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Approximate Trace Reconstruction via Median String (In Average-Case)

Authors Diptarka Chakraborty, Debarati Das, Robert Krauthgamer

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Author Details

Diptarka Chakraborty
  • National University of Singapore, Singapore
Debarati Das
  • Basic Algorithm Research Copenhagen (BARC), University of Copenhagen, Denmark
Robert Krauthgamer
  • Weizmann Institute of Science, Rehovot, Israel

Funding

  • Chakraborty, Diptarka: Work partially supported by NUS ODPRT Grant, WBS No. R-252-000-A94-133.
  • Krauthgamer, Robert: Work partially supported by ONR Award N00014-18-1-2364, the Israel Science Foundation grant #1086/18, and a Minerva Foundation grant, and by the Israeli Council for Higher Education (CHE) via the Weizmann Data Science Research Center.

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Diptarka Chakraborty, Debarati Das, and Robert Krauthgamer. Approximate Trace Reconstruction via Median String (In Average-Case). In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 11:1-11:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.FSTTCS.2021.11

Abstract

We consider an approximate version of the trace reconstruction problem, where the goal is to recover an unknown string s ∈ {0,1}ⁿ from m traces (each trace is generated independently by passing s through a probabilistic insertion-deletion channel with rate p). We present a deterministic near-linear time algorithm for the average-case model, where s is random, that uses only three traces. It runs in near-linear time Õ(n) and with high probability reports a string within edit distance Õ(p² n) from s, which significantly improves over the straightforward bound of O(pn).
Technically, our algorithm computes a (1+ε)-approximate median of the three input traces. To prove its correctness, our probabilistic analysis shows that an approximate median is indeed close to the unknown s. To achieve a near-linear time bound, we have to bypass the well-known dynamic programming algorithm that computes an optimal median in time O(n³).

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Trace Reconstruction
  • Approximation Algorithms
  • Edit Distance
  • String Median

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