Optimal bounds on a tree inference algorithm

TL;DR

The paper tightens the analysis of Hein's distance-based tree-inference algorithm, proving an optimal $O(n k \log_k n)$ query bound for trees with maximum degree $k$ and revealing cases where unbalanced trees admit $o(n k \log_k n)$ performance. It formalizes the tree-inference problem via leaf-leaf distances, introduces anchor calculations to place new leaves, and recasts complexity through rooted and unrooted recursion. By developing refined combinatorial bounds using $k$-ary and $g$-beanstalk structures, it shows the bound is tight in general and nearly optimal for several unbalanced classes, while contrasting with Brodal et al.'s $O(n k \log_k n)$ algorithm. The work highlights both asymptotic optimality and potential instance-optimality questions, and notes that implementation efficiency remains an open challenge with possible $O(n \log n)$ time solutions for fixed $k$. Overall, it advances understanding of the trade-offs between query richness and tree-imbalance in topology reconstruction from leaf distances.

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.

Figures (4)

• Figure 1: The red, green and blue lines represent $d(x,z)$, $d(x,y)$ and $d(y,z)$ respectively. Since $d(y,z)$ is subtracted from the anchor calculation, it will cancel out sections of the red and green lines, leaving us with $d(x,y) + d(x,z) - d(y,z) = 2d(x,a) = 2\sigma(x,y,z)$. Dividing by 2 gives the equation for this anchor calculation.
• Figure 2: A tree rooted at $v$, with leaf $x$ within $T_u$, and $y$ a leaf in $T_1$
• Figure 3: The red edge is the edge chosen to split on, and the nodes in bold are the roots of the rooted trees created by the splits. The complexity of this tree is $\max\{2,2\} + 2 = 4$.
• Figure 4: A filled tree with a layer-degree sequence of $(3,2)$

Theorems & Definitions (17)

• Theorem 3.1
• Lemma 3.2
• Lemma 3.3
• proof : Proof of Theorem \ref{['theorem:recursion']}
• Lemma 3.4
• Lemma 3.5
• Theorem 3.6
• proof
• Theorem 3.7
• proof