Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Isometric embeddings in trees and their use in the diameter problem   We prove that given a discrete space with $n$ points which is either embedded
in a system of $k$ trees, or the Cartesian product of $k$ trees, we can compute
all eccentricities in ${\cal O}(2^{{\cal O}(k\log{k})}(N+n)^{1+o(1)})$ time,
where $N$ is the cumulative total order over all these $k$ trees. This is near
optimal under the Strong Exponential-Time Hypothesis, even in the very special
case of an $n$-vertex graph embedded in a system of $\omega(\log{n})$ spanning
trees. However, given such an embedding in the strong product of $k$ trees,
there is a much faster ${\cal O}(N + kn)$-time algorithm for this problem. All
our positive results can be turned into approximation algorithms for the graphs
and finite spaces with a quasi isometric embedding in trees, if such embedding
is given as input, where the approximation factor (resp., the approximation
constant) depends on the distortion of the embedding (resp., of its stretch).
The existence of embeddings in the Cartesian product of finitely many trees has
been thoroughly investigated for cube-free median graphs. We give the
first-known quasi linear-time algorithm for computing the diameter within this
graph class. It does not require an embedding in a product of trees to be given
as part of the input. On our way, being given an $n$-node tree $T$, we propose
a data structure with ${\cal O}(n\log{n})$ pre-processing time in order to
compute in ${\cal O}(k\log^2{n})$ time the eccentricity of any subset of $k$
nodes. We combine the latter technical contribution, of independent interest,
with a recent distance-labeling scheme that was designed for cube-free median
graphs.

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The categories of this paper are: nucl-th
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The categories of this paper are: q-bio
2
The categories of this paper are: math
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The categories of this paper are: atom-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: chem-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: funct-an
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The categories of this paper are: eess
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The categories of this paper are: comp-gas
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The categories of this paper are: nlin
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The categories of this paper are: patt-sol
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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: nucl-ex
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The categories of this paper are: gr-qc