Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Simple Proof of a New Set Disjointness with Applications to Data
  Streams   The multiplayer promise set disjointness is one of the most widely used
problems from communication complexity in applications. In this problem there
are $k$ players with subsets $S^1, \ldots, S^k$, each drawn from $\{1, 2,
\ldots, n\}$, and we are promised that either the sets are (1) pairwise
disjoint, or (2) there is a unique element $j$ occurring in all the sets, which
are otherwise pairwise disjoint. The total communication of solving this
problem with constant probability in the blackboard model is $\Omega(n/k)$.
  We observe for most applications, it instead suffices to look at what we call
the ``mostly'' set disjointness problem, which changes case (2) to say there is
a unique element $j$ occurring in at least half of the sets, and the sets are
otherwise disjoint. This change gives us a much simpler proof of an
$\Omega(n/k)$ randomized total communication lower bound, avoiding Hellinger
distance and Poincare inequalities. Using this we show several new results for
data streams:
  \begin{itemize} \item for $\ell_2$-Heavy Hitters, any $O(1)$-pass streaming
algorithm in the insertion-only model for detecting if an $\eps$-$\ell_2$-heavy
hitter exists requires $\min(\frac{1}{\eps^2}\log \frac{\eps^2n}{\delta},
\frac{1}{\eps}n^{1/2})$ bits of memory, which is optimal up to a $\log n$
factor. For deterministic algorithms and constant $\eps$, this gives an
$\Omega(n^{1/2})$ lower bound, improving the prior $\Omega(\log n)$ lower
bound. We also obtain lower bounds for Zipfian distributions. \item for
$\ell_p$-Estimation, $p > 2$, we show an $O(1)$-pass $\Omega(n^{1-2/p}
\log(1/\delta))$ bit lower bound for outputting an $O(1)$-approximation with
probability $1-\delta$, in the insertion-only model. This is optimal, and the
best previous lower bound was $\Omega(n^{1-2/p} + \log(1/\delta))$.
\end{itemize}

21
The categories of this paper are: adap-org
11
The categories of this paper are: hep-lat
32
The categories of this paper are: atom-ph
30
The categories of this paper are: q-alg
26
The categories of this paper are: bayes-an
15
The categories of this paper are: nucl-ex
27
The categories of this paper are: comp-gas
5
The categories of this paper are: gr-qc
7
The categories of this paper are: hep-th
23
The categories of this paper are: solv-int
10
The categories of this paper are: q-bio
20
The categories of this paper are: acc-phys
8
The categories of this paper are: hep-ex
9
The categories of this paper are: nlin
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th