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Asymptotic distribution of two-protected nodes in ternary search trees
We study protected nodes in -ary search trees, by putting them in context
of generalised P\'olya urns. We show that the number of two-protected nodes
(the nodes that are neither leaves nor parents of leaves) in a random ternary
search tree is asymptotically normal. The methods apply in principle to -ary search trees with larger as well, although the size of the matrices
used in the calculations grow rapidly with ; we conjecture that the method
yields an asymptotically normal distribution for all .
The one-protected nodes, and their complement, i.e., the leaves, are easier
to analyze. By using a simpler P\'olya urn (that is similar to the one that has
earlier been used to study the total number of nodes in -ary search
trees), we prove normal limit laws for the number of one-protected nodes and
the number of leaves for all
Support and density of the limit -ary search trees distribution
The space requirements of an -ary search tree satisfies a well-known phase
transition: when , the second order asymptotics is Gaussian. When
, it is not Gaussian any longer and a limit of a complex-valued
martingale arises. We show that the distribution of has a square integrable
density on the complex plane, that its support is the whole complex plane, and
that it has finite exponential moments. The proofs are based on the study of
the distributional equation W\egalLoi\sum_{k=1}^mV_k^{\lambda}W_k, where
are the spacings of independent random variables
uniformly distributed on , are independent copies of W
which are also independent of and is a complex
number
Succinct Indexable Dictionaries with Applications to Encoding -ary Trees, Prefix Sums and Multisets
We consider the {\it indexable dictionary} problem, which consists of storing
a set for some integer , while supporting the
operations of \Rank(x), which returns the number of elements in that are
less than if , and -1 otherwise; and \Select(i) which returns
the -th smallest element in . We give a data structure that supports both
operations in O(1) time on the RAM model and requires bits to store a set of size , where {\cal B}(n,m) = \ceil{\lg
{m \choose n}} is the minimum number of bits required to store any -element
subset from a universe of size . Previous dictionaries taking this space
only supported (yes/no) membership queries in O(1) time. In the cell probe
model we can remove the additive term in the space bound,
answering a question raised by Fich and Miltersen, and Pagh.
We present extensions and applications of our indexable dictionary data
structure, including:
An information-theoretically optimal representation of a -ary cardinal
tree that supports standard operations in constant time,
A representation of a multiset of size from in bits that supports (appropriate generalizations of) \Rank
and \Select operations in constant time, and
A representation of a sequence of non-negative integers summing up to
in bits that supports prefix sum queries in constant
time.Comment: Final version of SODA 2002 paper; supersedes Leicester Tech report
2002/1
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