52,261 research outputs found
The average height of binary trees and other simple trees
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Extreme Value Statistics and Traveling Fronts: Various Applications
An intriguing connection between extreme value statistics and traveling
fronts has been found recently in a number of diverse problems. In this brief
review we outline a few such problems and consider their various applications.Comment: A brief review (6 pages, 2 figures) to appear in Physica A as part of
the proceedings of Statphys-Kolkata IV (2002
The shape of random tanglegrams
A tanglegram consists of two binary rooted trees with the same number of
leaves and a perfect matching between the leaves of the trees. We show that the
two halves of a random tanglegram essentially look like two independently
chosen random plane binary trees. This fact is used to derive a number of
results on the shape of random tanglegrams, including theorems on the number of
cherries and generally occurrences of subtrees, the root branches, the number
of automorphisms, and the height. For each of these, we obtain limiting
probabilities or distributions. Finally, we investigate the number of matched
cherries, for which the limiting distribution is identified as well
Drawing Binary Tanglegrams: An Experimental Evaluation
A binary tanglegram is a pair of binary trees whose leaf sets are in
one-to-one correspondence; matching leaves are connected by inter-tree edges.
For applications, for example in phylogenetics or software engineering, it is
required that the individual trees are drawn crossing-free. A natural
optimization problem, denoted tanglegram layout problem, is thus to minimize
the number of crossings between inter-tree edges.
The tanglegram layout problem is NP-hard and is currently considered both in
application domains and theory. In this paper we present an experimental
comparison of a recursive algorithm of Buchin et al., our variant of their
algorithm, the algorithm hierarchy sort of Holten and van Wijk, and an integer
quadratic program that yields optimal solutions.Comment: see
http://www.siam.org/proceedings/alenex/2009/alx09_011_nollenburgm.pd
The distribution of height and diameter in random non-plane binary trees
This study is dedicated to precise distributional analyses of the height of
non-plane unlabelled binary trees ("Otter trees"), when trees of a given size
are taken with equal likelihood. The height of a rooted tree of size is
proved to admit a limiting theta distribution, both in a central and local
sense, as well as obey moderate as well as large deviations estimates. The
approximations obtained for height also yield the limiting distribution of the
diameter of unrooted trees. The proofs rely on a precise analysis, in the
complex plane and near singularities, of generating functions associated with
trees of bounded height
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
Continuum Cascade Model of Directed Random Graphs: Traveling Wave Analysis
We study a class of directed random graphs. In these graphs, the interval
[0,x] is the vertex set, and from each y\in [0,x], directed links are drawn to
points in the interval (y,x] which are chosen uniformly with density one. We
analyze the length of the longest directed path starting from the origin. In
the large x limit, we employ traveling wave techniques to extract the
asymptotic behavior of this quantity. We also study the size of a cascade tree
composed of vertices which can be reached via directed paths starting at the
origin.Comment: 12 pages, 2 figures; figure adde
Random-bit optimal uniform sampling for rooted planar trees with given sequence of degrees and Applications
In this paper, we redesign and simplify an algorithm due to Remy et al. for
the generation of rooted planar trees that satisfies a given partition of
degrees. This new version is now optimal in terms of random bit complexity, up
to a multiplicative constant. We then apply a natural process
"simulate-guess-and-proof" to analyze the height of a random Motzkin in
function of its frequency of unary nodes. When the number of unary nodes
dominates, we prove some unconventional height phenomenon (i.e. outside the
universal square root behaviour.)Comment: 19 page
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