225 research outputs found
Exactly Solvable Balanced Tenable Urns with Random Entries via the Analytic Methodology
This paper develops an analytic theory for the study of some Polya urns with
random rules. The idea is to extend the isomorphism theorem in Flajolet et al.
(2006), which connects deterministic balanced urns to a differential system for
the generating function. The methodology is based upon adaptation of operators
and use of a weighted probability generating function. Systems of differential
equations are developed, and when they can be solved, they lead to
characterization of the exact distributions underlying the urn evolution. We
give a few illustrative examples.Comment: 23rd International Meeting on Probabilistic, Combinatorial, and
Asymptotic Methods for the Analysis of Algorithms (AofA'12), Montreal :
Canada (2012
The oscillatory distribution of distances in random tries
We investigate \Delta_n, the distance between randomly selected pairs of
nodes among n keys in a random trie, which is a kind of digital tree.
Analytical techniques, such as the Mellin transform and an excursion between
poissonization and depoissonization, capture small fluctuations in the mean and
variance of these random distances. The mean increases logarithmically in the
number of keys, but curiously enough the variance remains O(1), as n\to\infty.
It is demonstrated that the centered random variable
\Delta_n^*=\Delta_n-\lfloor2\log_2n\rfloor does not have a limit distribution,
but rather oscillates between two distributions.Comment: Published at http://dx.doi.org/10.1214/105051605000000106 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Analysis of Quickselect under Yaroslavskiy's Dual-Pivoting Algorithm
There is excitement within the algorithms community about a new partitioning
method introduced by Yaroslavskiy. This algorithm renders Quicksort slightly
faster than the case when it runs under classic partitioning methods. We show
that this improved performance in Quicksort is not sustained in Quickselect; a
variant of Quicksort for finding order statistics. We investigate the number of
comparisons made by Quickselect to find a key with a randomly selected rank
under Yaroslavskiy's algorithm. This grand averaging is a smoothing operator
over all individual distributions for specific fixed order statistics. We give
the exact grand average. The grand distribution of the number of comparison
(when suitably scaled) is given as the fixed-point solution of a distributional
equation of a contraction in the Zolotarev metric space. Our investigation
shows that Quickselect under older partitioning methods slightly outperforms
Quickselect under Yaroslavskiy's algorithm, for an order statistic of a random
rank. Similar results are obtained for extremal order statistics, where again
we find the exact average, and the distribution for the number of comparisons
(when suitably scaled). Both limiting distributions are of perpetuities (a sum
of products of independent mixed continuous random variables).Comment: full version with appendices; otherwise identical to Algorithmica
versio
Degrees in random -ary hooking networks
The theme in this paper is a composition of random graphs and P\'olya urns.
The random graphs are generated through a small structure called the seed. Via
P\'olya urns, we study the asymptotic degree structure in a random -ary
hooking network and identify strong laws. We further upgrade the result to
second-order asymptotics in the form of multivariate Gaussian limit laws. We
give a few concrete examples and explore some properties with a full
representation of the Gaussian limit in each case. The asymptotic covariance
matrix associated with the P\'olya urn is obtained by a new method that
originated in this paper and is reported in [25].Comment: 21 pages, 5 figure
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