人類基因體計畫在2003年完成，同時也產生人類基因序列草圖。任何兩個人類基因幾乎是同等的，只有非常小的差異造成每個人的不同。單一核

The Human Genome Project completed in 2003 and the draft of human genome sequences were also yielded. It has been known that any two human gnomes are almost identical, and only very little difference makes human diversities. Single nucleotide polymorphism (SNP) means that a single-base nucleotide changes in DNA. A SNP sequence from one of a pair of chromosomes is called a haplotype. In this thesis, we study how to reconstruct a pair of chromosomes from a given set of fragments obtained by DNA sequencing in an individual. We define a new problem, the chromosome pair assembly problem, for the chromosome reconstruction. The goal of the problem is to find a pair of sequences such that the pair of output sequences have the minimum mismatch with the input fragments and their lengths are minimum. We first transform the problem instance into a directed multigraph. And then we propose an efficient algorithm to solve the problem. We apply the ACO algorithm to optimize the ordering of input fragments and use dynamic programming to determine SNP sites. After the chromosome pair is reconstructed, the two haplotypes can also be determined. We perform our algorithm on some artificial test data. The experiments show that our results are near the optimal solutions of the test data.

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0
Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 SNPs and Haplotype . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Shotgun Sequencing and Sequence Assembly . . . . . . . . . . . . . . 1
1.3 The SNP Haplotype Assembly Problem . . . . . . . . . . . . . . . . . 2
Chapter 2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Previous Works for the Single Individual Haplotyping . . . . . . . . . 6
2.2 The MFR problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 The MSR Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Introduction to Ant Colony Optimization Algorithm . . . . . . . . . . 14
Chapter 3. Our Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1 The Definition of the Problem . . . . . . . . . . . . . . . . . . . . . . 17
3.2 The Overview of Our Algorithm . . . . . . . . . . . . . . . . . . . . . 18
3.3 Representation of a Candidate Solution . . . . . . . . . . . . . . . . . 20
3.4 The Consensus Process . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.5 An Algorithm for Determining Polymorphism . . . . . . . . . . . . . 30
3.6 The ACO Algorithm for Chromosome Pair Determination . . . . . . 34
3.7 The Time Complexity of Our Method . . . . . . . . . . . . . . . . . . 35
Chapter 4. Experimental Results and Discussion . . . . . . . . . . . . 36
4.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2 Discussion of Error Rate . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.3 Discussion of Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Chapter 5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

[1] V. Bafna, S. Istrail, G. Lancia, and R. Rizzi, “Polynomial and apx-hard casesof the individual haplotyping problem.,” Theor. Comput. Sci., Vol. 335, No. 1,pp. 109–125, 2005.
[2] C. Blum and A. Roli, “Metaheuristics in combinatorial optimization: Overviewand conceptual comparison,” ACM Comput. Surv., Vol. 35, No. 3, pp. 268–308, 2003.
[3] P. Bonizzoni, G. D. Vedova, R. Dondi, and J. Li, “The haplotyping problem:an overview of computational models and solutions,” J. Comput. Sci. Technol.,Vol. 18, No. 6, pp. 675–688, 2003.
[4] M. Dorigo, G. D. Caro, and L. M. Gambardella, “Ant algorithms for discreteoptimization,” Artificial Life, Vol. 5, No. 3, pp. 137–172, 1999.
[5] M. Dorigo and L. M. Gambardella, “Ant colonies for the traveling salesmanproblem,” BioSystems, No. 43, pp. 73–81, 1997.
[6] M. Dorigo and G. D. Di Caro, The Ant Colony Optimization Meta-Heuristic,pp. 11–32. London: McGraw-Hill, 1999.
[7] E. Eskin, E. Halperin, and R. M. Karp, “Efficient reconstruction of haplotypestructure via perfect phylogeny,” tech. rep., Berkeley, CA, USA, 2002.
[8] D. Fasulo, A. Halpern, I. Dew, and C. Mobarry,“Efficiently detecting polymorphismsduring the fragment assembly process,” Bioinformatics, Vol. 18,No. 90001, pp. 294S–302, 2002.
[9] S. Gabriel, S. Schaffner, H. Ngyen, J. Moore, J. Roy, B. Blumenstiel, J. Higgins,M. Deflice, A. Lochner, M. Faggart, S. N. Liu-Cordero, C. Rotimi, A. Adeyemo,R. Cooper, R. Ward, E. Lander, M. Daly, and D. Altshuler, “The structure ofhaplotype blocks in the human genome,” Science, Vol. 296, No. 21, pp. 2225–2229, 2002.
[10] H. J. Greenberg, W. E. Hart, and G. Lancia, “Opportunities for combinatorialoptimization in computational biology.,” INFORMS Journal on Computing,Vol. 16, No. 3, pp. 211–231, 2004.
[11] D. Gusfield, “Haplotyping as perfect phylogeny: conceptual framework andefficient solutions,” RECOMB ’02: Proceedings of the sixth annual internationalconference on Computational biology, New York, NY, USA, pp. 166–175, ACMPress, 2002.
[12] B. V. Halld’orsson, V. Bafna, N. Edwards, R. Lippert, S. Yooseph, and S. Istrail,“Combinatorial problems arising in snp and haplotype analysis.,” DMTCS,pp. 26–47, 2003.
[13] X. Huang and A. Madan, “CAP3: A DNA Sequence Assembly Program, ”Genome Res., Vol. 9, No. 9, pp. 868–877, 1999.
[14] S. Istrail, M. S. Waterman, and A. G. Clark, eds., Computational Methodsfor SNPs and Haplotype Inference, DIMACS/RECOMB Satellite Workshop,Piscataway, NJ, USA, November 21-22, 2002, Revised Papers, Vol. 2983 ofLecture Notes in Computer Science, Springer, 2004.
[15] G. Lancia, V. Bafna, S. Istrail, R. Lippert, and R. Schwartz, “Snps problems,complexity, and algorithms,” ESA ’01: Proceedings of the 9th Annual EuropeanSymposium on Algorithms, London, UK, pp. 182–193, Springer-Verlag, 2001.
[16] L. Li, J. H. Kim, and M. S. Waterman, “Haplotype reconstruction from snpalignment,” RECOMB ’03: Proceedings of the seventh annual internationalconference on Computational molecular biology, New York, NY, USA, pp. 207–216, ACM Press, 2003.
[17] R. Lippert, G. L. R. Schwartz, and S. Istrail, “Algorithmic Strategies for theSNPs Haplotype Assembly Problem,” Briefings in Bioinformatics, Vol. 3, No. 1,pp. 23–31, 2002.
[18] M. Olivier, “A haplotype map of the human genome,” Physiol. Genomics,Vol. 13, No. 1, pp. 3–9, 2003.
[19] Riva and I. S. Kohane, “SNPper: retrieval and analysis of human SNPs,” Bioin-formatics, Vol. 18, No. 12, pp. 1681–1685, 2002.
[20] R. Rizzi, V. Bafna, S. Istrail, and G. Lancia, “Practical algorithms and fixedparametertractability for the single individual snp haplotyping problem,”WABI ’02: Proceedings of the Second International Workshop on Algorithmsin Bioinformatics, London, UK, pp. 29–43, Spring-er-Verlag, 2002.
[21] B. S. Shastry, “SNP alleles in human disease and evolution,” Journal of HumanGenetics, Vol. 47, pp. 561–566, 2002.
[22] T. St‥utzle and H. Hoos, “MAX-MIN Ant System and local search for theTraveling Salesman Problem,” Proceedings of the fourth International Confer-ence on Evolutionary Computation (ICEC) (IEEE, ed.), pp. 308–313, IEEEPress, 1997.
[23] T. St‥utzle and H. Hoos, “Max-min ant system,” Future Gener. Comput. Syst.,Vol. 16, No. 9, pp. 889–914, 2000.
[24] L. Wang and Y. Xu, “Haplotype inference by maximum parsimony,” Bioinfor-matics, Vol. 19, No. 14, pp. 1773–1780, 2003.
[25] R.-S. Wang, L.-Y. Wu, Z.-P. Li, and X.-S. Zhang,“Haplotype reconstructionfrom SNP fragments by minimum error correction,” Bioinformatics, Vol. 21, No. 10, pp. 2456–2462, 2005.