117 research outputs found
Semi-Streaming Algorithms for Annotated Graph Streams
Considerable effort has been devoted to the development of streaming
algorithms for analyzing massive graphs. Unfortunately, many results have been
negative, establishing that a wide variety of problems require
space to solve. One of the few bright spots has been the development of
semi-streaming algorithms for a handful of graph problems -- these algorithms
use space .
In the annotated data streaming model of Chakrabarti et al., a
computationally limited client wants to compute some property of a massive
input, but lacks the resources to store even a small fraction of the input, and
hence cannot perform the desired computation locally. The client therefore
accesses a powerful but untrusted service provider, who not only performs the
requested computation, but also proves that the answer is correct.
We put forth the notion of semi-streaming algorithms for annotated graph
streams (semi-streaming annotation schemes for short). These are protocols in
which both the client's space usage and the length of the proof are . We give evidence that semi-streaming annotation schemes
represent a substantially more robust solution concept than does the standard
semi-streaming model. On the positive side, we give semi-streaming annotation
schemes for two dynamic graph problems that are intractable in the standard
model: (exactly) counting triangles, and (exactly) computing maximum matchings.
The former scheme answers a question of Cormode. On the negative side, we
identify for the first time two natural graph problems (connectivity and
bipartiteness in a certain edge update model) that can be solved in the
standard semi-streaming model, but cannot be solved by annotation schemes of
"sub-semi-streaming" cost. That is, these problems are just as hard in the
annotations model as they are in the standard model.Comment: This update includes some additional discussion of the results
proven. The result on counting triangles was previously included in an ECCC
technical report by Chakrabarti et al. available at
http://eccc.hpi-web.de/report/2013/180/. That report has been superseded by
this manuscript, and the CCC 2015 paper "Verifiable Stream Computation and
Arthur-Merlin Communication" by Chakrabarti et a
A Nearly Optimal Lower Bound on the Approximate Degree of AC
The approximate degree of a Boolean function is the least degree of a real polynomial that
approximates pointwise to error at most . We introduce a generic
method for increasing the approximate degree of a given function, while
preserving its computability by constant-depth circuits.
Specifically, we show how to transform any Boolean function with
approximate degree into a function on variables with approximate degree at least . In particular, if , then
is polynomially larger than . Moreover, if is computed by a
polynomial-size Boolean circuit of constant depth, then so is .
By recursively applying our transformation, for any constant we
exhibit an AC function of approximate degree . This
improves over the best previous lower bound of due to
Aaronson and Shi (J. ACM 2004), and nearly matches the trivial upper bound of
that holds for any function. Our lower bounds also apply to
(quasipolynomial-size) DNFs of polylogarithmic width.
We describe several applications of these results. We give:
* For any constant , an lower bound on the
quantum communication complexity of a function in AC.
* A Boolean function with approximate degree at least ,
where is the certificate complexity of . This separation is optimal
up to the term in the exponent.
* Improved secret sharing schemes with reconstruction procedures in AC.Comment: 40 pages, 1 figur
Lower Bounds for the Approximate Degree of Block-Composed Functions
We describe a new hardness amplification result for point-wise approximation of Boolean functions by low-degree polynomials.
Specifically, for any function f on N bits, define F(x_1,...,x_M) = OMB(f(x_1),...,f(x_M)) to be the function on M*N bits obtained by block-composing f with a function known as ODD-MAX-BIT. We show that, if f requires large degree to approximate to error 2/3 in a certain one-sided sense (captured by a complexity measure known as positive one-sided approximate degree), then F requires large degree to approximate even to error 1-2^{-M}. This generalizes a result of Beigel (Computational Complexity, 1994), who proved an identical result for the special case f=OR.
Unlike related prior work, our result implies strong approximate degree lower bounds even for many functions F that have low threshold degree. Our proof is constructive: we exhibit a solution to the dual of an appropriate linear program capturing the approximate degree of any function. We describe several applications, including improved separations between the complexity classes P^{NP} and PP in both the query and communication complexity settings. Our separations improve on work of Beigel (1994) and Buhrman, Vereshchagin, and de Wolf (CCC, 2007)
Parallel Peeling Algorithms
The analysis of several algorithms and data structures can be framed as a
peeling process on a random hypergraph: vertices with degree less than k are
removed until there are no vertices of degree less than k left. The remaining
hypergraph is known as the k-core. In this paper, we analyze parallel peeling
processes, where in each round, all vertices of degree less than k are removed.
It is known that, below a specific edge density threshold, the k-core is empty
with high probability. We show that, with high probability, below this
threshold, only (log log n)/log(k-1)(r-1) + O(1) rounds of peeling are needed
to obtain the empty k-core for r-uniform hypergraphs. Interestingly, we show
that above this threshold, Omega(log n) rounds of peeling are required to find
the non-empty k-core. Since most algorithms and data structures aim to peel to
an empty k-core, this asymmetry appears fortunate. We verify the theoretical
results both with simulation and with a parallel implementation using graphics
processing units (GPUs). Our implementation provides insights into how to
structure parallel peeling algorithms for efficiency in practice.Comment: Appears in SPAA 2014. Minor typo corrections relative to previous
versio
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