529 research outputs found
LRM-Trees: Compressed Indices, Adaptive Sorting, and Compressed Permutations
LRM-Trees are an elegant way to partition a sequence of values into sorted
consecutive blocks, and to express the relative position of the first element
of each block within a previous block. They were used to encode ordinal trees
and to index integer arrays in order to support range minimum queries on them.
We describe how they yield many other convenient results in a variety of areas,
from data structures to algorithms: some compressed succinct indices for range
minimum queries; a new adaptive sorting algorithm; and a compressed succinct
data structure for permutations supporting direct and indirect application in
time all the shortest as the permutation is compressible.Comment: 13 pages, 1 figur
Forbidden patterns and shift systems
The scope of this paper is two-fold. First, to present to the researchers in
combinatorics an interesting implementation of permutations avoiding
generalized patterns in the framework of discrete-time dynamical systems.
Indeed, the orbits generated by piecewise monotone maps on one-dimensional
intervals have forbidden order patterns, i.e., order patterns that do not occur
in any orbit. The allowed patterns are then those patterns avoiding the
so-called forbidden root patterns and their shifted patterns. The second scope
is to study forbidden patterns in shift systems, which are universal models in
information theory, dynamical systems and stochastic processes. Due to its
simple structure, shift systems are accessible to a more detailed analysis and,
at the same time, exhibit all important properties of low-dimensional chaotic
dynamical systems (e.g., sensitivity to initial conditions, strong mixing and a
dense set of periodic points), allowing to export the results to other
dynamical systems via order-isomorphisms.Comment: 21 pages, expanded Section 5 and corrected Propositions 3 and
Continuity of Limit Surfaces of Locally Uniform Random Permutations
A locally uniform random permutation is generated by sampling points
independently from some absolutely continuous distribution on the plane
and interpreting them as a permutation by the rule that maps to if the
th point from the left is the th point from below. As tends to
infinity, decreasing subsequences in the permutation will appear as curves in
the plane, and by interpreting these as level curves, a union of decreasing
subsequences gives rise to a surface. In a recent paper by the author it was
shown that, for any , under the correct scaling as tends to
infinity, the surface of the largest union of
decreasing subsequences approaches a limit in the sense that it will come close
to a maximizer of a specific variational integral (and, under reasonable
assumptions, that the maximizer is essentially unique). In the present paper we
show that there exists a continuous maximizer, provided that has bounded
density and support.
The key ingredient in the proof is a new theorem about real functions of two
variables that are increasing in both variables: We show that, for any constant
, any such function can be made continuous without increasing the diameter
of its image or decreasing anywhere the product of its partial derivatives
clipped by , that is the minimum of the product and
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