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Given a threshold $L$ and a set $\mathcal{R} = \{R_1, \ldots, R_m\}$ of $m$haplotype sequences, each having length $n$, the minimum segmentation problemfor founder reconstruction is to partition the sequences into disjoint segments$\mathcal{R}[i_1{+}1,i_2], \mathcal{R}[i_2{+}1, i_3], \ldots,\mathcal{R}[i_{r-1}{+}1, i_r]$, where $0 = i_1 < \cdots < i_r = n$ and$\mathcal{R}[i_{j-1}{+}1, i_j]$ is the set $\{R_1[i_{j-1}{+}1, i_j], \ldots,R_m[i_{j-1}{+}1, i_j]\}$, such that the length of each segment, $i_j -i_{j-1}$, is at least $L$ and $K = \max_j\{ |\mathcal{R}[i_{j-1}{+}1, i_j]| \}$is minimized. The distinct substrings in the segments $\mathcal{R}[i_{j-1}{+}1,i_j]$ represent founder blocks that can be concatenated to form $K$ foundersequences representing the original $\mathcal{R}$ such that crossovers happenonly at segment boundaries. We give an optimal $O(mn)$ time algorithm to solvethe problem, improving over earlier $O(mn^2)$. This improvement enables toexploit the algorithm on a pan-genomic setting of haplotypes being completehuman chromosomes, with a goal of finding a representative set of referencesthat can be indexed for read alignment and variant calling.

Minimum Segmentation for Pan-genomic Founder Reconstruction in Linear Time