Approximate Trace Reconstruction from a Single Trace

TL;DR

This paper investigates the approximate reconstruction of a binary string from a single trace affected by deletions, providing algorithms and bounds across different deletion regimes and source models.

Contribution

It introduces algorithms and lower bounds for approximate trace reconstruction from a single trace in various deletion regimes and source models.

Findings

Single trace can significantly improve worst-case reconstruction accuracy.

In average-case, a single trace offers limited improvement over no traces.

Efficient algorithms match lower bounds in several regimes.

Abstract

The well-known trace reconstruction problem is the problem of inferring an unknown source string $x \in \{0,1\}^n$ from independent "traces", i.e. copies of $x$ that have been corrupted by a $\delta$-deletion channel which independently deletes each bit of $x$ with probability $\delta$ and concatenates the surviving bits. The current paper considers the extreme data-limited regime in which only a single trace is provided to the reconstruction algorithm. In this setting exact reconstruction is of course impossible, and the question is to what accuracy the source string $x$ can be approximately reconstructed. We give a detailed study of this question, providing algorithms and lower bounds for the high, intermediate, and low deletion rate regimes in both the worst-case ($x$ is arbitrary) and average-case ($x$ is drawn uniformly from $\{0,1\}^n$) models. In several cases the lower bounds we establish are matched by computationally efficient algorithms that we provide. We highlight our results for the high deletion rate regime: roughly speaking, they show that - Having access to a single trace is already quite useful for worst-case trace reconstruction: an efficient algorithm can perform much more accurate reconstruction, given one trace that is even only a few bits long, than it could given no traces at all. But in contrast, - in the average-case setting, having access to a single trace is provably not very useful: no algorithm, computationally efficient or otherwise, can achieve significantly higher accuracy given one trace that is $o(n)$ bits long than it could with no traces.