Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Advice Complexity of Online Non-Crossing Matching   We study online matching in the Euclidean $2$-dimesional plane with
non-crossing constraint. The offline version was introduced by Atallah in 1985
and the online version was introduced and studied more recently by Bose et al.
The input to the problem consists of a sequence of points, and upon arrival of
a point an algorithm can match it with a previously unmatched point provided
that line segments corresponding to the matched edges do not intersect. The
decisions are irrevocable, and while an optimal offline solution always matches
all the points, an online algorithm cannot match all the points in the worst
case, unless it is given some side information, i.e., advice. We study two
versions of this problem -- monomchromatic (MNM) and bichromatic (BNM).
  We show that advice complexity of solving BNM optimally on a circle (or, more
generally, on inputs in a convex position) is tightly bounded by the logarithm
of the $n^\text{th}$ Catalan number from above and below. This result corrects
the previous claim of Bose et al. that the advice complexity is $\log(n!)$. At
the heart of the result is a connection between non-crossing constraint in
online inputs and $231$-avoiding property of permutations of $n$ elements We
also show a lower bound of $n/3-1$ and an upper bound of $3n$ on the advice
complexity for MNM on a plane. This gives an exponential improvement over the
previously best known lower bound and an improvement in the constant of the
leading term in the upper bound. In addition, we establish a lower bound of
$\frac{\alpha}{2}\infdiv{\frac{2(1-\alpha)}{\alpha}}{1/4}n$ on the advice
complexity for achieving competitive ratio $\alpha$ for MNM on a circle.
Standard tools from advice complexity, such as partition trees and reductions
from string guessing problem, do not seem to apply to MNM/BNM, so we have to
design our lower bounds from first principles.

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The categories of this paper are: nlin
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The categories of this paper are: chao-dyn
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The categories of this paper are: quant-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: supr-con
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The categories of this paper are: chem-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: econ
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The categories of this paper are: solv-int
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The categories of this paper are: ao-sci
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The categories of this paper are: stat
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The categories of this paper are: nucl-th
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The categories of this paper are: q-bio
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The categories of this paper are: q-alg