A Constrained Combinatorial Framework for Efficient N-mer Analysis of Microbial Communities

The analysis of microbial communities often requires examining combinations of taxa to capture higher-order relationships. However, the number of possible combinations increases rapidly with the number of taxa, making exhaustive enumeration computationally impractical. In this work, we present a hybrid N-mer generation framework that balances complete enumeration with practical limits on computational cost. The approach uses Gray code traversal to efficiently generate combinations, enabling full enumeration for samples with moderate numbers of taxa. For larger samples, where combinatorial growth is concentrated at intermediate values of k, the method introduces an adaptive cap that is applied only when the number of combinations becomes prohibitively large. Within these regions, combinations are prioritized using an abundance-based scoring scheme, favoring those supported by stronger quantitative signals. This design preserves full coverage of lower-order combinations while selectively reducing the number of high-order combinations that are often sparse and difficult to interpret. Applied to microbial abundance data, the framework maintains representative combinatorial structure while keeping runtime manageable. Overall, this work provides a scalable approach for exploring taxa combinations and a practical foundation for incorporating N-mer representations into downstream analyses.