On k-Mer-Based and Maximum Likelihood Estimation Algorithms for Trace Reconstruction

TL;DR

This paper investigates the limitations and performance of k-mer-based and maximum likelihood algorithms for trace reconstruction, establishing optimality bounds and analyzing their sample complexity in recovering original binary strings from traces.

Contribution

It proves the optimality of trace complexity for k-mer-based algorithms and analyzes the near-optimal performance of the maximum likelihood estimator in trace reconstruction.

Findings

k-mer algorithms require exponential traces, matching the lower bound

MLE achieves nearly optimal trace complexity, within a factor of n

Analysis techniques used are essentially tight, indicating need for new methods

Abstract

The goal of the trace reconstruction problem is to recover a string $x\in\{0,1\}^n$ given many independent {\em traces} of $x$, where a trace is a subsequence obtained from deleting bits of $x$ independently with some given probability $p\in [0,1).$ A recent result of Chase (STOC 2021) shows how $x$ can be determined (in exponential time) from $\exp(\widetilde{O}(n^{1/5}))$ traces. This is the state-of-the-art result on the sample complexity of trace reconstruction. In this paper we consider two kinds of algorithms for the trace reconstruction problem. Our first, and technically more involved, result shows that any $k$-mer-based algorithm for trace reconstruction must use $\exp(\Omega(n^{1/5}))$ traces, under the assumption that the estimator requires $poly(2^k, 1/\varepsilon)$ traces, thus establishing the optimality of this number of traces. The analysis of this result also shows that the analysis technique used by Chase (STOC 2021) is essentially tight, and hence new techniques are needed in order to improve the worst-case upper bound. Our second, simple, result considers the performance of the Maximum Likelihood Estimator (MLE), which specifically picks the source string that has the maximum likelihood to generate the samples (traces). We show that the MLE algorithm uses a nearly optimal number of traces, \ie, up to a factor of $n$ in the number of samples needed for an optimal algorithm, and show that this factor of $n$ loss may be necessary under general ``model estimation'' settings.