135 research outputs found
A limit process for partial match queries in random quadtrees and -d trees
We consider the problem of recovering items matching a partially specified
pattern in multidimensional trees (quadtrees and -d trees). We assume the
traditional model where the data consist of independent and uniform points in
the unit square. For this model, in a structure on points, it is known that
the number of nodes to visit in order to report the items matching
a random query , independent and uniformly distributed on ,
satisfies , where and
are explicit constants. We develop an approach based on the analysis of
the cost of any fixed query , and give precise estimates
for the variance and limit distribution of the cost . Our results
permit us to describe a limit process for the costs as varies in
; one of the consequences is that ; this settles a question of
Devroye [Pers. Comm., 2000].Comment: Published in at http://dx.doi.org/10.1214/12-AAP912 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org). arXiv admin note: text
overlap with arXiv:1107.223
A limit field for orthogonal range searches in two-dimensional random point search trees
We consider the cost of general orthogonal range queries in random quadtrees.
The cost of a given query is encoded into a (random) function of four variables
which characterize the coordinates of two opposite corners of the query
rectangle. We prove that, when suitably shifted and rescaled, the random cost
function converges uniformly in probability towards a random field that is
characterized as the unique solution to a distributional fixed-point equation.
We also state similar results for -d trees. Our results imply for instance
that the worst case query satisfies the same asymptotic estimates as a typical
query, and thereby resolve an old question of Chanzy, Devroye and Zamora-Cura
[\emph{Acta Inf.}, 37:355--383, 2000]Comment: 24 pages, 8 figure
On the non-holonomic character of logarithms, powers, and the n-th prime function
We establish that the sequences formed by logarithms and by "fractional"
powers of integers, as well as the sequence of prime numbers, are
non-holonomic, thereby answering three open problems of Gerhold [Electronic
Journal of Combinatorics 11 (2004), R87]. Our proofs depend on basic complex
analysis, namely a conjunction of the Structure Theorem for singularities of
solutions to linear differential equations and of an Abelian theorem. A brief
discussion is offered regarding the scope of singularity-based methods and
several naturally occurring sequences are proved to be non-holonomic.Comment: 13 page
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