Near-Optimal Trace Reconstruction for Mildly Separated Strings

TL;DR

This paper presents a polynomial-time, near-optimal solution for trace reconstruction of mildly separated binary strings with constant deletion probability, significantly improving previous bounds.

Contribution

It introduces an efficient method for trace reconstruction when the string has polylogarithmic separation, achieving near-linear trace complexity.

Findings

Reconstruction with O(n log n) traces for mildly separated strings

Polynomial-time algorithm for the problem under specified conditions

Improved bounds compared to previous exponential and polynomial lower bounds

Abstract

In the trace reconstruction problem our goal is to learn an unknown string $x\in \{0,1\}^n$ given independent traces of $x$. A trace is obtained by independently deleting each bit of $x$ with some probability $\delta$ and concatenating the remaining bits. It is a major open question whether the trace reconstruction problem can be solved with a polynomial number of traces when the deletion probability $\delta$ is constant. The best known upper bound and lower bounds are respectively $\exp(\tilde O(n^{1/5}))$ and $\tilde \Omega(n^{3/2})$ both by Chase [Cha21b,Cha21a]. Our main result is that if the string $x$ is mildly separated, meaning that the number of zeros between any two ones in $x$ is at least polylog$n$, and if $\delta$ is a sufficiently small constant, then the trace reconstruction problem can be solved with $O(n \log n)$ traces and in polynomial time.