Substring Density Estimation from Traces

TL;DR

This paper introduces a method to estimate the density map of substrings within a binary string from traces, requiring polynomially many traces, which improves understanding of trace reconstruction for certain string classes.

Contribution

It presents a novel approach to recover substring density maps from traces with polynomial trace complexity, extending trace reconstruction to a new density estimation problem.

Findings

Polynomial trace complexity for density map recovery

Effective density map estimation with error at most epsilon

Trace reconstruction for strings with large density map distance

Abstract

In the trace reconstruction problem, one seeks to reconstruct a binary string $s$ from a collection of traces, each of which is obtained by passing $s$ through a deletion channel. It is known that $\exp(\tilde O(n^{1/5}))$ traces suffice to reconstruct any length-$n$ string with high probability. We consider a variant of the trace reconstruction problem where the goal is to recover a "density map" that indicates the locations of each length-$k$ substring throughout $s$. We show that $\epsilon^{-2}\cdot \text{poly}(n)$ traces suffice to recover the density map with error at most $\epsilon$. As a result, when restricted to a set of source strings whose minimum "density map distance" is at least $1/\text{poly}(n)$, the trace reconstruction problem can be solved with polynomially many traces.