Mean-Based Trace Reconstruction over Oblivious Synchronization Channels

TL;DR

This paper extends mean-based trace reconstruction results to a broad class of oblivious synchronization channels, showing that the same exponential trace complexity bounds apply, thus unifying previous findings under a general framework.

Contribution

It introduces oblivious synchronization channels and proves that mean-based trace reconstruction complexity bounds hold for this general class, unifying prior results.

Findings

Mean-based trace reconstruction requires exponential traces for these channels.

The results apply to all channels with sub-exponential output length distributions.

The work generalizes previous specific channel models to a broad class.

Abstract

Mean-based reconstruction is a fundamental, natural approach to worst-case trace reconstruction over channels with synchronization errors. It is known that $\exp(O(n^{1/3}))$ traces are necessary and sufficient for mean-based worst-case trace reconstruction over the deletion channel, and this result was also extended to certain channels combining deletions and geometric insertions of uniformly random bits. In this work, we use a simple extension of the original complex-analytic approach to show that these results are examples of a much more general phenomenon. We introduce oblivious synchronization channels, which map each input bit to an arbitrarily distributed sequence of replications and insertions of random bits. This general class captures all previously considered synchronization channels. We show that for any oblivious synchronization channel whose output length follows a sub-exponential distribution either mean-based trace reconstruction is impossible or $\exp(O(n^{1/3}))$ traces suffice for this task.