Fundamental Limits of Multiple Sequence Reconstruction from Substrings

The problem of reconstructing a sequence from the set of its length-$k$ substrings has received considerable attention due to its various applications in genomics. We study an uncoded version of this problem where multiple random sources are to be simultaneously reconstructed from the union of their $k$-mer sets. We consider an asymptotic regime where $m = n^\alpha$ i.i.d. source sequences of length $n$ are to be reconstructed from the set of their substrings of length $k=\beta \log n$, and seek to characterize the $(\alpha,\beta)$ pairs for which reconstruction is information-theoretically feasible. We show that, as $n \to \infty$, the source sequences can be reconstructed if $\beta > \max(2\alpha+1,\alpha+2)$ and cannot be reconstructed if $\beta < \max( 2\alpha+1, \alpha+ \tfrac32)$, characterizing the feasibility region almost completely. Interestingly, our result shows that there are feasible $(\alpha,\beta)$ pairs where repeats across the source strings abound, and non-trivial reconstruction algorithms are needed to achieve the fundamental limit.