Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sublinear Average-Case Shortest Paths in Weighted Unit-Disk Graphs   We consider the problem of computing shortest paths in weighted unit-disk
graphs in constant dimension $d$. Although the single-source and all-pairs
variants of this problem are well-studied in the plane case, no non-trivial
exact distance oracles for unit-disk graphs have been known to date, even for
$d=2$.
  The classical result of Sedgewick and Vitter [Algorithmica '86] shows that
for weighted unit-disk graphs in the plane the $A^*$ search has average-case
performance superior to that of a standard shortest path algorithm, e.g.,
Dijkstra's algorithm. Specifically, if the $n$ corresponding points of a
weighted unit-disk graph $G$ are picked from a unit square uniformly at random,
and the connectivity radius is $r\in (0,1)$, $A^*$ finds a shortest path in $G$
in $O(n)$ expected time when $r=\Omega(\sqrt{\log n/n})$, even though $G$ has
$\Theta((nr)^2)$ edges in expectation. In other words, the work done by the
algorithm is in expectation proportional to the number of vertices and not the
number of edges.
  In this paper, we break this natural barrier and show even stronger sublinear
time results. We propose a new heuristic approach to computing point-to-point
exact shortest paths in unit-disk graphs. We analyze the average-case behavior
of our heuristic using the same random graph model as used by Sedgewick and
Vitter and prove it superior to $A^*$. Specifically, we show that, if we are
able to report the set of all $k$ points of $G$ from an arbitrary rectangular
region of the plane in $O(k + t(n))$ time, then a shortest path between
arbitrary two points of such a random graph on the plane can be found in
$O(1/r^2 + t(n))$ expected time. In particular, the state-of-the-art range
reporting data structures imply a sublinear expected bound for all
$r=\Omega(\sqrt{\log n/n})$ and $O(\sqrt{n})$ expected bound for
$r=\Omega(n^{-1/4})$ after only near-linear preprocessing of the point set.

24
The categories of this paper are: dg-ga
29
The categories of this paper are: funct-an
22
The categories of this paper are: chao-dyn
18
The categories of this paper are: econ
36
The categories of this paper are: cmp-lg
2
The categories of this paper are: math
14
The categories of this paper are: math-ph
6
The categories of this paper are: astro-ph
28
The categories of this paper are: alg-geom
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The categories of this paper are: q-alg
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The categories of this paper are: chem-ph
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The categories of this paper are: supr-con
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The categories of this paper are: q-bio
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The categories of this paper are: solv-int
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The categories of this paper are: mtrl-th
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The categories of this paper are: ao-sci