### Abstract

Results: We give an O(mn) time (i.e. linear time in the input size) algorithm to solve the minimum segmentation problem for founder reconstruction, improving over an earlier O(mn2).

Conclusions: Our improvement enables to apply the formulation on an input of thousands of complete human chromosomes. We implemented the new algorithm and give experimental evidence on its practicality. The implementation is available in https://github.com/tsnorri/founder-sequences.

Original language | English |
---|---|

Article number | 12 |

Journal | Algorithms for Molecular Biology |

Volume | 14 |

Number of pages | 15 |

ISSN | 1748-7188 |

DOIs | |

Publication status | Published - 17 May 2019 |

MoE publication type | A1 Journal article-refereed |

### Fields of Science

- Pan-genome indexing
- Founder reconstruction
- Dynamic programming
- Positional Burrows-Wheeler transform
- Range minimum query
- BURROWS-WHEELER TRANSFORM
- QUERIES
- STORAGE
- 113 Computer and information sciences
- 1182 Biochemistry, cell and molecular biology

### Cite this

}

**Linear time minimum segmentation enables scalable founder reconstruction.** / Norri, Tuukka; Cazaux, Bastien; Kosolobov, Dmitry; Mäkinen, Veli.

Research output: Contribution to journal › Article › Scientific › peer-review

TY - JOUR

T1 - Linear time minimum segmentation enables scalable founder reconstruction

AU - Norri, Tuukka

AU - Cazaux, Bastien

AU - Kosolobov, Dmitry

AU - Mäkinen, Veli

PY - 2019/5/17

Y1 - 2019/5/17

N2 - Background: We study a preprocessing routine relevant in pan-genomic analyses: consider a set of aligned haplotype sequences of complete human chromosomes. Due to the enormous size of such data, one would like to represent this input set with a few founder sequences that retain as well as possible the contiguities of the original sequences. Such a smaller set gives a scalable way to exploit pan-genomic information in further analyses (e.g. read alignment and variant calling). Optimizing the founder set is an NP-hard problem, but there is a segmentation formulation that can be solved in polynomial time, defined as follows. Given a threshold L and a set R={R1,...,Rm} of m strings (haplotype sequences), each having length n, the minimum segmentation problem for founder reconstruction is to partition [1,n] into set P of disjoint segments such that each segment [a,b]P has length at least L and the number d(a,b)=|{Ri[a,b]:1im}| of distinct substrings at segment [a,b] is minimized over [a,b]P. The distinct substrings in the segments represent founder blocks that can be concatenated to form max{d(a,b):[a,b]P} founder sequences representing the original R such that crossovers happen only at segment boundaries.Results: We give an O(mn) time (i.e. linear time in the input size) algorithm to solve the minimum segmentation problem for founder reconstruction, improving over an earlier O(mn2).Conclusions: Our improvement enables to apply the formulation on an input of thousands of complete human chromosomes. We implemented the new algorithm and give experimental evidence on its practicality. The implementation is available in https://github.com/tsnorri/founder-sequences.

AB - Background: We study a preprocessing routine relevant in pan-genomic analyses: consider a set of aligned haplotype sequences of complete human chromosomes. Due to the enormous size of such data, one would like to represent this input set with a few founder sequences that retain as well as possible the contiguities of the original sequences. Such a smaller set gives a scalable way to exploit pan-genomic information in further analyses (e.g. read alignment and variant calling). Optimizing the founder set is an NP-hard problem, but there is a segmentation formulation that can be solved in polynomial time, defined as follows. Given a threshold L and a set R={R1,...,Rm} of m strings (haplotype sequences), each having length n, the minimum segmentation problem for founder reconstruction is to partition [1,n] into set P of disjoint segments such that each segment [a,b]P has length at least L and the number d(a,b)=|{Ri[a,b]:1im}| of distinct substrings at segment [a,b] is minimized over [a,b]P. The distinct substrings in the segments represent founder blocks that can be concatenated to form max{d(a,b):[a,b]P} founder sequences representing the original R such that crossovers happen only at segment boundaries.Results: We give an O(mn) time (i.e. linear time in the input size) algorithm to solve the minimum segmentation problem for founder reconstruction, improving over an earlier O(mn2).Conclusions: Our improvement enables to apply the formulation on an input of thousands of complete human chromosomes. We implemented the new algorithm and give experimental evidence on its practicality. The implementation is available in https://github.com/tsnorri/founder-sequences.

KW - Pan-genome indexing

KW - Founder reconstruction

KW - Dynamic programming

KW - Positional Burrows-Wheeler transform

KW - Range minimum query

KW - BURROWS-WHEELER TRANSFORM

KW - QUERIES

KW - STORAGE

KW - 113 Computer and information sciences

KW - 1182 Biochemistry, cell and molecular biology

U2 - 10.1186/s13015-019-0147-6

DO - 10.1186/s13015-019-0147-6

M3 - Article

VL - 14

JO - Algorithms for Molecular Biology

JF - Algorithms for Molecular Biology

SN - 1748-7188

M1 - 12

ER -