Trace Reconstruction Revisited

The trace reconstruction problem is to reconstruct a string x of length n given m random subsequences where each subsequence is generated by deleting each character of x independently with probability p. Two natural questions are a) how large must m be as a function of n and p such that reconstruction is possible with high probability and b) how can this reconstruction be performed efficiently. Existing work considers the case when x is chosen uniformly at random and when x is arbitrary. In this paper, we relate the complexity of both cases; improve bounds by Holenstein et al. (SODA 2008) on the sufficient value of m in both cases; and present a significantly simpler analysis for some of the results proved by Viswanathan and Swaminathan (SODA 2008), Kannan and McGregor (ISIT 2005), and Batu et al. (SODA 2004). In particular, our work implies the first sub-polynomial upper bound (when the alphabet is polylogn) and super-logarithmic lower bound on the number of traces required when x is random and p is constant.