4,431 research outputs found
Regenerative tree growth: Binary self-similar continuum random trees and Poisson--Dirichlet compositions
We use a natural ordered extension of the Chinese Restaurant Process to grow
a two-parameter family of binary self-similar continuum fragmentation trees. We
provide an explicit embedding of Ford's sequence of alpha model trees in the
continuum tree which we identified in a previous article as a distributional
scaling limit of Ford's trees. In general, the Markov branching trees induced
by the two-parameter growth rule are not sampling consistent, so the existence
of compact limiting trees cannot be deduced from previous work on the sampling
consistent case. We develop here a new approach to establish such limits, based
on regenerative interval partitions and the urn-model description of sampling
from Dirichlet random distributions.Comment: Published in at http://dx.doi.org/10.1214/08-AOP445 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Optimization of Tree Modes for Parallel Hash Functions: A Case Study
This paper focuses on parallel hash functions based on tree modes of
operation for an inner Variable-Input-Length function. This inner function can
be either a single-block-length (SBL) and prefix-free MD hash function, or a
sponge-based hash function. We discuss the various forms of optimality that can
be obtained when designing parallel hash functions based on trees where all
leaves have the same depth. The first result is a scheme which optimizes the
tree topology in order to decrease the running time. Then, without affecting
the optimal running time we show that we can slightly change the corresponding
tree topology so as to minimize the number of required processors as well.
Consequently, the resulting scheme decreases in the first place the running
time and in the second place the number of required processors.Comment: Preprint version. Added citations, IEEE Transactions on Computers,
201
Random recursive trees and the Bolthausen-Sznitman coalescent
We describe a representation of the Bolthausen-Sznitman coalescent in terms
of the cutting of random recursive trees. Using this representation, we prove
results concerning the final collision of the coalescent restricted to [n]: we
show that the distribution of the number of blocks involved in the final
collision converges as n tends to infinity, and obtain a scaling law for the
sizes of these blocks. We also consider the discrete-time Markov chain giving
the number of blocks after each collision of the coalescent restricted to [n];
we show that the transition probabilities of the time-reversal of this Markov
chain have limits as n tends to infinity. These results can be interpreted as
describing a ``post-gelation'' phase of the Bolthausen-Sznitman coalescent, in
which a giant cluster containing almost all of the mass has already formed and
the remaining small blocks are being absorbed.Comment: 28 pages, 2 figures. Revised version with minor alterations. To
appear in Electron. J. Proba
Restricted exchangeable partitions and embedding of associated hierarchies in continuum random trees
We introduce the notion of a restricted exchangeable partition of
. We obtain integral representations, consider associated
fragmentations, embeddings into continuum random trees and convergence to such
limit trees. In particular, we deduce from the general theory developed here a
limit result conjectured previously for Ford's alpha model and its extension,
the alpha-gamma model, where restricted exchangeability arises naturally.Comment: 35 pages, 5 figure
Optimal Hierarchical Layouts for Cache-Oblivious Search Trees
This paper proposes a general framework for generating cache-oblivious
layouts for binary search trees. A cache-oblivious layout attempts to minimize
cache misses on any hierarchical memory, independent of the number of memory
levels and attributes at each level such as cache size, line size, and
replacement policy. Recursively partitioning a tree into contiguous subtrees
and prescribing an ordering amongst the subtrees, Hierarchical Layouts
generalize many commonly used layouts for trees such as in-order, pre-order and
breadth-first. They also generalize the various flavors of the van Emde Boas
layout, which have previously been used as cache-oblivious layouts.
Hierarchical Layouts thus unify all previous attempts at deriving layouts for
search trees.
The paper then derives a new locality measure (the Weighted Edge Product)
that mimics the probability of cache misses at multiple levels, and shows that
layouts that reduce this measure perform better. We analyze the various degrees
of freedom in the construction of Hierarchical Layouts, and investigate the
relative effect of each of these decisions in the construction of
cache-oblivious layouts. Optimizing the Weighted Edge Product for complete
binary search trees, we introduce the MinWEP layout, and show that it
outperforms previously used cache-oblivious layouts by almost 20%.Comment: Extended version with proofs added to the appendi
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