Polynomial-time trace reconstruction in the low deletion rate regime

TL;DR

This paper presents a polynomial-time algorithm for trace reconstruction that works effectively at higher deletion rates than previous methods, specifically for deletion probabilities up to n^{-(1/3 + ε)}.

Contribution

It introduces a new polynomial-time algorithm that improves the deletion rate threshold for trace reconstruction from n^{-(1/2 + ε)} to n^{-(1/3 + ε)}.

Findings

Successfully reconstructs source strings at higher deletion rates.

Combines alignment-based and novel subword length determination procedures.

Advances the theoretical understanding of trace reconstruction complexity.

Abstract

In the \emph{trace reconstruction problem}, an unknown source string $x \in \{0,1\}^n$ is transmitted through a probabilistic \emph{deletion channel} which independently deletes each bit with some fixed probability $\delta$ and concatenates the surviving bits, resulting in a \emph{trace} of $x$. The problem is to reconstruct $x$ given access to independent traces. Trace reconstruction of arbitrary (worst-case) strings is a challenging problem, with the current state of the art for poly$(n)$-time algorithms being the 2004 algorithm of Batu et al. \cite{BKKM04}. This algorithm can reconstruct an arbitrary source string $x \in \{0,1\}^n$ in poly$(n)$ time provided that the deletion rate $\delta$ satisfies $\delta \leq n^{-(1/2 + \varepsilon)}$ for some $\varepsilon > 0$. In this work we improve on the result of \cite{BKKM04} by giving a poly$(n)$-time algorithm for trace reconstruction for any deletion rate $\delta \leq n^{-(1/3 + \varepsilon)}$. Our algorithm works by alternating an alignment-based procedure, which we show effectively reconstructs portions of the source string that are not "highly repetitive", with a novel procedure that efficiently determines the length of highly repetitive subwords of the source string.