A Generalized Trace Reconstruction Problem: Recovering a String of Probabilities

TL;DR

This paper generalizes the trace reconstruction problem to probabilistic strings with deletion, establishing bounds for worst-case scenarios and demonstrating efficient recovery for random strings under certain conditions.

Contribution

It introduces a new probabilistic trace reconstruction model and provides both lower bounds for worst-case strings and efficient algorithms for random strings.

Findings

No algorithms can approximate worst-case strings with fewer than 2^{Ω(√n)} traces for certain deletion probabilities.

Random strings can be reconstructed efficiently with polynomially many traces when deletion probability is small.

Fourier analysis is used to establish indistinguishability and lower bounds in the model.

Abstract

We introduce the following natural generalization of trace reconstruction, parameterized by a deletion probability $\delta \in (0,1)$ and length $n$: There is a length $n$ string of probabilities, $S=p_1,\ldots,p_n,$ and each "trace" is obtained by 1) sampling a length $n$ binary string whose $i$th coordinate is independently set to 1 with probability $p_i$ and 0 otherwise, and then 2) deleting each of the binary values independently with probability $\delta$, and returning the corresponding binary string of length $\le n$. The goal is to recover an estimate of $S$ from a set of independently drawn traces. In the case that all $p_i \in \{0,1\}$ this is the standard trace reconstruction problem. We show two complementary results. First, for worst-case strings $S$ and any deletion probability at least order $1/\sqrt{n}$, no algorithm can approximate $S$ to constant $\ell_\infty$ distance or $\ell_1$ distance $o(\sqrt n)$ using fewer than $2^{\Omega(\sqrt{n})}$ traces. Second -- as in the case for standard trace reconstruction -- reconstruction is easy for random $S$: for any sufficiently small constant deletion probability, and any $\epsilon>0$, drawing each $p_i$ independently from the uniform distribution over $[0,1]$, with high probability $S$ can be recovered to $\ell_1$ error $\epsilon$ using $\mathrm{poly}(n,1/\epsilon)$ traces and computation time. We show indistinguishability in our lower bound by regarding a complicated alternating sum (comparing two distributions) as the Fourier transformation of some function evaluated at $\pm \pi,$ and then showing that the Fourier transform decays rapidly away from zero by analyzing its moment generating function.