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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.

Cite AsGet BibTex

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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