78,494 research outputs found
Almost sure asymptotics for the random binary search tree
We consider a (random permutation model) binary search tree with n nodes and
give asymptotics on the loglog scale for the height H_n and saturation level
h_n of the tree as n\to\infty, both almost surely and in probability. We then
consider the number F_n of particles at level H_n at time n, and show that F_n
is unbounded almost surely.Comment: 12 pages, 2 figure
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
Flows on Graphs with Random Capacities
We investigate flows on graphs whose links have random capacities. For binary
trees we derive the probability distribution for the maximal flow from the root
to a leaf, and show that for infinite trees it vanishes beyond a certain
threshold that depends on the distribution of capacities. We then examine the
maximal total flux from the root to the leaves. Our methods generalize to
simple graphs with loops, e.g., to hierarchical lattices and to complete
graphs.Comment: 8 pages, 6 figure
Simple Derivation of the Lifetime and the Distribution of Faces for a Binary Subdivision Model
The iterative random subdivision of rectangles is used as a generation model
of networks in physics, computer science, and urban planning. However, these
researches were independent. We consider some relations in them, and derive
fundamental properties for the average lifetime depending on birth-time and the
balanced distribution of rectangle faces.Comment: 2 figure
Random Sequential Generation of Intervals for the Cascade Model of Food Webs
The cascade model generates a food web at random. In it the species are
labeled from 0 to , and arcs are given at random between pairs of the
species. For an arc with endpoints and (), the species is
eaten by the species labeled . The chain length (height), generated at
random, models the length of food chain in ecological data. The aim of this
note is to introduce the random sequential generation of intervals as a Poisson
model which gives naturally an analogous behavior to the cascade model
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