49,301 research outputs found
Fringe trees, Crump-Mode-Jagers branching processes and -ary search trees
This survey studies asymptotics of random fringe trees and extended fringe
trees in random trees that can be constructed as family trees of a
Crump-Mode-Jagers branching process, stopped at a suitable time. This includes
random recursive trees, preferential attachment trees, fragmentation trees,
binary search trees and (more generally) -ary search trees, as well as some
other classes of random trees.
We begin with general results, mainly due to Aldous (1991) and Jagers and
Nerman (1984). The general results are applied to fringe trees and extended
fringe trees for several particular types of random trees, where the theory is
developed in detail. In particular, we consider fringe trees of -ary search
trees in detail; this seems to be new.
Various applications are given, including degree distribution, protected
nodes and maximal clades for various types of random trees. Again, we emphasise
results for -ary search trees, and give for example new results on protected
nodes in -ary search trees.
A separate section surveys results on height, saturation level, typical depth
and total path length, due to Devroye (1986), Biggins (1995, 1997) and others.
This survey contains well-known basic results together with some additional
general results as well as many new examples and applications for various
classes of random trees
On weighted depths in random binary search trees
Following the model introduced by Aguech, Lasmar and Mahmoud [Probab. Engrg.
Inform. Sci. 21 (2007) 133-141], the weighted depth of a node in a labelled
rooted tree is the sum of all labels on the path connecting the node to the
root. We analyze weighted depths of nodes with given labels, the last inserted
node, nodes ordered as visited by the depth first search process, the weighted
path length and the weighted Wiener index in a random binary search tree. We
establish three regimes of nodes depending on whether the second order
behaviour of their weighted depths follows from fluctuations of the keys on the
path, the depth of the nodes, or both. Finally, we investigate a random
distribution function on the unit interval arising as scaling limit for
weighted depths of nodes with at most one child
A functional limit theorem for the profile of -ary trees
In this paper we prove a functional limit theorem for the weighted profile of
a -ary tree. For the proof we use classical martingales connected to
branching Markov processes and a generalized version of the profile-polynomial
martingale. By embedding, choosing weights and a branch factor in a right way,
we finally rediscover the profiles of some well-known discrete time trees.Comment: Published in at http://dx.doi.org/10.1214/09-AAP640 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Understanding Search Trees via Statistical Physics
We study the random m-ary search tree model (where m stands for the number of
branches of a search tree), an important problem for data storage in computer
science, using a variety of statistical physics techniques that allow us to
obtain exact asymptotic results. In particular, we show that the probability
distributions of extreme observables associated with a random search tree such
as the height and the balanced height of a tree have a traveling front
structure. In addition, the variance of the number of nodes needed to store a
data string of a given size N is shown to undergo a striking phase transition
at a critical value of the branching ratio m_c=26. We identify the mechanism of
this phase transition, show that it is generic and occurs in various other
problems as well. New results are obtained when each element of the data string
is a D-dimensional vector. We show that this problem also has a phase
transition at a critical dimension, D_c= \pi/\sin^{-1}(1/\sqrt{8})=8.69363...Comment: 11 pages, 8 .eps figures included. Invited contribution to
STATPHYS-22 held at Bangalore (India) in July 2004. To appear in the
proceedings of STATPHYS-2
DeltaTree: A Practical Locality-aware Concurrent Search Tree
As other fundamental programming abstractions in energy-efficient computing,
search trees are expected to support both high parallelism and data locality.
However, existing highly-concurrent search trees such as red-black trees and
AVL trees do not consider data locality while existing locality-aware search
trees such as those based on the van Emde Boas layout (vEB-based trees), poorly
support concurrent (update) operations.
This paper presents DeltaTree, a practical locality-aware concurrent search
tree that combines both locality-optimisation techniques from vEB-based trees
and concurrency-optimisation techniques from non-blocking highly-concurrent
search trees. DeltaTree is a -ary leaf-oriented tree of DeltaNodes in which
each DeltaNode is a size-fixed tree-container with the van Emde Boas layout.
The expected memory transfer costs of DeltaTree's Search, Insert, and Delete
operations are , where are the tree size and the unknown
memory block size in the ideal cache model, respectively. DeltaTree's Search
operation is wait-free, providing prioritised lanes for Search operations, the
dominant operation in search trees. Its Insert and {\em Delete} operations are
non-blocking to other Search, Insert, and Delete operations, but they may be
occasionally blocked by maintenance operations that are sometimes triggered to
keep DeltaTree in good shape. Our experimental evaluation using the latest
implementation of AVL, red-black, and speculation friendly trees from the
Synchrobench benchmark has shown that DeltaTree is up to 5 times faster than
all of the three concurrent search trees for searching operations and up to 1.6
times faster for update operations when the update contention is not too high
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