1,045 research outputs found
Information Cost Tradeoffs for Augmented Index and Streaming Language Recognition
This paper makes three main contributions to the theory of communication
complexity and stream computation. First, we present new bounds on the
information complexity of AUGMENTED-INDEX. In contrast to analogous results for
INDEX by Jain, Radhakrishnan and Sen [J. ACM, 2009], we have to overcome the
significant technical challenge that protocols for AUGMENTED-INDEX may violate
the "rectangle property" due to the inherent input sharing. Second, we use
these bounds to resolve an open problem of Magniez, Mathieu and Nayak [STOC,
2010] that asked about the multi-pass complexity of recognizing Dyck languages.
This results in a natural separation between the standard multi-pass model and
the multi-pass model that permits reverse passes. Third, we present the first
passive memory checkers that verify the interaction transcripts of priority
queues, stacks, and double-ended queues. We obtain tight upper and lower bounds
for these problems, thereby addressing an important sub-class of the memory
checking framework of Blum et al. [Algorithmica, 1994]
Augmented Index and Quantum Streaming Algorithms for DYCK(2)
We show how two recently developed quantum information theoretic tools can be applied to obtain lower bounds on quantum information complexity. We also develop new tools with potential for broader applicability, and use them to establish a lower bound on the quantum information complexity for the Augmented Index function on an easy distribution. This approach allows us to handle superpositions rather than distributions over inputs, the main technical challenge faced previously. By providing a quantum generalization of the argument of Jain and Nayak [IEEE TIT\u2714], we leverage this to obtain a lower bound on the space complexity of multi-pass, unidirectional quantum streaming algorithms for the DYCK(2) language
Patterns of Scalable Bayesian Inference
Datasets are growing not just in size but in complexity, creating a demand
for rich models and quantification of uncertainty. Bayesian methods are an
excellent fit for this demand, but scaling Bayesian inference is a challenge.
In response to this challenge, there has been considerable recent work based on
varying assumptions about model structure, underlying computational resources,
and the importance of asymptotic correctness. As a result, there is a zoo of
ideas with few clear overarching principles.
In this paper, we seek to identify unifying principles, patterns, and
intuitions for scaling Bayesian inference. We review existing work on utilizing
modern computing resources with both MCMC and variational approximation
techniques. From this taxonomy of ideas, we characterize the general principles
that have proven successful for designing scalable inference procedures and
comment on the path forward
Incidence Geometries and the Pass Complexity of Semi-Streaming Set Cover
Set cover, over a universe of size , may be modelled as a data-streaming
problem, where the sets that comprise the instance are to be read one by
one. A semi-streaming algorithm is allowed only space to process this stream. For each , we give a very
simple deterministic algorithm that makes passes over the input stream and
returns an appropriately certified -approximation to the
optimum set cover. More importantly, we proceed to show that this approximation
factor is essentially tight, by showing that a factor better than
is unachievable for a -pass semi-streaming
algorithm, even allowing randomisation. In particular, this implies that
achieving a -approximation requires
passes, which is tight up to the factor. These results extend to a
relaxation of the set cover problem where we are allowed to leave an
fraction of the universe uncovered: the tight bounds on the best
approximation factor achievable in passes turn out to be
. Our lower bounds are based
on a construction of a family of high-rank incidence geometries, which may be
thought of as vast generalisations of affine planes. This construction, based
on algebraic techniques, appears flexible enough to find other applications and
is therefore interesting in its own right.Comment: 20 page
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