Optimal bounds on a tree inference algorithm

TL;DR

This paper provides tight bounds on the number of path-length queries needed to infer a weighted tree's topology, establishing both upper and lower bounds and analyzing algorithmic performance.

Contribution

It improves the analysis of Hein's 1989 algorithm, establishing optimal bounds and demonstrating the existence of trees with better inference performance.

Findings

Number of queries needed is O(n k log_k n)

Lower bound matches the upper bound for query complexity

Existence of trees with asymptotically better inference performance

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 $O(n k \log_k n)$, 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 $O(n k \log_k n)$ algorithm.