Prokrustean Graph: A substring index for rapid k-mer size analysis

The widespread adoption of k-mers in bioinformatics has led to efficient methods utilizing genomic sequences in a variety of biological tasks. However, understanding the influence of k-mer sizes within these methods remains a persistent challenge, as the outputs of complex bioinformatics pipelines obscure this influence with various noisy factors. The choice of k-mer size is often arbitrary, with justification frequently omitted in the literature and method tutorials. Furthermore, recent methods employing multiple k-mer sizes encounter significant computational challenges. Nevertheless, most methods are built on well-defined objects related to k-mers, such as de Bruijn graphs, Jaccard similarity, Bray-Curtis dissimilarity, and k-mer spectra. The role of k-mer sizes within these objects is more intuitive and can be described by numerous quantities and metrics. Therefore, exploring these objects across k-mer sizes opens opportunities for robust analyses and new applications. However, the evolution of k-mer objects with respect to k-mer sizes is surprisingly elusive. We introduce a novel substring index, the Prokrustean graph, that elucidates the transformation of k-mer sets across k-mer sizes. Our framework built upon this index rapidly computes k-mer-based quantities for all k-mer sizes, with computational complexity independent of the size range and dependent only on maximal repeats. For example, counting maximal simple paths in de Bruijn graphs for $k=1,\dots ,100$ is achieved in seconds using our index on a gigabase-scale dataset. We present a variety of such experiments relevant to pangenomics and metagenomics. The Prokrustean graph is space-efficiently constructed from the Burrows-Wheeler Transform. Through this construction, it becomes evident that other modern substring indices inherently face difficulties in exploring k-mer objects across sizes, which motivated our data structure. Our implementation is available at: https://github.com/KoslickiLab/prokrustean.