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The space complexity of approximating the frequency moments

by Noga Alon, Yossi Matias, Mario Szegedy - JOURNAL OF COMPUTER AND SYSTEM SCIENCES , 1996
"... The frequency moments of a sequence containing mi elements of type i, for 1 ≤ i ≤ n, are the numbers Fk = �n i=1 mki. We consider the space complexity of randomized algorithms that approximate the numbers Fk, when the elements of the sequence are given one by one and cannot be stored. Surprisingly, ..."
Abstract - Cited by 845 (12 self) - Add to MetaCart
The frequency moments of a sequence containing mi elements of type i, for 1 ≤ i ≤ n, are the numbers Fk = �n i=1 mki. We consider the space complexity of randomized algorithms that approximate the numbers Fk, when the elements of the sequence are given one by one and cannot be stored. Surprisingly

The fundamental properties of natural numbers

by Grzegorz Bancerek - Journal of Formalized Mathematics , 1989
"... Summary. Some fundamental properties of addition, multiplication, order relations, exact division, the remainder, divisibility, the least common multiple, the greatest common divisor are presented. A proof of Euclid algorithm is also given. MML Identifier:NAT_1. WWW:http://mizar.org/JFM/Vol1/nat_1.h ..."
Abstract - Cited by 688 (73 self) - Add to MetaCart
.html The articles [4], [6], [1], [2], [5], and [3] provide the notation and terminology for this paper. A natural number is an element of N. For simplicity, we use the following convention: x is a real number, k, l, m, n are natural numbers, h, i, j are natural numbers, and X is a subset of R

Near Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?

by Emmanuel J. Candès , Terence Tao , 2004
"... Suppose we are given a vector f in RN. How many linear measurements do we need to make about f to be able to recover f to within precision ɛ in the Euclidean (ℓ2) metric? Or more exactly, suppose we are interested in a class F of such objects— discrete digital signals, images, etc; how many linear m ..."
Abstract - Cited by 1513 (20 self) - Add to MetaCart
as the class F of those elements whose entries obey the power decay law |f | (n) ≤ C · n −1/p. We take measurements 〈f, Xk〉, k = 1,..., K, where the Xk are N-dimensional Gaussian

On limits of wireless communications in a fading environment when using multiple antennas

by G. J. Foschini, M. J. Gans - Wireless Personal Communications , 1998
"... Abstract. This paper is motivated by the need for fundamental understanding of ultimate limits of bandwidth efficient delivery of higher bit-rates in digital wireless communications and to also begin to look into how these limits might be approached. We examine exploitation of multi-element array (M ..."
Abstract - Cited by 2426 (14 self) - Add to MetaCart
the capacity scales with increasing SNR for a large but practical number, n, of antenna elements at both transmitter and receiver. We investigate the case of independent Rayleigh faded paths between antenna elements and find that with high probability extraordinary capacity is available. Compared

New tight frames of curvelets and optimal representations of objects with piecewise C² singularities

by Emmanuel J. Candès, David L. Donoho - COMM. ON PURE AND APPL. MATH , 2002
"... This paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along C2 edges. Conceptually, the curvelet transform is a multiscale pyramid with many directions and positions at each length scale, and needle-shap ..."
Abstract - Cited by 428 (21 self) - Add to MetaCart
for discontinuities along C 2 curves and is essentially optimal. In comparison, the squared error of n-term wavelet approximations only converges as n −1 as n → ∞, which is considerably worst than the optimal behavior.

Trading Group Theory for Randomness

by László Babai , 1985
"... In a previous paper [BS] we proved, using the elements of the Clwory of nilyotenf yroupu, that some of the /undamcn-la1 computational problems in mat & proup, belong to NP. These problems were also ahown to belong to CONP, assuming an unproven hypofhedi.9 concerning finilc simple Q ’ oup,. The a ..."
Abstract - Cited by 353 (9 self) - Add to MetaCart
In a previous paper [BS] we proved, using the elements of the Clwory of nilyotenf yroupu, that some of the /undamcn-la1 computational problems in mat & proup, belong to NP. These problems were also ahown to belong to CONP, assuming an unproven hypofhedi.9 concerning finilc simple Q ’ oup

Efficiency of a Good But Not Linear Set Union Algorithm

by Robert Endre Tarjan , 1975
"... Two types of instructmns for mampulating a family of disjoint sets which partitmn a umverse of n elements are considered FIND(x) computes the name of the (unique) set containing element x UNION(A, B, C) combines sets A and B into a new set named C. A known algorithm for implementing sequences of the ..."
Abstract - Cited by 321 (15 self) - Add to MetaCart
Two types of instructmns for mampulating a family of disjoint sets which partitmn a umverse of n elements are considered FIND(x) computes the name of the (unique) set containing element x UNION(A, B, C) combines sets A and B into a new set named C. A known algorithm for implementing sequences

Min-wise Independent Permutations

by Andrei Z. Broder, Moses Charikar, Alan M. Frieze, Michael Mitzenmacher - Journal of Computer and System Sciences , 1998
"... We define and study the notion of min-wise independent families of permutations. We say that F ⊆ Sn is min-wise independent if for any set X ⊆ [n] and any x ∈ X, when π is chosen at random in F we have Pr(min{π(X)} = π(x)) = 1 |X |. In other words we require that all the elements of any fixed set ..."
Abstract - Cited by 276 (11 self) - Add to MetaCart
We define and study the notion of min-wise independent families of permutations. We say that F ⊆ Sn is min-wise independent if for any set X ⊆ [n] and any x ∈ X, when π is chosen at random in F we have Pr(min{π(X)} = π(x)) = 1 |X |. In other words we require that all the elements of any fixed set

Maintaining Stream Statistics over Sliding Windows (Extended Abstract)

by Mayur Datar, Aristides Gionis, Piotr Indyk, Rajeev Motwani , 2002
"... We consider the problem of maintaining aggregates and statistics over data streams, with respect to the last N data elements seen so far. We refer to this model as the sliding window model. We consider the following basic problem: Given a stream of bits, maintain a count of the number of 1's i ..."
Abstract - Cited by 269 (9 self) - Add to MetaCart
We consider the problem of maintaining aggregates and statistics over data streams, with respect to the last N data elements seen so far. We refer to this model as the sliding window model. We consider the following basic problem: Given a stream of bits, maintain a count of the number of 1&apos

Succinct indexable dictionaries with applications to encoding k-ary trees and multisets

by Rajeev Raman, Venkatesh Raman, S. Srinivasa Rao - In Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA
"... We consider the indexable dictionary problem, which consists of storing a set S ⊆ {0,...,m − 1} for some integer m, while supporting the operations of rank(x), which returns the number of elements in S that are less than x if x ∈ S, and −1 otherwise; and select(i) which returns the i-th smallest ele ..."
Abstract - Cited by 259 (16 self) - Add to MetaCart
element in S. We give a data structure that supports both operations in O(1) time on the RAM model and requires B(n,m)+ o(n)+O(lg lg m) bits to store a set of size n, where B(n,m) = ⌈ lg ( m) ⌉ n is the minimum number of bits required to store any n-element subset from a universe of size m. Previous
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