computing the quartet distance between general trees

More documents

Recommendations

Info

viiiCONTENTS5.3 Implementing leaf set algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 275.3.1 Subtree leaf set sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.3.2 Shared leaf set sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 Cubic time algorithm 316.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.1.1 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 Sub-cubic time algorithm 377.1 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.1.1 Basic preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.1.2 Counting shared butterflies . . . . . . . . . . . . . . . . . . . . . . . . 407.1.3 Counting different butterflies . . . . . . . . . . . . . . . . . . . . . . 427.1.4 How to count butterflies in constant time . . . . . . . . . . . . . . . 447.1.4.1 The calculation of I ′′′ . . . . . . . . . . . . . . . . . . . . . 457.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.2.1 Prototype . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487.2.2 Introducing a library for matrix multiplication . . . . . . . . . . . . 497.2.3 The final version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 Results and discussion 598.1 Large numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609 Conclusion 639.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Bibliography 65A Preprocessing for the sub-cubic algorithm 69B Obtaining and running the programs 71C Real-life application of the algorithm 73
Chapter 1IntroductionThe topic of quartet distance computation is studied by computer scientists, but has itsorigin within evolutionary biology and requires some amount of motivation. The followingintroduction should be sufficient to understand why the topic is relevant and has itsapplications within other areas than computer science. However, it will by no means becomprehensive and should not be seen as a general introduction to phylogenetics.1.1 Phylogenetic treesAn evolutionary or phylogenetic tree is a hierarchical structure, commonly used to expressthe interrelationship, with regards to inheritance, among a set of evolutionary units(EUs), e.g. different species, and has found its use within the field of bioinformatics. Thetree consists of a set of nodes connected by edges, and the configuration of these decidethe structure or topology of the tree. By definition, there is exactly one path between anytwo nodes in a tree. The outermost nodes of the tree – the ones that are only incident to asingle edge – are called leaves and represent EUs, while branching points in the tree representspeciation events. A speciation event is a splitting of lineage. Some evolutionarytrees are rooted in one of the branching points, which adds a direction to the evolutionand the root is then said to be the most recent common ancestor to all species in the tree.In this case we picture the tree with the root on top and all edges as directed and pointingdownwards away from the root. Each branching point or internal node is the root of asubtree, and is viewed as a common ancestor to the EUs in this subtree. Two EUs downone branch are closer related than each of them are to any EU found in another branchof the same branching point. When each branching point in a rooted tree has exactlytwo immediate descendants, the tree is said to be binary. In a binary unrooted tree, eachnode has exactly three connections to other nodes. The number of connections a node1
Page 1: Master’s thesisCOMPUTING THE QUAR
Page 5: AcknowledgementsFirst of all I woul
Page 10 and 11: 2 CHAPTER 1. INTRODUCTIONhas is cal
Page 12 and 13: 4 CHAPTER 1. INTRODUCTIONIt is evid
Page 14 and 15: 6 CHAPTER 1. INTRODUCTIONsub-cubic
Page 17 and 18: Chapter 2PrerequisitesFirst, this c
Page 19: 2.2. CHOICE OF LANGUAGE AND TEST EN
Page 22 and 23: 14 CHAPTER 3. EXPERIMENTAL APPROACH
Page 28 and 29: 20 CHAPTER 4. QUARTIC TIME ALGORITH
Page 30 and 31: 22 CHAPTER 4. QUARTIC TIME ALGORITH
Page 33 and 34: Chapter 5Calculating leaf set sizes
Page 35 and 36: 5.3. IMPLEMENTING LEAF SET ALGORITH
Page 37 and 38: 5.3. IMPLEMENTING LEAF SET ALGORITH
Page 39 and 40: Chapter 6Cubic time algorithmHere I
Page 41 and 42: 6.1. IMPLEMENTATION 33that we can l
Page 43 and 44: 6.1. IMPLEMENTATION 35carried out o
Page 45 and 46: Chapter 7Sub-cubic time algorithmIn
Page 47 and 48: 7.1. THE ALGORITHM 39CcabADFigure 7
Page 49 and 50: 7.1. THE ALGORITHM 41reflecting Eq.
Page 51 and 52: 7.1. THE ALGORITHM 43choice of indi
Page 53 and 54: 7.1. THE ALGORITHM 45More interesti
Page 55 and 56: 7.2. IMPLEMENTATION 477.2 Implement
Page 57 and 58: 7.2. IMPLEMENTATION 49tween the num
Page 59 and 60:
7.2. IMPLEMENTATION 51oretic approa
Page 61 and 62:
7.2. IMPLEMENTATION 53well. My impl
Page 63 and 64:
7.2. IMPLEMENTATION 55Performance o
Page 65:
7.2. IMPLEMENTATION 57ble sort, wil
Page 68 and 69:
60 CHAPTER 8. RESULTS AND DISCUSSIO
Page 71 and 72:
Chapter 9ConclusionThe focus of thi
Page 73 and 74:
Bibliography[1] Mukul S. Bansal, Ji
Page 75:
BIBLIOGRAPHY 67[20] M.S. Waterman a
Page 78 and 79:
70 APPENDIX A. PREPROCESSING FOR TH
Page 81 and 82:
Appendix CReal-life application of
Page 83:
75t29t25t54t20t27 t13 t1t3t17t24t41
show all

computing the quartet distance between general trees

Create successful ePaper yourself

Delete template?

Save as template?