Masked superstrings as a unified framework for textualk-mer set representations

Abstract

The popularity of 𝑘-mer-based methods has recently led to the development of compact 𝑘-mer-set representations, such as simplitigs/Spectrum-Preserving String Sets (SPSS), matchtigs, and eulertigs. These aim to represent 𝑘-mer sets via strings that contain individual 𝑘-mers as substrings more efficiently than the traditional unitigs. Here, we demonstrate that all such representations can be viewed as superstrings of input 𝑘-mers, and as such can be generalized into a unified framework that we call the masked superstring of 𝑘-mers. We study the complexity of masked superstring computation and prove NP- hardness for both 𝑘-mer superstrings and their masks. We then design local and global greedy heuristics for efficient computation of masked superstrings, implement them in a program called KmerCamel🐫 , and evaluate their performance using selected genomes and pan-genomes. Overall, masked superstrings unify the theory and practice of textual 𝑘-mer set representations and provide a useful framework for optimizing representations for specific bioinformatics applications.

Towards Efficientk-Mer Set Operations via Function-Assigned Masked Superstrings

“…The last 𝑘 − 1 positions of every mask are always set to 0. When discussing specific masked superstrings, we present them in their masked-cased superstring encoding (enc2 in [61]), where each character of the superstring is upper-case if it corresponds to 1 in the mask, and lower-case if it corresponds to 0.…”

“…superstring using the local greedy algorithm [61] executed on the FMS-index, as described in Appendix D for the uni-directional model.…”

“…Union. As implicitly shown in [61], concatenating masked superstrings, which are or - masked superstrings in our notation, acts as union on the represented sets, that is, the resulting represented set is the union of the original represented sets. This allows or -masked superstrings to generalize (r)SPSS representations, since any set of k -mers in the (r)SPSS representation can be directly viewed as an or -masked superstring by concatenating the individual simplitigs/matchtigs.…”

“…In our previous work [61], we have developed the concept of masked superstring for k -mer set representations. Masked superstrings are based on the overlap graph of k -mers, instead of the de Bruijn graph, and thus removing the dependency on the SLP.…”

“…First, in Section 3 , we develop the theoretical foundations of function-assigned masked superstrings ( f -MS), which can be seen as an algebraic generalization of masked superstrings from our earlier work [61]. While our previous work [61] considered a k -long substring of K represented by ( S , M ) if any of its occurrences in S is unmasked, that is, the or of corresponding mask bits equals 1, here we use other functions f for determining the presence or absence of a k -mer from ( S , M ), allowing besides or -masked superstrings also xor -MS, and -MS, and many others types.…”