Reconstructing trees from traces

Davies, Sami; Rácz, Miklós Z.; Rashtchian, Cyrus

doi:10.1214/21-aap1662

Cited by 11 publications

(32 citation statements)

References 26 publications

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“…Variants of trace reconstruction have been proposed in order to develop new techniques to close the exponential gap in the original problem formulation. Davies et al [10] investigated a new problem of reconstructing trees from their traces. The authors investigated two classes of trees; complete k-ary trees and spider graphs.…”

Section: Related Workmentioning

confidence: 99%

“…In this paper, we study the complexity of reconstructing trees from traces under the tree edit distance model, as defined by Davies et al [10]. Tree trace reconstruction is a generalization of the traditional trace reconstruction problem, where we observe noisy samples of a binary string after passing it through a deletion channel several times [1].…”

Section: Introductionmentioning

confidence: 99%

“…This naturally leads to the tree trace reconstruction problem posed by Davies et al [10]. In their model, a tree of known topology has n nodes, and each node v also carries one hidden bit of information (v) ∈ {0, 1}.…”

Section: Introductionmentioning

confidence: 99%

“…Given this tree trace reconstruction setting, in this paper we focus primarily on the tree edit distance (TED) deletion model for trees, as presented by Davies et al [10]. In the TED model, when a node is deleted, all of its children become children of its parent.…”

Section: Introductionmentioning

confidence: 99%

“…The tree trace reconstruction problem trivially generalizes the string reconstruction problems; for example, a path graph is a tree, and under node deletions is equivalent to the string reconstruction problem. Some tree structures simplify the problem; for example, Davies et al [10] showed significant improvements over the Ω(exp(Cn 1/3 )) bound for the cases of k-ary trees and spider graphs. However, it was a priori possible that some tree structures may present harder problems and require even more samples than their string counterparts.…”

Section: Introductionmentioning

confidence: 99%

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Reconstructing Arbitrary Trees from Traces in the Tree Edit Distance Model

Maranzatto¹

2021

Preprint

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In this paper, we consider the problem of reconstructing trees from traces in the tree edit distance model. Previous work by Davies et al. [10] 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 [19]. We conclude by discussing why arbitrary trees require exponentially many samples under the left propagation model.

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Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Reconstructing Arbitrary Trees from Traces in the Tree Edit Distance Model

Maranzatto¹

2021

Preprint

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Tree trace reconstruction using subtraces

Brailovskaya

Rácz

2022

J. Appl. Probab.

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Tree trace reconstruction aims to learn the binary node labels of a tree, given independent samples of the tree passed through an appropriately defined deletion channel. In recent work, Davies, Rácz, and Rashtchian [10] used combinatorial methods to show that $\exp({\mathrm{O}} (k \log_{k} n))$ samples suffice to reconstruct a complete k-ary tree with n nodes with high probability. We provide an alternative proof of this result, which allows us to generalize it to a broader class of tree topologies and deletion models. In our proofs we introduce the notion of a subtrace, which enables us to connect with and generalize recent mean-based complex analytic algorithms for string trace reconstruction.

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