520 research outputs found
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
The dual tree of a recursive triangulation of the disk
In the recursive lamination of the disk, one tries to add chords one after
another at random; a chord is kept and inserted if it does not intersect any of
the previously inserted ones. Curien and Le Gall [Ann. Probab. 39 (2011)
2224-2270] have proved that the set of chords converges to a limit
triangulation of the disk encoded by a continuous process . Based
on a new approach resembling ideas from the so-called contraction method in
function spaces, we prove that, when properly rescaled, the planar dual of the
discrete lamination converges almost surely in the Gromov-Hausdorff sense to a
limit real tree , which is encoded by . This confirms
a conjecture of Curien and Le Gall.Comment: Published in at http://dx.doi.org/10.1214/13-AOP894 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
The combinatorics of the colliding bullets problem
The finite colliding bullets problem is the following simple problem:
consider a gun, whose barrel remains in a fixed direction; let be an i.i.d.\ family of random variables with uniform distribution on
; shoot bullets one after another at times , where the
th bullet has speed . When two bullets collide, they both annihilate.
We give the distribution of the number of surviving bullets, and in some
generalisation of this model. While the distribution is relatively simple (and
we found a number of bold claims online), our proof is surprisingly intricate
and mixes combinatorial and geometric arguments; we argue that any rigorous
argument must very likely be rather elaborate.Comment: 29 page
Longest path distance in random circuits
We study distance properties of a general class of random directed acyclic
graphs (DAGs). In a DAG, many natural notions of distance are possible, for
there exists multiple paths between pairs of nodes. The distance of interest
for circuits is the maximum length of a path between two nodes. We give laws of
large numbers for the typical depth (distance to the root) and the minimum
depth in a random DAG. This completes the study of natural distances in random
DAGs initiated (in the uniform case) by Devroye and Janson (2009+). We also
obtain large deviation bounds for the minimum of a branching random walk with
constant branching, which can be seen as a simplified version of our main
result.Comment: 21 pages, 2 figure
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
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