1,684 research outputs found
Dynamic Data Structures for Document Collections and Graphs
In the dynamic indexing problem, we must maintain a changing collection of
text documents so that we can efficiently support insertions, deletions, and
pattern matching queries. We are especially interested in developing efficient
data structures that store and query the documents in compressed form. All
previous compressed solutions to this problem rely on answering rank and select
queries on a dynamic sequence of symbols. Because of the lower bound in
[Fredman and Saks, 1989], answering rank queries presents a bottleneck in
compressed dynamic indexing. In this paper we show how this lower bound can be
circumvented using our new framework. We demonstrate that the gap between
static and dynamic variants of the indexing problem can be almost closed. Our
method is based on a novel framework for adding dynamism to static compressed
data structures. Our framework also applies more generally to dynamizing other
problems. We show, for example, how our framework can be applied to develop
compressed representations of dynamic graphs and binary relations
Integration of Oscillatory and Subanalytic Functions
We prove the stability under integration and under Fourier transform of a
concrete class of functions containing all globally subanalytic functions and
their complex exponentials. This paper extends the investigation started in
[J.-M. Lion, J.-P. Rolin: "Volumes, feuilles de Rolle de feuilletages
analytiques et th\'eor\`eme de Wilkie" Ann. Fac. Sci. Toulouse Math. (6) 7
(1998), no. 1, 93-112] and [R. Cluckers, D. J. Miller: "Stability under
integration of sums of products of real globally subanalytic functions and
their logarithms" Duke Math. J. 156 (2011), no. 2, 311-348] to an enriched
framework including oscillatory functions. It provides a new example of
fruitful interaction between analysis and singularity theory.Comment: Final version. Accepted for publication in Duke Math. Journal.
Changes in proofs: from Section 6 to the end, we now use the theory of
continuously uniformly distributed modulo 1 functions that provides a uniform
technical point of view in the proofs of limit statement
Space hierarchy theorem revised
AbstractWe show that, for an arbitrary function h(n) and each recursive function â(n), that are separated by a nondeterministically fully space constructible g(n), such that h(n)âΩ(g(n)) but â(n)âΩ(g(n)), there exists a unary language L in NSPACE(h(n)) that is not contained in NSPACE(â(n)). The same holds for the deterministic case.The main contribution to the well-known Space Hierarchy Theorem is that (i) the language L separating the two space classes is unary (tally), (ii) the hierarchy is independent of whether h(n) or â(n) are in Ω(logn) or in o(logn), (iii) the functions h(n) or â(n) themselves need not be space constructible nor monotone increasing, (iv) the hierarchy is established both for strong and weak space complexity classes. This allows us to present unary languages in such complexity classes as, for example, NSPACE(loglogn·logân)â§čNSPACE(loglogn), using a plain diagonalization
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