Optimal Construction of Hierarchical Overlap Graphs

Genome assembly is a fundamental problem in Bioinformatics, where for a given set of overlapping substrings of a genome, the aim is to reconstruct the source genome. The classical approaches to solving this problem use assembly graphs, such as de Bruijn graphs or overlap graphs, which maintain partial information about such overlaps. For genome assembly algorithms, these graphs present a trade-off between overlap information stored and scalability. Thus, Hierarchical Overlap Graph (HOG) was proposed to overcome the limitations of both these approaches. For a given set $P$ of $n$ strings, the first algorithm to compute HOG was given by Cazaux and Rivals [IPL20] requiring $O(||P||+n^2)$ time using superlinear space, where $||P||$ is the cumulative sum of the lengths of strings in $P$. This was improved by Park et al. [SPIRE20] to $O(||P||\log n)$ time and $O(||P||)$ space using segment trees, and further to $O(||P||\frac{\log n}{\log \log n})$ for the word RAM model. Both these results described an open problem to compute HOG in optimal $O(||P||)$ time and space. In this paper, we achieve the desired optimal bounds by presenting a simple algorithm that does not use any complex data structures. At its core, our solution improves the classical result [IPL92] for a special case of the All Pairs Suffix Prefix (APSP) problem from $O(||P||+n^2)$ time to optimal $O(||P||)$ time, which may be of independent interest.