單點基因多型(Single Nucleotide Polymorphism)為目前頗受關注的問題。這個在人類基因中的多型現象，被認為與追蹤某些重要的疾病相關，且在各個生醫領域有許多的應用。所以如何正確的取得單點基因多型的資訊，成了重要的課題。
完整單點基因多型的序列資訊稱之為”Haplotye”。目前有一個基於散彈槍定序法的方法，能夠有效針對單一個體收集完整單點基因多型的序列資訊。由於每一個源自於散彈槍定序法的DNA片斷皆含有些許單點基因多型的資訊，如何有效且正確的重組這一些DNA片段，成了這個方法的核心問題。如果以計算機最佳化的觀點來探討這個核心問題，我們可以定義出兩個組合最佳化的問題，Minimal Fragments Removal 與 Minimal SNPs Removal。本篇論文的主要結果，其一為針對Minimal SNPs Removal 這個問題，在允許一些DNA片段有遺失部份的的情況下，我提出一個較快的計算方法。其二為針對Minimal Fragments Removal這個問題，我修改原有的計算方法使其能應用在多倍體生物上。

Single nucleotide polymorphisms (SNPs) is one of the most considered topics. This phenomenon of genetic polymorphism is the most frequent human genetic variation and corresponding to numerous applications such as medical diagnosis, drug design and phylogenis. It is also helpful for tracking disease genes.
The complete sequence of SNP varieties from a single copy of chromosomes is called a haplotype. To determine haplotypes for a single individual, one alternative method proposed in [1, 2] is based on the DNA fragments and the methodology of Shotgun Sequencing Assembly. Every DNA fragment contains several SNPs information. After an appropriate assembly of the fragments, we can get the haplotypes for a single individuals. But it is difficult to get error-free fragments in the begining, how to remove errors to obtain valid assembly of all corrected fragments becomes the first problem. Since different error types are considered, two version of the problem, Minimal
Fragments Removal(MFR) and Minimal SNPs Removal(MSR), were
introduced in [1]. In this paper, we revised the original algorithm for MSR on fragments at most k holes. Although the original one was claimed to run in O(mn2k+2) [3], with more careful analysis, we found that it should be an O(mn2 + n2k+2) algorithm. Moreover, the existing algorithms for MFR only use the fragments from diploid genomes as input data. We extended the algorithm such that it also works robustly on the gapless fragments from polyploid genomes.

References
[1] R. Lippert, R. Schtwartz, G. Lancia, and S. Istrail. Algorithmic strategies
for the SNPs haplotype assembly problem. Briefings in Bioinformatics,
Vol. 3(1), pages 23-31, 2002.
[2] G. Lancia, V. Bafna, S. Istrail, R. Lippert, and R. Schwartz. SNPs
Problems, complexity and algorithms. In Proceedings of Annual Europran
Symposium on Algorithms(ESA), Vol. 2161 of Lecture Notes in
Computer Science, pages 182-193. Springer, 2001.
[3] R. Rizzi, V. Bafna, S. Istrail, and G. Lancia. Practical algorithms and
fixed-parameter tractability for the single individual SNP haplotyping
problem. Vol. 2452 of Lecture Notes in Computer Science pages 29-43.
Springer, 2002.
[4] J. Terwilliger, and K. Weiss. Linkage disquilibrium mapping of complex
disease: Fantasy and reality? Curr. Opin. Biotechnology, pages 579-594,
1998.
[5] M. Hoehe, K. Kopke, B. Wendel, K. Rohde, C. Flachmeier, K. Kidd,
W. Berrettini, and G. Church. Sequence variability and candidate gene
analysis in complex disease: Association of μ opioid receptor gene variation
with substance dependence. Human Molecular Genitics, Vol. 9,
pages 2895-2908, 2000.
[6] A. ChakraVarti. It’s raining SNP, hallelujah? Nature Genetics, Vol. 19,
pages 216-217,1998
19
[7] J. C. Stephens, J. A. Schneider, D. A. Tanduay, and et al. Haplotype
variation and linkage disequilibrium in 313 human genes. Science, Vol.
293, pages 489-493, 2001
[8] S. L. C. Woo, ‘Collation of RFLP haplotypes at the human phenylalanine
hydroxylase (PAH) locus, Amer. J. Hum. Genet., Vol. 43, pages
781-783, 1988.
[9] A. G. Clark, Inference of haplotypes from PCR amplified samples of
diploid populations. Molecular Biology and Evolution, Vol.7, pages 111-
122, 1990.
[10] D. Gusfield, Haplotyping as perfect phylogeny:Conceptual framework
and efficient solutions. In Proceedings of 6th Annual International Conference
on Research in Computational Molecular Biology (RECOMB).
ACM Press, pages 166-175, 2002
[11] L. S. Wang, and Y. Xu, Haploptye inference by maximum parsimony.
Bioinformatics, Vol. 19, pages 1773-1780, 2003
[12] E. Halperin, and E. Eskin, Haplotype reconstruction from genotype
data using imperfect phylogeny. Bioinformatics, Vol.20, pages 1842-
1849, 2004.
20