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Down the Rabbit Hole: Robust Proximity Search and Density Estimation in Sublinear Space
For a set of points in , and parameters and \eps, we present
a data structure that answers (1+\eps,k)-\ANN queries in logarithmic time.
Surprisingly, the space used by the data-structure is \Otilde (n /k); that
is, the space used is sublinear in the input size if is sufficiently large.
Our approach provides a novel way to summarize geometric data, such that
meaningful proximity queries on the data can be carried out using this sketch.
Using this, we provide a sublinear space data-structure that can estimate the
density of a point set under various measures, including:
\begin{inparaenum}[(i)]
\item sum of distances of closest points to the query point, and
\item sum of squared distances of closest points to the query point.
\end{inparaenum}
Our approach generalizes to other distance based estimation of densities of
similar flavor. We also study the problem of approximating some of these
quantities when using sampling. In particular, we show that a sample of size
\Otilde (n /k) is sufficient, in some restricted cases, to estimate the above
quantities. Remarkably, the sample size has only linear dependency on the
dimension
Prime divisors of sequences associated to elliptic curves
We consider the primes which divide the denominator of the x-coordinate of a
sequence of rational points on an elliptic curve. It is expected that for every
sufficiently large value of the index, each term should be divisible by a
primitive prime divisor, one that has not appeared in any earlier term. Proofs
of this are known in only a few cases. Weaker results in the general direction
are given, using a strong form of Siegel's Theorem and some congruence
arguments. Our main result is applied to the study of prime divisors of Somos
sequences
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