A memory-efficient data structure representing exact-match overlap graphs with application for next generation DNA assembly

An exact-match overlap graph of $n$ given strings of length $\ell$ is an edge-weighted graph in which each vertex is associated with a string and there is an edge $(x,y)$ of weight $\omega = \ell - |ov_{max}(x,y)|$ if and only if $\omega \leq \lambda$, where $|ov_{max}(x,y)|$ is the length of $ov_{max}(x,y)$ and $\lambda$ is a given threshold. In this paper, we show that the exact-match overlap graphs can be represented by a compact data structure that can be stored using at most $(2\lambda -1 )(2\lceil\log n\rceil + \lceil\log\lambda\rceil)n$ bits with a guarantee that the basic operation of accessing an edge takes $O(\log \lambda)$ time. Exact-match overlap graphs have been broadly used in the context of DNA assembly and the \emph{shortest super string problem} where the number of strings $n$ ranges from a couple of thousands to a couple of billions, the length $\ell$ of the strings is from 25 to 1000, depending on DNA sequencing technologies. However, many DNA assemblers using overlap graphs are facing a major problem of constructing and storing them. Especially, it is impossible for these DNA assemblers to handle the huge amount of data produced by the next generation sequencing technologies where the number of strings $n$ is usually very large ranging from hundred million to a couple of billions. In fact, to our best knowledge there is no DNA assemblers that can handle such a large number of strings. Fortunately, with our compact data structure, the major problem of constructing and storing overlap graphs is practically solved since it only requires linear time and and linear memory. As a result, it opens the door of possibilities to build a DNA assembler that can handle large-scale datasets efficiently.