Lower bounds for trace reconstruction

TL;DR

This paper establishes new lower bounds on the number of traces needed for reconstructing bit strings in the trace reconstruction problem, showing that significantly more samples are necessary than previously known, especially for worst-case and random strings.

Contribution

It proves improved lower bounds on the sample complexity for trace reconstruction, advancing understanding of the fundamental difficulty of the problem.

Findings

At least c n^{5/4}/√log n traces are needed for certain string pairs.

Improved lower bound for random string reconstruction from c log^2 n to c log^{9/4} n / √log log n.

Demonstrates fundamental limits on trace reconstruction sample complexity.

Abstract

In the trace reconstruction problem, an unknown bit string ${\bf x}\in\{0,1 \}^n$ is sent through a deletion channel where each bit is deleted independently with some probability $q\in(0,1)$, yielding a contracted string $\widetilde{\bf x}$. How many i.i.d.\ samples of $\widetilde{\bf x}$ are needed to reconstruct $\bf x$ with high probability? We prove that there exist ${\bf x},{\bf y} \in\{0,1 \}^n$ such that at least $c\, n^{5/4}/\sqrt{\log n}$ traces are required to distinguish between ${\bf x}$ and ${\bf y}$ for some absolute constant $c$, improving the previous lower bound of $c\,n$. Furthermore, our result improves the previously known lower bound for reconstruction of random strings from $c \log^2 n$ to $c \log^{9/4}n/\sqrt{\log \log n} $.