BIKE: A Binary $K$-Mer Exact Counter With Alphabet-Independent Memory and Deterministic Parallelism

$K$-mer counting is a fundamental computational task in bioinformatics, underpinning genome assembly, metagenomic classification, error correction, and similarity analysis. Existing exact-counting methods rely on hash tables or static allocation strategies whose memory requirements grow exponentially with the alphabet size and substring length, rendering them impractical for amino acid sequences at moderate-to-large values of $k$. We propose BIKE (Binary $K$-mer Exact Counter), a novel exact $k$-mer counting algorithm whose memory footprint depends exclusively on the input sequence length $n$, independently of the alphabet size $m$ or the $k$-mer length $k$. BIKE decomposes the counting problem into $n-1$ mutually independent pivot-based comparison blocks operating entirely on binary matrices, requiring only one bit per entry, and employs a union-find aggregation mechanism that guarantees exact counts for $k$-mers of arbitrary multiplicity. This structural regularity yields a fully deterministic degree of parallelism, enabling closed-form analytical models that provide accurate execution-time predictions under ideal parallel execution assumptions. Experimental results on real biological sequences confirm functional correctness and demonstrate memory reductions of up to three orders of magnitude over classical exact methods for amino acid alphabets. Analytical performance projections, derived from the closed-form parallel model, indicate that an FPGA realisation of BIKE would be expected to outperform CPU-based dynamic allocation at moderate sequence lengths; however, these remain theoretical estimates pending hardware implementation. BIKE is therefore presented as a theoretical and data-structural contribution, establishing a new algorithmic foundation for alphabet-independent, exactly-counted, and deterministically parallel $k$-mer analysis.