Trace reconstruction with varying deletion probabilities

TL;DR

This paper extends the theoretical understanding of trace reconstruction by analyzing the problem under varying deletion probabilities, providing bounds for reconstruction when deletion rates differ across positions or symbols.

Contribution

It generalizes existing results to scenarios with non-uniform deletion probabilities, covering both position-dependent and symbol-dependent cases.

Findings

Reconstruction bounds are extended to variable deletion probabilities.

The analysis applies to both known and unknown deletion probabilities.

The results demonstrate the feasibility of trace reconstruction under more realistic deletion models.

Abstract

In the trace reconstruction problem an unknown string ${\bf x}=(x_0,\dots,x_{n-1})\in\{0,1,...,m-1\}^n$ is observed through the deletion channel, which deletes each $x_k$ with a certain probability, yielding a contracted string $\widetilde{\bf X}$. Earlier works have proved that if each $x_k$ is deleted with the same probability $q\in[0,1)$, then $\exp(O(n^{1/3}))$ independent copies of the contracted string $\widetilde{\bf X}$ suffice to reconstruct $\bf x$ with high probability. We extend this upper bound to the setting where the deletion probabilities vary, assuming certain regularity conditions. First we consider the case where $x_k$ is deleted with some known probability $q_k$. Then we consider the case where each letter $\zeta\in \{0,1,...,m-1\}$ is associated with some possibly unknown deletion probability $q_\zeta$.