Read-Consistent Minimum Unique Substrings: A Parameter-Free, Linear-Time Framework for Genomic Sequence Representation

Fixed-length k-mers have been the standard unit of genomic sequence representation for over two decades. However, they impose a uniform resolution on genomes whose complexity varies across loci. We introduce Minimum Unique Substrings (MUSs), variable-length sequence units defined by the local uniqueness structure of the genome rather than predefined parameters. We first extend MUS theory from single contiguous strings to fragmented sequencing reads by formalizing a definition of uniqueness that is consistent with these reads. Next, we present a linear-time extraction algorithm that runs in O(n) time using the generalized suffix tree. In this context, we introduce outpost nodes, topological anchors within the suffix tree that accurately localize MUS boundaries in fragmented sequencing reads. Finally, we empirically characterize the distributions of MUS lengths in E. coli K-12 and human chromosome 11. Our results demonstrate that MUS lengths naturally mirror genomic architectural complexity without the need for user-defined parameters. Notably, the MUS framework achieves 100% unique positional coverage with a mean length of only 36.08 bp. In contrast, fixed-length k=61 coverage reaches only 69.4%, despite being 1.69 times the MUS average. We show that increasing k from 21 to 61 triples the unique k-mer count from 2.35M to 6.86M. This k-paradox occurs because repetitive sequences are fragmented into spuriously unique tokens without improving true genomic resolution. MUSs escape this artifact entirely by adapting dynamically to local sequence complexity. These results establish MUSs as a biologically grounded, computationally tractable foundation for parameter-free genome assembly, repeat characterization, and alignment-free genomics.