Faster Iterative $\phi$ Queries on the Positional BWT

TL;DR

This paper introduces a new decomposition scheme for the Positional BWT that enables faster iterative $phi$ queries, crucial for haplotype analysis, with improved space-time tradeoffs over previous methods.

Contribution

It proposes a novel segmentation approach that reduces query time and space complexity for iterative $phi$ operations on the PBWT, enhancing genomic data analysis efficiency.

Findings

Supports $k$ iterative $phi$ queries in $O( ext{log log}_w ext{min}(m,h) + k)$ time with $O(( ilde{r}+h) ext{log} n)$ bits.

Provides a more space-efficient structure using $O( ilde{r} ext{log} h + h ext{log} n)$ bits, with $O(k ext{log log}_w h)$ query time.

Expected to be practical for genomic datasets where haplotypes are fewer than sites.

Abstract

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 $\phi_j(i)$ query, which returns the haplotype index immediately preceding $S_i$ in co-lexicographic order at site $j$. Efficient support for $k$ iterative queries $\phi^1, \dots, \phi^k$ 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 $S_{\phi_e(i)}$, 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 $\phi$ 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.