Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Some Bounds on Communication Complexity of Gap Hamming Distance   In this paper we obtain some bounds on communication complexity of Gap
Hamming Distance problem ($\mathsf{GHD}^n_{L, U}$): Alice and Bob are given
binary string of length $n$ and they are guaranteed that Hamming distance
between their inputs is either $\le L$ or $\ge U$ for some $L < U$. They have
to output 0, if the first inequality holds, and 1, if the second inequality
holds.
  In this paper we study the communication complexity of $\mathsf{GHD}^n_{L,
U}$ for probabilistic protocols with one-sided error and for deterministic
protocols. Our first result is a protocol which communicates
$O\left(\left(\frac{s}{U}\right)^\frac{1}{3} \cdot n\log n\right)$ bits and has
one-sided error probability $e^{-s}$ provided $s \ge \frac{(L +
\frac{10}{n})^3}{U^2}$.
  Our second result is about deterministic communication complexity of
$\mathsf{GHD}^n_{0,\, t}$. Surprisingly, it can be computed with logarithmic
precision: $$\mathrm{D}(\mathsf{GHD}^n_{0,\, t}) = n - \log_2 V_2\left(n,
\left\lfloor\frac{t}{2}\right\rfloor\right) + O(\log n),$$ where $V_2(n, r)$
denotes the size of Hamming ball of radius $r$.
  As an application of this result for every $c < 2$ we prove a
$\Omega\left(\frac{n(2 - c)^2}{p}\right)$ lower bound on the space complexity
of any $c$-approximate deterministic $p$-pass streaming algorithm for computing
the number of distinct elements in a data stream of length $n$ with tokens
drawn from the universe $U = \{1, 2, \ldots, n\}$. Previously that lower bound
was known for $c < \frac{3}{2}$ and for $c < 2$ but with larger $|U|$.

23
The categories of this paper are: solv-int
11
The categories of this paper are: hep-lat
21
The categories of this paper are: adap-org
28
The categories of this paper are: alg-geom
37
The categories of this paper are: supr-con
9
The categories of this paper are: nlin
7
The categories of this paper are: hep-th
15
The categories of this paper are: nucl-ex
30
The categories of this paper are: q-alg
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The categories of this paper are: dg-ga
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The categories of this paper are: atom-ph
12
The categories of this paper are: nucl-th
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The categories of this paper are: comp-gas
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: q-fin