6,278 research outputs found
Lower Bounds on Time-Space Trade-Offs for Approximate Near Neighbors
We show tight lower bounds for the entire trade-off between space and query
time for the Approximate Near Neighbor search problem. Our lower bounds hold in
a restricted model of computation, which captures all hashing-based approaches.
In articular, our lower bound matches the upper bound recently shown in
[Laarhoven 2015] for the random instance on a Euclidean sphere (which we show
in fact extends to the entire space using the techniques from
[Andoni, Razenshteyn 2015]).
We also show tight, unconditional cell-probe lower bounds for one and two
probes, improving upon the best known bounds from [Panigrahy, Talwar, Wieder
2010]. In particular, this is the first space lower bound (for any static data
structure) for two probes which is not polynomially smaller than for one probe.
To show the result for two probes, we establish and exploit a connection to
locally-decodable codes.Comment: 47 pages, 2 figures; v2: substantially revised introduction, lots of
small corrections; subsumed by arXiv:1608.03580 [cs.DS] (along with
arXiv:1511.07527 [cs.DS]
Optimal Hashing-based Time-Space Trade-offs for Approximate Near Neighbors
[See the paper for the full abstract.]
We show tight upper and lower bounds for time-space trade-offs for the
-Approximate Near Neighbor Search problem. For the -dimensional Euclidean
space and -point datasets, we develop a data structure with space and query time for
every such that: \begin{equation} c^2 \sqrt{\rho_q} +
(c^2 - 1) \sqrt{\rho_u} = \sqrt{2c^2 - 1}. \end{equation}
This is the first data structure that achieves sublinear query time and
near-linear space for every approximation factor , improving upon
[Kapralov, PODS 2015]. The data structure is a culmination of a long line of
work on the problem for all space regimes; it builds on Spherical
Locality-Sensitive Filtering [Becker, Ducas, Gama, Laarhoven, SODA 2016] and
data-dependent hashing [Andoni, Indyk, Nguyen, Razenshteyn, SODA 2014] [Andoni,
Razenshteyn, STOC 2015].
Our matching lower bounds are of two types: conditional and unconditional.
First, we prove tightness of the whole above trade-off in a restricted model of
computation, which captures all known hashing-based approaches. We then show
unconditional cell-probe lower bounds for one and two probes that match the
above trade-off for , improving upon the best known lower bounds
from [Panigrahy, Talwar, Wieder, FOCS 2010]. In particular, this is the first
space lower bound (for any static data structure) for two probes which is not
polynomially smaller than the one-probe bound. To show the result for two
probes, we establish and exploit a connection to locally-decodable codes.Comment: 62 pages, 5 figures; a merger of arXiv:1511.07527 [cs.DS] and
arXiv:1605.02701 [cs.DS], which subsumes both of the preprints. New version
contains more elaborated proofs and fixed some typo
Towards Bulk Metric Reconstruction from Extremal Area Variations
The Ryu-Takayanagi and Hubeny-Rangamani-Takayanagi formulae suggest that bulk
geometry emerges from the entanglement structure of the boundary theory. Using
these formulae, we build on a result of Alexakis, Balehowsky, and Nachman to
show that in four bulk dimensions, the entanglement entropies of boundary
regions of disk topology uniquely fix the bulk metric in any region foliated by
the corresponding HRT surfaces. More generally, for a bulk of any dimension , knowledge of the (variations of the) areas of two-dimensional
boundary-anchored extremal surfaces of disk topology uniquely fixes the bulk
metric wherever these surfaces reach. This result is covariant and not reliant
on any symmetry assumptions; its applicability thus includes regions of strong
dynamical gravity such as the early-time interior of black holes formed from
collapse. While we only show uniqueness of the metric, the approach we present
provides a clear path towards an explicit spacetime metric reconstruction.Comment: 33+4 pages, 7 figures; v2: addressed referee comment
Lower Bounds for 2-Query LCCs over Large Alphabet
A locally correctable code (LCC) is an error correcting code that allows correction of any arbitrary coordinate of a corrupted codeword by querying only a few coordinates. We show that any 2-query locally correctable code C:{0,1}^k -> Sigma^n that can correct a constant fraction of corrupted symbols must have n >= exp(k/log|Sigma|) under the assumption that the LCC is zero-error. We say that an LCC is zero-error if there exists a non-adaptive corrector algorithm that succeeds with probability 1 when the input is an uncorrupted codeword. All known constructions of LCCs are zero-error.
Our result is tight upto constant factors in the exponent. The only previous lower bound on the length of 2-query LCCs over large alphabet was Omega((k/log|Sigma|)^2) due to Katz and Trevisan (STOC 2000). Our bound implies that zero-error LCCs cannot yield 2-server private information retrieval (PIR) schemes with sub-polynomial communication. Since there exists a 2-server PIR scheme with sub-polynomial communication (STOC 2015) based on a zero-error 2-query locally decodable code (LDC), we also obtain a separation between LDCs and LCCs over large alphabet
The Interplay between Chemistry and Mechanics in the Transduction of a Mechanical Signal into a Biochemical Function
There are many processes in biology in which mechanical forces are generated.
Force-bearing networks can transduce locally developed mechanical signals very
extensively over different parts of the cell or tissues. In this article we
conduct an overview of this kind of mechanical transduction, focusing in
particular on the multiple layers of complexity displayed by the mechanisms
that control and trigger the conversion of a mechanical signal into a
biochemical function. Single molecule methodologies, through their capability
to introduce the force in studies of biological processes in which mechanical
stresses are developed, are unveiling subtle intertwining mechanisms between
chemistry and mechanics and in particular are revealing how chemistry can
control mechanics. The possibility that chemistry interplays with mechanics
should be always considered in biochemical studies.Comment: 50 pages, 18 figure
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