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Computer Science, Data Structures and Algorithms, cs.DS
Abstract:
We show that the eccentricities, diameter, radius, and Wiener index of an
undirected $n$-vertex graph with nonnegative edge lengths can be computed in
time $O(n\cdot \binom{k+\lceil\log n\rceil}{k} \cdot 2^k k^2 \log n)$, where
$k$ is the treewidth of the graph. For every $\epsilon>0$, this bound is
$n^{1+\epsilon}\exp O(k)$, which matches a hardness result of Abboud,
Vassilevska Williams, and Wang (SODA 2015) and closes an open problem in the
multivariate analysis of polynomial-time computation. To this end, we show that
the analysis of an algorithm of Cabello and Knauer (Comp. Geom., 2009) in the
regime of non-constant treewidth can be improved by revisiting the analysis of
orthogonal range searching, improving bounds of the form $\log^d n$ to
$\binom{d+\lceil\log n\rceil}{d}$, as originally observed by Monier (J. Alg.
1980).
We also investigate the parameterization by vertex cover number.
