computing the quartet distance between general trees

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16 CHAPTER 3. EXPERIMENTAL APPROACHWorst-case trees (wc) In a worst-case tree there is one internal node of degree n 2 and n 2internal nodes of degree three, see Figure3.4. The tree has its name from Christiansenand Randers [5], where it was invented with the purpose of having a tree that has O(n)internal nodes and O(n) internal edges connected to an internal node of degree O(n).More interesting to this thesis is the total number of edges, which is |E| = 3 2n. The nameshould not be taken literally, since we do not yet know how the algorithms in this thesiswill perform.Figure 3.4: A wc tree3.2 Tree constructionSince this thesis mainly encompasses an experimental approach to evaluating algorithms,I find it natural to describe my choice of data format and data construction.3.2.1 Newick formatThe programs are reading in tree data from files, described in the Newick tree format 3which is a commonly used format for representation of phylogenetic trees. It is a simpletext format based on parentheses and commas. The grammar in Fig 3.5 provides a formaldescription for parsing the Newick format.Tree → Subtree ";"| Branch ";"Subtree → Leaf | InternalLeaf → NameInternal → "("BranchSet ")"NameBranchSet → Branch | BranchSet ","BranchBranch → Subtree LengthName → empty | stringLength → empty | ":"numberFigure 3.5: A context-free grammar for the Newick format3 Wikipedia article on the Newick format: http://en.wikipedia.org/wiki/Newick_format (accessedOct. 3, 2010)
3.2. TREE CONSTRUCTION 17Formal descriptions are good, however, this one has some small restrictions, e.g.parentheses inside Name are prohibited. Nevertheless, this grammar more than satisfiesthe needs for this thesis. Note that a tree either descends from nowhere, or it is rootedin an internal node. Thus, the Newick format is incapable of describing unrooted trees.This is not a problem, since the tree can merely be rooted in an arbitrary internal nodewhich, when parsing, will be treated as any other internal node.The two trees from the example in Fig. 1.1 would be written as follows in Newick treeformat. Note that there is only one named Internal node in each tree, namely the root(Panthera):(((Tiger, Snow Leopard), ((Lion, Leopard), Jaguar)), Clouded Leopard) Panthera;(((Lion, Leopard), Snow Leopard, Jaguar), (Tiger, Clouded Leopard)) Panthera;Data construction Data files have been generated in Newick format using a range ofsimple Python-scripts for each class of artificial tree described in 3.1. They are foundalong with the code and all the data files used, see Appendix B.3.2.2 Parsing data files and building a tree data structureOne way of parsing data described by a grammar, like the one in Fig. 3.5, is by using arecursive descent parser. As the name indicates, a recursive descent parser is workingtop-down, by repeatedly identifying one construct in the data that corresponds to a productionrule in the grammar and then process this piece of data according to the rule.A number of mutually recursive procedures will typically represent the rules defined bythe grammar.For the Python implementation I have made use of a parser based on the Toy ParserGenerator framework 4 (TPG), and written by Thomas Mailund 5 , with a few modifications.For the C++ implementation I wrote my own parser from scratch. I wrote proceduresParse(), ParseSubtree(), ParseInternalNode(), ParseLeafNode(), ParseBranchSet()and ParseBranch(), corresponding to the six first rules of the grammar of the Newick format.Both parsers are capable of parsing all the formats suggested by the grammar inSec. 3.2.1 and they meet the needs of this thesis.While the algorithms in general deal with non-rooted trees, some parts view the treeas rooted, in which cases it turns out more convenient to have a rooted, top-down representation.Furthermore, the Newick format dictates that some random node should4 TPG framework: http://christophe.delord.free.fr/tpg/index.html5 Mailunds parser: www.mailund.dk/index.php/2009/01/19/yet-another-newick-parser/
Page 1: Master’s thesisCOMPUTING THE QUAR
Page 5: AcknowledgementsFirst of all I woul
Page 8 and 9: viiiCONTENTS5.3 Implementing leaf s
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 26 and 27: 18 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
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BIBLIOGRAPHY 67[20] M.S. Waterman a
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70 APPENDIX A. PREPROCESSING FOR TH
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Appendix CReal-life application of
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computing the quartet distance between general trees

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