Algorithms for reconstruction over single and multiple deletion channels

TL;DR

This paper investigates sequence reconstruction over deletion channels, focusing on maximum likelihood and posterior estimation, providing new algorithms and tools for better understanding and solving these complex problems in DNA storage and sequencing.

Contribution

The paper introduces a relaxation of the ML estimation problem, computes exact posterior distributions, and develops visualization tools for deletion channel analysis.

Findings

ML estimation over single deletion channel is equivalent to its relaxation.

Exact symbolwise posterior distributions are computed for single and multiple deletion channels.

New visualization tools for error event analysis are proposed.

Abstract

Recent advances in DNA sequencing technology and DNA storage systems have rekindled the interest in deletion channels. Multiple recent works have looked at variants of sequence reconstruction over a single and over multiple deletion channels, a notoriously difficult problem due to its highly combinatorial nature. Although works in theoretical computer science have provided algorithms which guarantee perfect reconstruction with multiple independent observations from the deletion channel, they are only applicable in the large blocklength regime and more restrictively, when the number of observations is also large. Indeed, with only a few observations, perfect reconstruction of the input sequence may not even be possible in most cases. In such situations, maximum likelihood (ML) and maximum aposteriori (MAP) estimates for the deletion channels are natural questions that arise and these have remained open to the best of our knowledge. In this work, we take steps to answer the two aforementioned questions. Specifically: 1. We show that solving for the ML estimate over the single deletion channel (which can be cast as a discrete optimization problem) is equivalent to solving its relaxation, a continuous optimization problem; 2. We exactly compute the symbolwise posterior distributions (under some assumptions on the priors) for both the single as well as multiple deletion channels. As part of our contributions, we also introduce tools to visualize and analyze error events, which we believe could be useful in other related problems concerning deletion channels.