Random input helps searching predecessors

Autor: Belazzougui, D, Kaporis, AC, Spirakis, PG
Rok vydání: 2011
Předmět:
Zdroj: CEUR Workshop Proceedings
DOI: 10.48550/arxiv.1104.4353
Popis: A data structure problem consists of the finite sets: D of data, Q of queries, A of query answers, associated with a function f: D x Q → A. The data structure of file X is "static" ("dynamic") if we "do not" ("do") require quick updates as X changes. An important goal is to compactly encode a file X ϵ D, such that for each query y ϵ Q, function f (X, y) requires the minimum time to compute an answer in A. This goal is trivial if the size of D is large, since for each query y ϵ Q, it was shown that f(X,y) requires O(1) time for the most important queries in the literature. Hence, this goal becomes interesting to study as a trade off between the "storage space" and the "query time", both measured as functions of the file size n = \X\. The ideal solution would be to use linear O(n) = O(\X\) space, while retaining a constant O(1) query time. However, if f (X, y) computes the static predecessor search (find largest x ϵ X: x ≤ y), then Ajtai [Ajt88] proved a negative result. By using just n0(1) = [IX]0(1) data space, then it is not possible to evaluate f(X,y) in O(1) time Ay ϵ Q. The proof exhibited a bad distribution of data D, such that Ey∗ ϵ Q (a "difficult" query y∗), that f(X,y∗) requires ω(1) time. Essentially [Ajt88] is an existential result, resolving the worst case scenario. But, [Ajt88] left open the question: do we typically, that is, with high probability (w.h.p.)1 encounter such "difficult" queries y ϵ Q, when assuming reasonable distributions with respect to (w.r.t.) queries and data? Below we make reasonable assumptions w.r.t. the distribution of the queries y ϵ Q, as well as w.r.t. the distribution of data X ϵ D. In two interesting scenarios studied in the literature, we resolve the typical (w.h.p.) query time.
Databáze: OpenAIRE
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