95,970 research outputs found
Combinatorial Optimization of Subsequence Patterns in Words
Packing patterns in words concerns finding a word with the maximum number of a prescribed pattern. The majority of the work done thus far is on packing patterns into permutations. In 2002, Albert, Atkinson, Handley, Holton and Stromquist showed that there always exists a layered permutation containing the maximum number of a layered pattern among all permutations of length n. Consequently, the packing density for all but two (up to equivalence) permutation patterns up to length 4 can be obtained. In this thesis we consider the analogous question for colored patterns and permutations. By introducing the concept of colored blocks we characterize the optimal permutations with the maximum number of a given colored pattern when it contains at most three colored blocks. As examples, we apply this characterization to find the optimal permutations of various colored patterns and subsequently obtain their corresponding packing densities
On Packing Densities of Set Partitions
We study packing densities for set partitions, which is a generalization of
packing words. We use results from the literature about packing densities for
permutations and words to provide packing densities for set partitions. These
results give us most of the packing densities for partitions of the set
. In the final section we determine the packing density of the set
partition .Comment: 12 pages, to appear in the Permutation Patterns edition of the
Australasian Journal of Combinatoric
Approximating Bin Packing within O(log OPT * log log OPT) bins
For bin packing, the input consists of n items with sizes s_1,...,s_n in
[0,1] which have to be assigned to a minimum number of bins of size 1. The
seminal Karmarkar-Karp algorithm from '82 produces a solution with at most OPT
+ O(log^2 OPT) bins.
We provide the first improvement in now 3 decades and show that one can find
a solution of cost OPT + O(log OPT * log log OPT) in polynomial time. This is
achieved by rounding a fractional solution to the Gilmore-Gomory LP relaxation
using the Entropy Method from discrepancy theory. The result is constructive
via algorithms of Bansal and Lovett-Meka
Improving bounds on packing densities of 4-point permutations
We consolidate what is currently known about packing densities of 4-point
permutations and in the process improve the lower bounds for the packing
densities of 1324 and 1342. We also provide rigorous upper bounds for the
packing densities of 1324, 1342, and 2413. All our bounds are within
of the true packing densities. Together with the known bounds, this gives us a
fairly complete picture of all 4-point packing densities. We also provide new
upper bounds for several small permutations of length at least five. Our main
tool for the upper bounds is the framework of flag algebras introduced by
Razborov in 2007.Comment: journal style, 18 page
Avian photoreceptor patterns represent a disordered hyperuniform solution to a multiscale packing problem
Optimal spatial sampling of light rigorously requires that identical
photoreceptors be arranged in perfectly regular arrays in two dimensions.
Examples of such perfect arrays in nature include the compound eyes of insects
and the nearly crystalline photoreceptor patterns of some fish and reptiles.
Birds are highly visual animals with five different cone photoreceptor
subtypes, yet their photoreceptor patterns are not perfectly regular. By
analyzing the chicken cone photoreceptor system consisting of five different
cell types using a variety of sensitive microstructural descriptors, we find
that the disordered photoreceptor patterns are ``hyperuniform'' (exhibiting
vanishing infinite-wavelength density fluctuations), a property that had
heretofore been identified in a unique subset of physical systems, but had
never been observed in any living organism. Remarkably, the photoreceptor
patterns of both the total population and the individual cell types are
simultaneously hyperuniform. We term such patterns ``multi-hyperuniform''
because multiple distinct subsets of the overall point pattern are themselves
hyperuniform. We have devised a unique multiscale cell packing model in two
dimensions that suggests that photoreceptor types interact with both short- and
long-ranged repulsive forces and that the resultant competition between the
types gives rise to the aforementioned singular spatial features characterizing
the system, including multi-hyperuniformity.Comment: 31 pages, 12 figure
MaTrEx: the DCU machine translation system for IWSLT 2007
In this paper, we give a description of the machine translation system developed at DCU that was used for our second participation in the evaluation campaign of the International Workshop on Spoken Language Translation (IWSLT 2007). In this participation, we focus on some new methods to improve system quality. Specifically, we try our word packing technique for different language pairs, we smooth our translation tables with out-of-domain word translations for the Arabic–English and Chinese–English tasks in order to solve the high number of out of vocabulary items, and finally we deploy a translation-based model for case and punctuation restoration
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