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 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 has length at least L and the number of distinct substrings at segment [ a , b ] is minimized over . The distinct substrings in the segments represent founder blocks that can be concatenated to form founder sequences representing the original 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 . 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 .
Given a language L that is online recognizable in linear time and space, we construct a linear time and space online recognition algorithm for the language L · Pal, where Pal is the language of all nonempty palindromes. Hence for every fixed positive k, Pal k is online recognizable in linear time and space. Thus we solve an open problem posed by Galil and Seiferas in 1978.
We present an algorithm that constructs the LZ-End parsing (a variation of LZ77) of a given string of length n in O(n log ℓ) expected time and O(z + ℓ) space, where z is the number of phrases in the parsing and ℓ is the length of the longest phrase. As an option, we can fix ℓ (e.g., to the size of RAM) thus obtaining a reasonable LZ-End approximation with the same functionality and the length of phrases restricted by ℓ. This modified algorithm constructs the parsing in streaming fashion in one left to right pass on the input string w.h.p. and performs one right to left pass to verify the correctness of the result. Experimentally comparing this version to other LZ77-based analogs, we show that it is of practical interest.
We study a lossy compression scheme linked to the biological problem of founder reconstruction: The goal in founder reconstruction is to replace a set of strings with a smaller set of founders such that the original connections are maintained as well as possible. A general formulation of this problem is NP-hard, but when limiting to reconstructions that form a segmentation of the input strings, polynomial time solutions exist. We proposed in our earlier work (WABI 2018) a linear time solution to a formulation where minimum segment length was bounded, but it was left open if the same running time can be obtained when the targeted compression level (number of founders) is bounded and lossyness is minimized. This optimization is captured by the Maximum Segmentation problem: Given a threshold M and a set R = {R1, . . . , Rm} of strings of the same length n, find a minimum cost partition P where for each segment [i, j] ∈ P , the compression level |{R k [i, j] : 1 ≤ k ≤ m}| is bounded from above by M . We give linear time algorithms to solve the problem for two different (compression quality) measures on P : the average length of the intervals of the partition and the length of the minimal interval of the partition. These algorithms make use of positional Burrows-Wheeler transform and the range maximum queue, an extension of range maximum queries to the case where the input string can be operated as a queue. For the latter, we present a new solution that may be of independent interest. The solutions work in a streaming model where one column of the input strings is introduced at a time.
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