NDLI: Distributional Results for Costs of Partial Match Queries in Asymmetric K-Dimensional Tries

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Distributional Results for Costs of Partial Match Queries in Asymmetric K-Dimensional Tries

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Schachinger, Werner
Copyright Year	2004
Abstract	In this paper we study the costs CN of partial match retrievals in K-dimensional tries (K-d tries), constructed from N records. The probabilistic model that we assume is the asymmetric Bernoulli model: keys are sequences of independently and identically distributed random variables, which assume the values 0 and 1 with probability $p \neq \frac{1}{2}$ and 1-p, and are pairwise independent. We determine the extremal asymptotic orders that the sequence of expectations $(\mathbb{E}\,C_N)_{N \ge 0}$ may have for different fixed queries, as well as the narrow region that contains $(\mathbb{E}\,C_N)_{N \geq 0}$ for almost every query. Furthermore we show that $(\mathbb{E}\,C_N)_{N \geq 0}$ and $({\rm Var}\,C_N)_{N \ge 0}$ have the same asymptotics up to a logarithmic factor and, employing a central limit theorem for martingale difference arrays, we prove asymptotic normality of $\frac{C_N - \mathbb{E}\,C_N}{\sqrt{{\rm Var}\,C_N}}$. For random queries, assumed to be independent of the keys and having their specified components distributed according to the same Bernoulli model, no limiting distribution for CN exists, but we can prove asymptotic normality of $\ln C_N$, when appropriately normalized, and determine ${\rm Var}\,C_N$ up to a logarithmic factor, where now $(\mathbb{E}\,C_N)^2 = o({\rm Var}\,C_N)$.
Starting Page	952
Ending Page	983
Page Count	32
File Format	PDF
ISSN	00975397
DOI	10.1137/S0097539703425873
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	4
Volume Number	33
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2004-01-01
Access Restriction	Subscribed
Subject Keyword	asymptotic normality Information storage and retrieval multidimensional data K-d trie martingale Data structures Searching and sorting partial match retrieval Analysis of algorithms Central limit and other weak theorems
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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