The hybrid k-deck problem: Reconstructing sequences from short and long traces

We introduce a new variant of the k-deck problem, which in its traditional formulation asks for determining the smallest k that allows one to reconstruct any binary sequence of length n from the multiset of its k-length subsequences. In our version of the problem, termed the hybrid k-deck problem, one is given a certain number of special subsequences of the sequence of length n - t, t > 0, and the question of interest is to determine the smallest value of k such that the k-deck, along with the subsequences, allows for reconstructing the original sequence in an error-free manner. We first consider the case that one is given a single subsequence of the sequence of length n - t, obtained by deleting zeros only, and seek the value of k that allows for hybrid reconstruction. We prove that in this case, k [log t + 2, min{t + 1, O(√n)}]. We then proceed to extend the single-subsequence setup to the case where one is given M subsequences of length n - t obtained by deleting zeroes only. In this case, we first aggregate the asymmetric traces and then invoke the single-trace results. The analysis and problem at hand are motivated by nanopore sequencing problems for DNA-based data storage.