Dagstuhl Seminar Proceedings, Volume 8341
Dagstuhl Seminar Proceedings
DagSemProc
https://www.dagstuhl.de/dagpub/1862-4405
https://dblp.org/db/series/dagstuhl
1862-4405
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
8341
2008
https://drops.dagstuhl.de/entities/volume/DagSemProc-volume-8341
08341 Abstracts Collection – Sublinear Algorithms
From August 17 to August 22, 2008, the Dagstuhl Seminar 08341 ``Sublinear Algorithms'' was held in the International Conference and Research Center (IBFI),
Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.
Sublinear algorithms
property testing
data streaming
graph algorithms
approximation algorithms
1-19
Regular Paper
Artur
Czumaj
Artur Czumaj
S. Muthu
Muthukrishnan
S. Muthu Muthukrishnan
Ronitt
Rubinfeld
Ronitt Rubinfeld
Christian
Sohler
Christian Sohler
10.4230/DagSemProc.08341.1
Creative Commons Attribution 4.0 International license
https://creativecommons.org/licenses/by/4.0/legalcode
08341 Executive Summary – Sublinear Algorithms
This report summarizes the content and structure of the Dagstuhl seminar `Sublinear Algorithms', which was held from 17.8.2008 to 22.8.2008 in Schloss Dagstuhl, Germany.
Sublinear algorithms
property testing
data streaming
graph algorithms
approximation algorithms
1-2
Regular Paper
Artur
Czumaj
Artur Czumaj
S. Muthu
Muthukrishnan
S. Muthu Muthukrishnan
Ronitt
Rubinfeld
Ronitt Rubinfeld
Christian
Sohler
Christian Sohler
10.4230/DagSemProc.08341.2
Creative Commons Attribution 4.0 International license
https://creativecommons.org/licenses/by/4.0/legalcode
Breaking the $\epsilon$-Soundness Bound of the Linearity Test over GF(2)
For Boolean functions that are $epsilon$-far from the set of linear functions,
we study the lower bound on the rejection probability (denoted by $extsc{rej}(epsilon)$) of the linearity test suggested by Blum, Luby and Rubinfeld.
This problem is arguably the most fundamental and extensively studied problem in property testing of Boolean functions.
The previously best bounds for $extsc{rej}(epsilon)$ were obtained by Bellare,
Coppersmith, H{{a}}stad, Kiwi and Sudan. They used Fourier analysis
to show that $ extsc{rej}(epsilon) geq e$ for every $0 leq epsilon leq
frac{1}{2}$. They also conjectured that this bound might not be tight for
$epsilon$'s which are close to $1/2$. In this paper we show that this indeed is
the case. Specifically, we improve the lower bound of $ extsc{rej}(epsilon) geq
epsilon$ by an additive constant that depends only on $epsilon$:
$extsc{rej}(epsilon) geq epsilon + min {1376epsilon^{3}(1-2epsilon)^{12},
frac{1}{4}epsilon(1-2epsilon)^{4}}$, for every $0 leq epsilon leq frac{1}{2}$.
Our analysis is based on a relationship between $extsc{rej}(epsilon)$ and the
weight distribution of a coset of the Hadamard code. We use both Fourier
analysis and coding theory tools to estimate this weight distribution.
Linearity test
Fourier analysis
coding theory
1-0
Regular Paper
Tali
Kaufman
Tali Kaufman
Simon
Litsyn
Simon Litsyn
Ning
Xie
Ning Xie
10.4230/DagSemProc.08341.3
Creative Commons Attribution 4.0 International license
https://creativecommons.org/licenses/by/4.0/legalcode
Lower bound for estimating frequency for update data streams
We consider general update streams, where, the stream is a sequence of updates of the form $(index, i, v)$, where, $i in {1,2 ldots, n}$ and $v in {-1,+1}$, signifying deletion or insertion, respectively of an instance of $i$. The frequency of $i in {1,2,ldots, n}$ is given as the sum of the updates to $i$, that is,
$f_i(sigma) = sum_{(index,i,v) in sigma} v $. The $n$-dimensional vector $f(sigma)$ with $i$th coordinate $f_i(sigma)$ is called the frequency vector of the stream $sigma$. We consider the problem of finding an n-dimensional integer vector $hat{f}(sigma)$ that estimates the frequency vector $f(sigma)$ of an input stream $sigma$ in the following sense:
orm{hat{f} (sigma)- f(sigma)} le epsilon
orm{f(sigma)}_p
For $p=1$ and $2$, there are randomized algorithms known with space bound $ ilde{O}(epsilon^{-p})$. A space lower bound of $Omega(epsilon^{-1} log (nepsilon))$ is also known. However, the deterministic space upper bound is $ ilde{O}(epsilon^{-2})$.
In this work, we present a deterministic space lower bound of $Omega(n^{2-2/p}epsilon^{-2} log |{sigma}|)$, for $1le p < 2$ and $1/4 le epsilon = Omega(n^{1/2-1/p})$. For $p ge 2$, we show an $Omega(n)$ space lower bound for all $epsilon < 1/4$.
The results are obtained using a new characterization of data stream computations, that show that any uniform computation over a data stream may be viewed as an appropriate linear map.
Data stream
lower bound
frequency estimation
stream automata
linear map
1-15
Regular Paper
Sumit
Ganguly
Sumit Ganguly
10.4230/DagSemProc.08341.4
Creative Commons Attribution 4.0 International license
https://creativecommons.org/licenses/by/4.0/legalcode