Reconstructing Arbitrary Trees from Traces in the Tree Edit Distance Model

TL;DR

This paper advances the understanding of reconstructing arbitrary trees from traces in the tree edit distance model, providing reductions, bounds, and algorithms that extend prior work on special cases.

Contribution

It introduces a reduction from tree trace reconstruction to string reconstruction, establishes lower bounds, and presents a general algorithm for learning arbitrary tree topologies.

Findings

Reduction from tree trace to string reconstruction when topology is known

Lower bounds on learning arbitrary tree topologies

Algorithm for topology learning using Nazarov and Peres techniques

Abstract

In this paper, we consider the problem of reconstructing trees from traces in the tree edit distance model. Previous work by Davies et al. (2019) analyzed special cases of reconstructing labeled trees. In this work, we significantly expand our understanding of this problem by giving general results in the case of arbitrary trees. Namely, we give: a reduction from the tree trace reconstruction problem to the more classical string reconstruction problem when the tree topology is known, a lower bound for learning arbitrary tree topologies, and a general algorithm for learning the topology of any tree using techniques of Nazarov and Peres (2017). We conclude by discussing why arbitrary trees require exponentially many samples under the left propagation model.