Faster Iterative $ϕ$ Queries on the Positional BWT
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2026
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| author | Bonizzoni, Paola Gagie, Travis Gao, Younan |
| author_facet | Bonizzoni, Paola Gagie, Travis Gao, Younan |
| contents | The Positional Burrows-Wheeler Transform (PBWT) is a fundamental data structure for the efficient representation and analysis of large-scale haplotype panels. For a panel of $h$ sequences $\{S_1, \dots, S_h\}$ over $m$ sites, a key operation is the $ϕ_j(i)$ query, which returns the haplotype index immediately preceding $S_i$ in co-lexicographic order at site $j$. Efficient support for $k$ iterative queries $ϕ^1, \dots, ϕ^k$ is essential for haplotype matching and variation analysis.
In this work, we introduce a simple and novel decomposition scheme that decomposes each haplotype row into sub-intervals, called refined segments, within which a haplotype's co-lexicographic predecessor for the sites remains unchanged. We show that refined segments satisfy two key properties: (i) each segment $[b,e]$ associated with $S_i$ overlaps with at most a constant number of segments of $S_{ϕ_e(i)}$, and (ii) the total number of segments is bounded by $O(\tilde{r} + h)$, where $\tilde{r}$ denotes the number of runs in the PBWT. Building on this decomposition, we present two space-time tradeoffs for supporting $k$ iterative $ϕ$ queries: (i) a structure using $O((\tilde{r} + h)\log n)$ bits of space that answers $k$ iterative queries in $O(\log \log_w \min(m,h) + k)$ time, where $n = m \cdot h$, and (ii) a more compact structure using $O(\tilde{r} \log h + h \log n)$ bits of space that supports queries in $O(k \log \log_w h)$ time.
Prior to our work, supporting these queries required $O((\tilde{r} + h)\log n)$ bits of space and $O(k \cdot \log \log_w m)$ time. Our second tradeoff is expected to be effective in practice for modern genomic datasets, where the number $h$ of haplotypes is typically much smaller than the number $m$ of sites. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_04244 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Faster Iterative $ϕ$ Queries on the Positional BWT Bonizzoni, Paola Gagie, Travis Gao, Younan Data Structures and Algorithms The Positional Burrows-Wheeler Transform (PBWT) is a fundamental data structure for the efficient representation and analysis of large-scale haplotype panels. For a panel of $h$ sequences $\{S_1, \dots, S_h\}$ over $m$ sites, a key operation is the $ϕ_j(i)$ query, which returns the haplotype index immediately preceding $S_i$ in co-lexicographic order at site $j$. Efficient support for $k$ iterative queries $ϕ^1, \dots, ϕ^k$ is essential for haplotype matching and variation analysis. In this work, we introduce a simple and novel decomposition scheme that decomposes each haplotype row into sub-intervals, called refined segments, within which a haplotype's co-lexicographic predecessor for the sites remains unchanged. We show that refined segments satisfy two key properties: (i) each segment $[b,e]$ associated with $S_i$ overlaps with at most a constant number of segments of $S_{ϕ_e(i)}$, and (ii) the total number of segments is bounded by $O(\tilde{r} + h)$, where $\tilde{r}$ denotes the number of runs in the PBWT. Building on this decomposition, we present two space-time tradeoffs for supporting $k$ iterative $ϕ$ queries: (i) a structure using $O((\tilde{r} + h)\log n)$ bits of space that answers $k$ iterative queries in $O(\log \log_w \min(m,h) + k)$ time, where $n = m \cdot h$, and (ii) a more compact structure using $O(\tilde{r} \log h + h \log n)$ bits of space that supports queries in $O(k \log \log_w h)$ time. Prior to our work, supporting these queries required $O((\tilde{r} + h)\log n)$ bits of space and $O(k \cdot \log \log_w m)$ time. Our second tradeoff is expected to be effective in practice for modern genomic datasets, where the number $h$ of haplotypes is typically much smaller than the number $m$ of sites. |
| title | Faster Iterative $ϕ$ Queries on the Positional BWT |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2605.04244 |