Bee Identification Problem for DNA Strands

Motivated by DNA-based applications, we generalize the bee identification problem proposed by Tandon et al. (2019). In this setup, we transmit all M codewords from a codebook over some channel and each codeword results in N noisy outputs. Then our task is to identify each codeword from the MN noisy outputs.First, via a reduction to a minimum-cost flow problem on a related flow network ${\mathcal{G}_N}$, we show that the problem can be solved in O(M
³
) time in the worst case. Next, we consider the deletion channel and study the expected number of edges in the network ${\mathcal{G}_N}$. Specifically, we obtain closed expressions for this quantity for certain codebooks and when the codebook comprises all binary words, we show that this quantity is sub-quadratic when the deletion probability is less than 1/2. This then implies that the expected running time for this codebook is o(M
³
). For other codebooks, we develop methods to compute the expected number of edges efficiently. Finally, we adapt classical peeling-decoding techniques to reduce the number of nodes and edges in ${\mathcal{G}_N}$.