Optimal bounds on a tree inference algorithm

Abstract

This paper tightens the best known analysis of Hein's 1989 algorithm to infer the topology of a weighted tree based on the lengths of paths between its leaves. It shows that the number of length queries required for a degree-$k$ tree of $n$ leaves is $O(nk{\log }_{k}n)$, which is the lower bound. It also presents a family of trees for which the performance is asymptotically better, and shows that no such family exists for a competing $O(nk{\log }_{k}n)$ algorithm.