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# Partitioning and indexing architecture

An index is split into a fixed number of partitions, each handling an independent, disjoint slice of the kmer space. Partitioning keeps the working set of each stage small enough to process efficiently and enables parallel construction and querying.

## Routing

The canonical minimizer of a super-kmer (see Minimizer selection) is hashed to produce a \(p\)-bit routing value that selects the destination partition:

    canonical minimizer → hash(minimizer) → p-bit value → partition index

The routing value is recomputed whenever it is needed (during construction and again at query time) rather than stored — it is not part of the on-disk super-kmer representation.

Within a partition, kmers are indexed as plain values via a minimal perfect hash function (see On-disk storage); the minimizer plays no further role once a super-kmer has reached its partition.

## Why hashing is necessary

A canonical minimizer is an m-mer (\(m \in \{9, 11, 13, 15\}\)), and its distribution over all possible m-mer values is not uniform — as the lexicographic minimum of a window, small values are systematically over-represented (Golan & Shur 2025; Kille et al. 2023; Pan & Reinert 2024; Zheng et al. 2020, 2021)1 2 3 4 5. Routing directly on the raw minimizer value would therefore produce badly unbalanced partitions.

Hashing the minimizer before routing redistributes this skewed distribution uniformly across partitions. This works reliably because the number of partition-index bits \(p\) is chosen well below the number of bits available in the minimizer (\(2m\)): even with strong bias in the minimizer distribution, the hash has enough entropy margin to absorb it, provided the number of distinct minimizers actually observed is much larger than the number of partitions.

## Parameter guidance

Minimizer size \(m\) | Minimizer bits (\(2m\)) | Typical partition-index bits \(p\) | Partitions
--- | --- | --- | ---
9 | 18 | 6–8 | 64–256
11 | 22 | 8–10 | 256–1 024
13 | 26 | 10–12 | 1 024–4 096
15 | 30 | 10–14 | 1 024–16 384

The number of partitions must satisfy \(p \le 2m\), and in practice \(p\) is chosen well below that bound to leave a comfortable entropy margin. For \(k=31\), \(m=13\), \(p=10\) (1024 partitions), partition load is well balanced on real genomic data.

1. Zheng, H., Kingsford, C. & Marçais, G. (2020). Improved design and analysis of practical minimizers. Bioinformatics (Oxford, England), 36, i119--i127. ↩

2. Zheng, H., Kingsford, C. & Marçais, G. (2021). Sequence-specific minimizers via polar sets. Bioinformatics (Oxford, England), 37, i187--i195. ↩

3. Pan, C. & Reinert, K. (2024). A simple refined DNA minimizer operator enables 2-fold faster computation. Bioinformatics (Oxford, England), 40. ↩

4. Kille, B., Garrison, E., Treangen, T.J. & Phillippy, A.M. (2023). Minmers are a generalization of minimizers that enable unbiased local jaccard estimation. Bioinformatics (Oxford, England), 39. ↩

5. Golan, S. & Shur, A.M. (2025). Expected density of random minimizers. In: Lecture notes in computer science, Lecture notes in computer science. Springer Nature Switzerland, Cham, pp. 347--360. ↩