Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Subquadratic-time algorithm for the diameter and all eccentricities on
  median graphs   On sparse graphs, Roditty and Williams [2013] proved that no
$O(n^{2-\varepsilon})$-time algorithm achieves an approximation factor smaller
than $\frac{3}{2}$ for the diameter problem unless SETH fails. In this article,
we solve an open question formulated in the literature: can we use the
structural properties of median graphs to break this global quadratic barrier?
  We propose the first combinatiorial algorithm computing exactly all
eccentricities of a median graph in truly subquadratic time. Median graphs
constitute the family of graphs which is the most studied in metric graph
theory because their structure represents many other discrete and geometric
concepts, such as CAT(0) cube complexes. Our result generalizes a recent one,
stating that there is a linear-time algorithm for all eccentricities in median
graphs with bounded dimension $d$, i.e. the dimension of the largest induced
hypercube. This prerequisite on $d$ is not necessarily anymore to determine all
eccentricities in subquadratic time. The execution time of our algorithm is
$O(n^{1.6456}\log^{O(1)} n)$.
  We provide also some satellite outcomes related to this general result. In
particular, restricted to simplex graphs, this algorithm enumerates all
eccentricities with a quasilinear running time. Moreover, an algorithm is
proposed to compute exactly all reach centralities in time
$O(2^{3d}n\log^{O(1)}n)$.

24
The categories of this paper are: dg-ga
25
The categories of this paper are: patt-sol
23
The categories of this paper are: solv-int
29
The categories of this paper are: funct-an
21
The categories of this paper are: adap-org
13
The categories of this paper are: quant-ph
28
The categories of this paper are: alg-geom
16
The categories of this paper are: stat
7
The categories of this paper are: hep-th
15
The categories of this paper are: nucl-ex
12
The categories of this paper are: nucl-th
4
The categories of this paper are: cond-mat
9
The categories of this paper are: nlin
37
The categories of this paper are: supr-con
10
The categories of this paper are: q-bio
32
The categories of this paper are: atom-ph