Almost optimal algorithms for diameter-optimally augmenting trees

We consider the problem of augmenting an $n$-vertex tree with one shortcut in order to minimize the diameter of the resulting graph. The tree is embedded in an unknown space and we have access to an oracle that, when queried on a pair of vertices $u$ and $v$, reports the weight of the shortcut $(u,v)$ in constant time. Previously, the problem was solved in $O(n^2 \log^3 n)$ time for general weights [Oh and Ahn, ISAAC 2016], in $O(n^2 \log n)$ time for trees embedded in a metric space [Große et al., {\tt arXiv:1607.05547}], and in $O(n \log n)$ time for paths embedded in a metric space [Wang, WADS 2017]. Furthermore, a $(1+\varepsilon)$-approximation algorithm running in $O(n+1/\varepsilon^{3})$ has been designed for paths embedded in $\mathbb{R}^d$, for constant values of $d$ [Große et al., ICALP 2015]. The contribution of this paper is twofold: we address the problem for trees (not only paths) and we also improve upon all known results. More precisely, we design a {\em time-optimal} $O(n^2)$ time algorithm for general weights. Moreover, for trees embedded in a metric space, we design (i) an exact $O(n \log n)$ time algorithm and (ii) a $(1+\varepsilon)$-approximation algorithm that runs in $O\big(n+ \varepsilon^{-1}\log \varepsilon^{-1}\big)$ time.