Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Improved Weighted Additive Spanners   Graph spanners and emulators are sparse structures that approximately
preserve distances of the original graph. While there has been an extensive
amount of work on additive spanners, so far little attention was given to
weighted graphs. Only very recently [ABSKS20] extended the classical +2
(respectively, +4) spanners for unweighted graphs of size $O(n^{3/2})$ (resp.,
$O(n^{7/5})$) to the weighted setting, where the additive error is $+2W$
(resp., $+4W$). This means that for every pair $u,v$, the additive stretch is
at most $+2W_{u,v}$, where $W_{u,v}$ is the maximal edge weight on the shortest
$u-v$ path. In addition, [ABSKS20] showed an algorithm yielding a $+8W_{max}$
spanner of size $O(n^{4/3})$, here $W_{max}$ is the maximum edge weight in the
entire graph.
  In this work we improve the latter result by devising a simple deterministic
algorithm for a $+(6+\varepsilon)W$ spanner for weighted graphs with size
$O(n^{4/3})$ (for any constant $\varepsilon>0$), thus nearly matching the
classical +6 spanner of size $O(n^{4/3})$ for unweighted graphs. Furthermore,
we show a $+(2+\varepsilon)W$ subsetwise spanner of size $O(n\cdot\sqrt{|S|})$,
improving the $+4W_{max}$ result of [ABSKS20] (that had the same size). We also
show a simple randomized algorithm for a $+4W$ emulator of size
$\tilde{O}(n^{4/3})$.
  In addition, we show that our technique is applicable for very sparse
additive spanners, that have linear size. For weighted graphs, we use a variant
of our simple deterministic algorithm that yields a linear size
$+\tilde{O}(\sqrt{n}\cdot W)$ spanner, and we also obtain a tradeoff between
size and stretch.
  Finally, generalizing the technique of [DHZ00] for unweighted graphs, we
devise an efficient randomized algorithm producing a $+2W$ spanner for weighted
graphs of size $\tilde{O}(n^{3/2})$ in $\tilde{O}(n^2)$ time.

25
The categories of this paper are: patt-sol
17
The categories of this paper are: q-fin
7
The categories of this paper are: hep-th
29
The categories of this paper are: funct-an
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The categories of this paper are: solv-int
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The categories of this paper are: bayes-an
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The categories of this paper are: chem-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: nlin
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The categories of this paper are: acc-phys
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The categories of this paper are: math-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: q-alg
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The categories of this paper are: alg-geom
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The categories of this paper are: econ
2
The categories of this paper are: math