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

TL;DR

This paper introduces a near-linear time algorithm for approximate trace reconstruction in the average-case, using only three traces to recover a string within a small edit distance, significantly improving efficiency and accuracy.

Contribution

It presents the first near-linear time algorithm for approximate trace reconstruction with three traces in the average-case, utilizing median string computation.

Findings

Algorithm runs in near-linear time $ ilde O(n)$

Reports a string within $O( ilde{p} n)$ edit distance from the original

Uses only three traces for reconstruction

Abstract

We consider an \emph{approximate} version of the trace reconstruction problem, where the goal is to recover an unknown string $s\in\{0,1\}^n$ 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 \emph{three} traces. It runs in near-linear time $\tilde O(n)$ and with high probability reports a string within edit distance $O(\epsilon p n)$ from $s$ for $\epsilon=\tilde O(p)$, which significantly improves over the straightforward bound of $O(pn)$. Technically, our algorithm computes a $(1+\epsilon)$-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 $O(n^3)$.