Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Better Distance Preservers and Additive Spanners   We study two popular ways to sketch the shortest path distances of an input
graph. The first is distance preservers, which are sparse subgraphs that agree
with the distances of the original graph on a given set of demand pairs. Prior
work on distance preservers has exploited only a simple structural property of
shortest paths, called consistency, stating that one can break shortest path
ties such that no two paths intersect, split apart, and then intersect again
later. We prove that consistency alone is not enough to understand distance
preservers, by showing both a lower bound on the power of consistency and a new
general upper bound that polynomially surpasses it. Specifically, our new upper
bound is that any $p$ demand pairs in an $n$-node undirected unweighted graph
have a distance preserver on $O(n^{2/3}p^{2/3} + np^{1/3})$ edges. We leave a
conjecture that the right bound is $O(n^{2/3}p^{2/3} + n)$ or better.
  The second part of this paper leverages these distance preservers in a new
construction of additive spanners, which are subgraphs that preserve all
pairwise distances up to an additive error function. We give improved error
bounds for spanners with relatively few edges; for example, we prove that all
graphs have spanners on $O(n)$ edges with $+O(n^{3/7 + \varepsilon})$ error.
Our construction can be viewed as an extension of the popular path-buying
framework to clusters of larger radii.

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The categories of this paper are: funct-an
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The categories of this paper are: nucl-th
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The categories of this paper are: adap-org
8
The categories of this paper are: hep-ex
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The categories of this paper are: nlin
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The categories of this paper are: alg-geom
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The categories of this paper are: math-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: supr-con
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The categories of this paper are: gr-qc
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The categories of this paper are: bayes-an
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The categories of this paper are: dg-ga
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The categories of this paper are: stat
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The categories of this paper are: astro-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: physics