Walks on SPR Neighborhoods

TL;DR

This paper establishes tight bounds on the shortest sequences of NNI moves that visit all trees in an SPR neighborhood, resolving a longstanding conjecture in phylogenetics.

Contribution

It provides the first tight bounds on NNI-walk lengths visiting all trees in an SPR neighborhood, confirming Bryant's Second Combinatorial Conjecture.

Findings

Shortest NNI-walks are (n^2) steps longer than the number of trees in the neighborhood.

The bounds are tight and exact for unrooted binary trees.

Answers a 2011 conjecture in phylogenetics.

Abstract

A nearest-neighbor-interchange (NNI) walk is a sequence of unrooted phylogenetic trees, T_0, T_1, T_2,... where each consecutive pair of trees differ by a single NNI move. We give tight bounds on the length of the shortest NNI-walks that visit all trees in an subtree-prune-and-regraft (SPR) neighborhood of a given tree. For any unrooted, binary tree, T, on n leaves, the shortest walk takes {\theta}(n^2) additional steps than the number of trees in the SPR neighborhood. This answers Bryant's Second Combinatorial Conjecture from the Phylogenetics Challenges List, the Isaac Newton Institute, 2011, and the Penny Ante Problem List, 2009.