Limitations of Mean-Based Algorithms for Trace Reconstruction at Small Distance

Trace reconstruction considers the task of recovering an unknown string x is an element of {0,1}(n) given a number of independent "traces", i.e., subsequences of x obtained by randomly and independently deleting every symbol of x with some probability p. The information-theoretic limit of the number of traces needed to recover a string of length n are still unknown. This limit is essentially the same as the number of traces needed to determine, given strings x and y and traces of one of them, which string is the source. The most studied class of algorithms for the worst-case version of the problem are "mean-based" algorithms. These are a restricted class of distinguishers that only use the mean value of each coordinate on the given samples. In this work we study limitations of mean-based algorithms on strings at small Hamming or edit distance. We show on the one hand that distinguishing strings that are nearby in Hamming distance is "easy" for such distinguishers. On the other hand, we show that distinguishing strings that are nearby in edit distance is "hard" for mean-based algorithms. Along the way we also describe a connection to the famous Prouhet-Tarry-Escott (PTE) problem, which shows a barrier to finding explicit hard-to-distinguish strings: namely such strings would imply explicit short solutions to the PTE problem, a well-known difficult problem in number theory. Our techniques rely on complex analysis arguments that involve careful trigonometric estimates, and algebraic techniques that include applications of Descartes' rule of signs for polynomials over the reals.