Refined Buneman Trees

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Abstract The field of bioinformatics needs efficient methods for analysing data. New and improved technologies increase the efficiency of e.g. DNA sequencing, and the amount of data coming out of biological research laboratories grows at an increasing rate. It is therefore important to be able to handle these huge amounts of data with limited computational power. Evolutionary tree reconstruction is a fundamental research problem in biology, with a wide range of applications. The perfect phylogeny problem is NP-hard, and large dataset need to be analysed quickly and accurately. Current methods are very one-sided in their trade-off when tackling this problem, being either slow, but precise, or fast, but inaccurate. The refined Buneman method is a relatively new and certainly untested tree reconstruction method. Earlier algorithms computing the refined buneman tree would run in time O(n 5 )andspaceO(n 4 ), making the method infeasible to run for large datasets, but with the advent of the algorithm described in [BFÖ+ 03], the method has complexity bounds of O(n 3 )andO(n 2 ) for time and space, respectively. Suddenly the method becomes useful in practice, and is directly comparable to the popular Neighbor-Joining method which has precisely the same complexity bounds. This thesis describes the first implementation of the cubic time algorithm computing the refined Buneman tree. We verify the complexity bounds and perform an initial comparitive study of the refined Buneman algorithm and the Neighbor-Joining method. We describe how the implementation can easily be integrated into the widely used JSplits software package, thus making the method instantly and easily available. And we ponder over the biological accurary of the refined Buneman method. 1
This thesis is dedicated to my family Estrid, René and Morten and to my dear Khay Ling 2
Page 1: Refined Buneman Trees Lasse Westh-N
Page 5 and 6: Contents 1 Introduction 7 1.1 Docum
Page 7 and 8: 13.3 Correctness of the reference i
Page 9 and 10: The theory of evolution has also be
Page 11 and 12: Chapter 2 Definitions This chapter
Page 13 and 14: A C B Figure 2.1: An evolutionary t
Page 15 and 16: 2.4 Quartets To every set of four s
Page 17 and 18: 2.6 Splits The partition of a finit
Page 19 and 20: evolutionary tree gives an invaluab
Page 21 and 22: time in the algorithm. In [BFÖ+ 03
Page 23 and 24: Figure 3.1: A tree of life. 22
Page 25 and 26: knowing its origins, but how does h
Page 27 and 28: anging from huge time complexity to
Page 29 and 30: Algorithm 2 The Neighbor-Joining al
Page 31 and 32: A, C, T 1 2 3 T A, C, T T C, T A, T
Page 33 and 34: Part II Implementing Refined Bunema
Page 35 and 36: Algorithm 3 Overapproximating the r
Page 37 and 38: the pseudo-code for the algorithm i
Page 39 and 40: AE DE D B e E BC A C BE AC Figure 5
Page 41 and 42: construction, but we still have to
Page 43 and 44: Chapter 6 TheTreeDataStructure This
Page 45 and 46: incidentedge Figure 6.2: The world
Page 47 and 48: interface EdgeIterator { boolean ha
Page 49 and 50: Figure 6.7: An example a node which
Page 51 and 52: So how do we find σ ′ We start
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Algorithm 5. Offhand, the algorithm
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Algorithm 6 The algorithm that calc
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a b c d root ab cd Figure 8.2: Upda
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6000 Quad Tree performance characte
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30000 Quad Tree performance charact
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sets A, B, C and D by scanning the
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Chapter 10 The Selection Algorithm
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is O(n 2 ). The algorithm uses a di
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Chapter 11 JSplits Figure 11.1: A s
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implementing an algorithm with a hi
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Chapter 12 Source Code The source c
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Chapter 13 The Reference Implementa
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• the splits that are generated.
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Chapter 14 Correctness This chapter
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The best possible way of testing wo
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100000 90000 Performance of the ref
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of the size of the heap during the
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140000 Space complexity best fit: x
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Chapter 16 Comparing Evolutionary T
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Figure 16.1: The size of the set B(
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Figure 16.2: The total number of sp
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that it over-induces splits, and th
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efined Buneman therefore suffers a
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Speedups might be achieved using op
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Appendix A Correctness of the Refer
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Quartet: 0 0 | 1 4 -0.1263501163396
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Appendix B Garbage Collector Log 0.
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Bibliography [AJL + 02] Bruce Alber
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[Kim80] M. Kimura. A simple model f
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