9,559 research outputs found
Binary Patterns in Binary Cube-Free Words: Avoidability and Growth
The avoidability of binary patterns by binary cube-free words is investigated
and the exact bound between unavoidable and avoidable patterns is found. All
avoidable patterns are shown to be D0L-avoidable. For avoidable patterns, the
growth rates of the avoiding languages are studied. All such languages, except
for the overlap-free language, are proved to have exponential growth. The exact
growth rates of languages avoiding minimal avoidable patterns are approximated
through computer-assisted upper bounds. Finally, a new example of a
pattern-avoiding language of polynomial growth is given.Comment: 18 pages, 2 tables; submitted to RAIRO TIA (Special issue of Mons
Days 2012
Avoidability index for binary patterns with reversal
For every pattern over the alphabet , we specify the
least such that is -avoidable.Comment: 15 pages, 1 figur
Tower-type bounds for unavoidable patterns in words
A word is said to contain the pattern if there is a way to substitute
a nonempty word for each letter in so that the resulting word is a subword
of . Bean, Ehrenfeucht and McNulty and, independently, Zimin characterised
the patterns which are unavoidable, in the sense that any sufficiently long
word over a fixed alphabet contains . Zimin's characterisation says that a
pattern is unavoidable if and only if it is contained in a Zimin word, where
the Zimin words are defined by and . We
study the quantitative aspects of this theorem, obtaining essentially tight
tower-type bounds for the function , the least integer such that any
word of length over an alphabet of size contains . When , the first non-trivial case, we determine up to a constant factor,
showing that .Comment: 17 page
Counting Houses of Pareto Optimal Matchings in the House Allocation Problem
Let with and be two sets. We assume that every
element has a reference list over all elements from . We call an
injective mapping from to a matching. A blocking coalition of
is a subset of such that there exists a matching that
differs from only on elements of , and every element of
improves in , compared to according to its preference list. If
there exists no blocking coalition, we call the matching an exchange
stable matching (ESM). An element is reachable if there exists an
exchange stable matching using . The set of all reachable elements is
denoted by . We show This is
asymptotically tight. A set is reachable (respectively exactly
reachable) if there exists an exchange stable matching whose image
contains as a subset (respectively equals ). We give bounds for the
number of exactly reachable sets. We find that our results hold in the more
general setting of multi-matchings, when each element of is matched
with elements of instead of just one. Further, we give complexity
results and algorithms for corresponding algorithmic questions. Finally, we
characterize unavoidable elements, i.e., elements of that are used by all
ESM's. This yields efficient algorithms to determine all unavoidable elements.Comment: 24 pages 2 Figures revise
Avoiding Patterns in the Abelian Sense
We classify all 3 letter patterns that are avoidable in the abelian sense. A short list of four letter patterns for which abelian avoidance is undecided is given. Using a generalization of Zimin words we deduce some properties of ω-words avoiding these patterns.Research of both authors supported by NSERC Operating Grants.https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/avoiding-patterns-in-the-abelian-sense/42148B0781A38A6618A537AAD7D39B8
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