Average-Case to (shifted) Worst-Case Reduction for the Trace Reconstruction Problem

The {\em insertion-deletion channel} takes as input a binary string $x \in\{0, 1\}^n$, and outputs a string $\widetilde{x}$ where some of the bits have been deleted and others inserted independently at random. In the {\em trace reconstruction problem}, one is given many outputs (called {\em traces}) of the insertion-deletion channel on the same input message $x$, and asked to recover the input message. Nazarov and Peres (STOC 2017), and De, Odonnell and Servido (STOC 2017) showed that any string $x$ can be reconstructed from $\exp(O(n^{1/3}))$ traces. Holden, Pemantle, Peres and Zhai (COLT 2018) adapt the techniques used to prove this upper bound, to an algorithm for the average-case trace reconstruction with a sample complexity of $\exp(O(\log^{1/3} n))$. However, it is not clear how to apply their techniques more generally and in particular for the recent worst-case upper bound of $\exp(\widetilde{O}(n^{1/5}))$ shown by Chase (STOC 2021) for the deletion-channel. We prove a general reduction from the average-case to smaller instances of a problem similar to worst-case. Using this reduction and a generalization of Chase's bound, we construct an improved average-case algorithm with a sample complexity of $\exp(\widetilde{O}(\log^{1/5} n))$. Additionally, we show that Chase's upper-bound holds for the insertion-deletion channel as well.