13,272 research outputs found
Partial fillup and search time in LC tries
Andersson and Nilsson introduced in 1993 a level-compressed trie (in short:
LC trie) in which a full subtree of a node is compressed to a single node of
degree being the size of the subtree. Recent experimental results indicated a
'dramatic improvement' when full subtrees are replaced by partially filled
subtrees. In this paper, we provide a theoretical justification of these
experimental results showing, among others, a rather moderate improvement of
the search time over the original LC tries. For such an analysis, we assume
that n strings are generated independently by a binary memoryless source with p
denoting the probability of emitting a 1. We first prove that the so called
alpha-fillup level (i.e., the largest level in a trie with alpha fraction of
nodes present at this level) is concentrated on two values with high
probability. We give these values explicitly up to O(1), and observe that the
value of alpha (strictly between 0 and 1) does not affect the leading term.
This result directly yields the typical depth (search time) in the alpha-LC
tries with p not equal to 1/2, which turns out to be C loglog n for an
explicitly given constant C (depending on p but not on alpha). This should be
compared with recently found typical depth in the original LC tries which is C'
loglog n for a larger constant C'. The search time in alpha-LC tries is thus
smaller but of the same order as in the original LC tries.Comment: 13 page
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 Steiner subtrees of Random Trees for Traceroute Algorithms
We consider in this paper the problem of discovering, via a traceroute
algorithm, the topology of a network, whose graph is spanned by an infinite
branching process. A subset of nodes is selected according to some criterion.
As a measure of efficiency of the algorithm, the Steiner distance of the
selected nodes, i.e. the size of the spanning sub-tree of these nodes, is
investigated. For the selection of nodes, two criteria are considered: A node
is randomly selected with a probability, which is either independent of the
depth of the node (uniform model) or else in the depth biased model, is
exponentially decaying with respect to its depth. The limiting behavior the
size of the discovered subtree is investigated for both models
Training a perceptron in a discrete weight space
On-line and batch learning of a perceptron in a discrete weight space, where
each weight can take different values, are examined analytically and
numerically. The learning algorithm is based on the training of the continuous
perceptron and prediction following the clipped weights. The learning is
described by a new set of order parameters, composed of the overlaps between
the teacher and the continuous/clipped students. Different scenarios are
examined among them on-line learning with discrete/continuous transfer
functions and off-line Hebb learning. The generalization error of the clipped
weights decays asymptotically as / in the case of on-line learning with binary/continuous activation
functions, respectively, where is the number of examples divided by N,
the size of the input vector and is a positive constant that decays
linearly with 1/L. For finite and , a perfect agreement between the
discrete student and the teacher is obtained for . A crossover to the generalization error ,
characterized continuous weights with binary output, is obtained for synaptic
depth .Comment: 10 pages, 5 figs., submitted to PR
Asymptotic genealogy of a critical branching process
Consider a continuous-time binary branching process conditioned to have
population size n at some time t, and with a chance p for recording each
extinct individual in the process. Within the family tree of this process, we
consider the smallest subtree containing the genealogy of the extant
individuals together with the genealogy of the recorded extinct individuals. We
introduce a novel representation of such subtrees in terms of a point-process,
and provide asymptotic results on the distribution of this point-process as the
number of extant individuals increases. We motivate the study within the scope
of a coherent analysis for an a priori model for macroevolution.Comment: 30 page
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