Circular Trace Reconstruction

Trace Reconstruction is the problem of learning an unknown string $x$ from independent traces of $x$, where traces are generated by independently deleting each bit of $x$ with some deletion probability $q$. In this paper, we initiate the study of Circular Trace Reconstruction, where the unknown string $x$ is circular and traces are now rotated by a random cyclic shift. Trace reconstruction is related to many computational biology problems studying DNA, which is a primary motivation for this problem as well, as many types of DNA are known to be circular. Our main results are as follows. First, we prove that we can reconstruct arbitrary circular strings of length $n$ using $\exp\big(\tilde{O}(n^{1/3})\big)$ traces for any constant deletion probability $q$, as long as $n$ is prime or the product of two primes. For $n$ of this form, this nearly matches the best known bound of $\exp\big(O(n^{1/3})\big)$ for standard trace reconstruction. Next, we prove that we can reconstruct random circular strings with high probability using $n^{O(1)}$ traces for any constant deletion probability $q$. Finally, we prove a lower bound of $\tilde{\Omega}(n^3)$ traces for arbitrary circular strings, which is greater than the best known lower bound of $\tilde{\Omega}(n^{3/2})$ in standard trace reconstruction.