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December 2021 Reconstructing trees from traces
Sami Davies, Miklós Z. Rácz, Cyrus Rashtchian
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Ann. Appl. Probab. 31(6): 2772-2810 (December 2021). DOI: 10.1214/21-AAP1662

Abstract

We study the problem of learning a node-labeled tree given independent traces from an appropriately defined deletion channel. This problem, tree trace reconstruction, generalizes string trace reconstruction, which corresponds to the tree being a path. For many classes of trees, including complete trees and spiders, we provide algorithms that reconstruct the labels using only a polynomial number of traces. This exhibits a stark contrast to known results on string trace reconstruction, which require exponentially many traces, and where a central open problem is to determine whether a polynomial number of traces suffice. Our techniques combine novel combinatorial and complex analytic methods.

Funding Statement

The research of S.D. was supported by NSF CAREER Grant 1651861 and the David & Lucile Packard Foundation. The research of M.Z.R. was supported in part by NSF Grant DMS-1811724.

Acknowledgments

We thank Nina Holden for helpful discussions relating to Lemma 5.3, and Bichlien Nguyen and Karin Strauss for pointing us to connections on branched DNA and recent work in this area. We also thank Alyshia Olsen for help designing the figures. Finally, we thank Tatiana Brailovskaya and an anonymous referee for their careful reading of the paper and their numerous helpful questions and suggestions that helped improve the paper.

An extended abstract of this paper appears in the Proceedings of the 32nd Conference on Learning Theory (COLT), 2019 [13].

Citation

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Sami Davies. Miklós Z. Rácz. Cyrus Rashtchian. "Reconstructing trees from traces." Ann. Appl. Probab. 31 (6) 2772 - 2810, December 2021. https://doi.org/10.1214/21-AAP1662

Information

Received: 1 June 2019; Revised: 1 August 2020; Published: December 2021
First available in Project Euclid: 13 December 2021

Digital Object Identifier: 10.1214/21-AAP1662

Subjects:
Primary: 60C05 , 68Q32
Secondary: 30C80

Keywords: deletion channel , Littlewood polynomials , Trace reconstruction , tree trace reconstruction

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 6 • December 2021
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