Trace reconstruction with varying deletion probabilities

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 $\tilde{\mathbf{X}}$. Earlier works have proved that if each $x_k$ is deleted with the same probability $q\in [0,1)$, then $\exp (O(n^{1\mathrm{/}3}))$ independent copies of the contracted string $\tilde{\mathbf{X}}$ suffice to reconstruct $\mathbf{x}$ 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 $\zeta\in {0,1,…,m-1}$isassociatedwithsomepossiblyunknowndeletionprobability$q_\zeta$.