The next TCS+ talk will take place next Wednesday, September 27th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). Yuval Peres (Microsoft Research) will will speak about “Trace reconstruction for the deletion channel” (abstract below).
Please make sure you reserve a spot for your group to join us live by signing up on the online form. As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.
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 recently, the best upper bound 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. (Similar results were obtained independently and concurrently by De, O’Donnell and Servedio). If the string $x$ is random and $q<1/2$, we can show a subpolynomial number of traces suffices by comparison to a biased random walk.
(Joint works with Fedor Nazarov, STOC 2017 and with Alex Zhai, FOCS 2017).