this PFA to a PFA on two symbols keeping exponential shortest
synchronizing word length, yielding a better bound than applying a similar
known transformation.Comment: 16 pages, 2 figures source code adde
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