Yuval Peres: Trace reconstruction for the deletion channel

Abstract: In the trace reconstruction problem, an unknown string $x$ of $n$ bits is observed through  the deletion channel, which deletes each bit with some constant probability $q$, yielding a contracted string.  How many independent outputs (traces) of the deletion channel are needed to reconstruct $x$ with high probability?

The best lower bound known is linear in $n$.  Until 2016, the best upper bound (due to Holenstein, Mitzenmacher, Panigrahy and Wieder 2008) was exponential in the square root of $n$. We improve the square root to a cube root using statistics of individual output bits and some complex analysis;  this bound is sharp for reconstruction algorithms that only use this statistical information. (Joint work with Fedor Nazarov, STOC 2017; similar results were obtained independently and concurrently by De, O’Donnell and Servedio).  In very recent work with Alex Zhai, we showed that If the string $x$  is random and $q<1/2$, then  a subpolynomial number of traces suffices.