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(n2log3⁡n) time for general weights [Oh and Ahn, ISAAC 2016], in O(n2log⁡n) time for trees embedded in a metric space [Große et al., Int. J. FCS], and in O(nlog⁡n) time for paths embedded in a metric space [Wang, Comput. Geom.]. Very recently an algorithm for general weights requiring O(n2log⁡n) time and O(n) space has been developed [Wang and Zhao, Theor. Comp. Science]. Finally, a (1+ε)-approximation algorithm running in O(n+1/ε3) has been designed for paths embedded in Rd, for constant values of d [Große et al., Int. J. FCS]. In this paper we design an asymptotic time-optimal algorithm for trees and general edge-weights that requires O(n2) time and O(nlog⁡n) space. Moreover, for trees embedded in a metric space, we design (i) an exact O(nlog⁡n)-time algorithm and (ii) a (1+ε)-approximation algorithm that runs in O(n+ε−1log⁡ε−1) time.