8,664 research outputs found
Universal Layered Permutations
We establish an exact formula for the length of the shortest permutation containing all layered permutations of length n, proving a conjecture of Gray
Some open problems on permutation patterns
This is a brief survey of some open problems on permutation patterns, with an
emphasis on subjects not covered in the recent book by Kitaev, \emph{Patterns
in Permutations and words}. I first survey recent developments on the
enumeration and asymptotics of the pattern 1324, the last pattern of length 4
whose asymptotic growth is unknown, and related issues such as upper bounds for
the number of avoiders of any pattern of length for any given . Other
subjects treated are the M\"obius function, topological properties and other
algebraic aspects of the poset of permutations, ordered by containment, and
also the study of growth rates of permutation classes, which are containment
closed subsets of this poset.Comment: 20 pages. Related to upcoming talk at the British Combinatorial
Conference 2013. To appear in London Mathematical Society Lecture Note Serie
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
One-way permutations, computational asymmetry and distortion
Computational asymmetry, i.e., the discrepancy between the complexity of
transformations and the complexity of their inverses, is at the core of one-way
transformations. We introduce a computational asymmetry function that measures
the amount of one-wayness of permutations. We also introduce the word-length
asymmetry function for groups, which is an algebraic analogue of computational
asymmetry. We relate boolean circuits to words in a Thompson monoid, over a
fixed generating set, in such a way that circuit size is equal to word-length.
Moreover, boolean circuits have a representation in terms of elements of a
Thompson group, in such a way that circuit size is polynomially equivalent to
word-length. We show that circuits built with gates that are not constrained to
have fixed-length inputs and outputs, are at most quadratically more compact
than circuits built from traditional gates (with fixed-length inputs and
outputs). Finally, we show that the computational asymmetry function is closely
related to certain distortion functions: The computational asymmetry function
is polynomially equivalent to the distortion of the path length in Schreier
graphs of certain Thompson groups, compared to the path length in Cayley graphs
of certain Thompson monoids. We also show that the results of Razborov and
others on monotone circuit complexity lead to exponential lower bounds on
certain distortions.Comment: 33 page
Layered Label Propagation: A MultiResolution Coordinate-Free Ordering for Compressing Social Networks
We continue the line of research on graph compression started with WebGraph,
but we move our focus to the compression of social networks in a proper sense
(e.g., LiveJournal): the approaches that have been used for a long time to
compress web graphs rely on a specific ordering of the nodes (lexicographical
URL ordering) whose extension to general social networks is not trivial. In
this paper, we propose a solution that mixes clusterings and orders, and devise
a new algorithm, called Layered Label Propagation, that builds on previous work
on scalable clustering and can be used to reorder very large graphs (billions
of nodes). Our implementation uses overdecomposition to perform aggressively on
multi-core architecture, making it possible to reorder graphs of more than 600
millions nodes in a few hours. Experiments performed on a wide array of web
graphs and social networks show that combining the order produced by the
proposed algorithm with the WebGraph compression framework provides a major
increase in compression with respect to all currently known techniques, both on
web graphs and on social networks. These improvements make it possible to
analyse in main memory significantly larger graphs
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