computing the quartet distance between general trees

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10 CHAPTER 2. PREREQUISITESidentifies the subtree ¯F , which includes every part of the tree behind e. The quantity |F |is called the leaf set size and will be used to denote the number of leaves in the subtree F.Therefore, |F | + | ¯F | = n. This is a slight abuse of the mathematical notation for sets andone could argue that F should merely denote the set of leaves in the subtree, however,I will stick to this notation as it is used in the background literature. Most often, thealgorithms will deal with two subtrees, one from each tree, where F is a subtree of T andG is a subtree of T ′ . Since T and T ′ contain the same set of leaves, F and G might alsohave some leaves in common. This is written |F ∩G| and will be referred to as the sharedleaf set size.The concept of quartets of four leaves a, b, c and d has been described in Sec. 1.2.1.The three butterfly topologies, illustrated in Fig. 1.2 (a)–(c), are written ab|cd, ac|bd andad|bc respectively, while the star quartet in Fig. 1.2(d) is written a b ×c d .2.2 Choice of language and test environmentsMy first choice of implementation language was the Python programming language.Python is not among the most efficient languages, and usually not used in critical algorithmicor mathematical applications. However it is a popular language for fast prototyping,because it supports clarity and simplicity and a very short workcycle. This was agood match for me, during the early parts of the implementation process, where I gainedcomplete understanding through the experimental work. In addition, I did not know theactual time needed for quartet distance calculation and therefore productivity was moreimportant than performance.Later on, I made a decision to implement the algorithms in C++ as well. We haveseen, in Sec. 1.3.1, that the results of the Python implementation were indeed informative,however, the practical running times were rather slow. The time-consuming calculationswere carried out on remote servers and despite the fact that this thesis is notabout optimizations, I found it interesting to see if I could bring down the running timeto a level where the experiments could be carried out on my own laptop. Furthermoreanother language with other properties might give the whole study a new perspective.C++ does indeed have other properties. It is compiled and, with support for a set oflow-level language features, considerably closer to the machine level and often used foralgorithms and other time critical calculations.These two tracks of implementation will be described in parallel, and in some casesnot distinguished between. However, they should not be compared in a one-to-one relationand the resulting programs will merely be used as two different opportunities toexperiment with the theoretic results.
2.2. CHOICE OF LANGUAGE AND TEST ENVIRONMENTS 11Details about the two programming languages, libraries and physical environmentsused are listed below. For a guide on how to obtain, compile and run the code, see App. B.Python programming environmentplatform:PCprocessor:Intel Xeon 3.00 GHzmemory:1 GBoperating system: Red Hat Linux (kernel: 2.6.18)language: Python (v. 2.4.3)libraries: NumPy (v. 1.2.1)SciPy (v. 0.6.0)BLAS (preinstalled, v. 3.1.1)C++ programming environmentplatform: MacBook Pro 13”model:MacBookPro5,5processor:Intel Core 2 Duo 2.26 GHzmemory:4 GBoperating system:Mac OS X 10.6.4 (10F569)language: C++ (g++ from gcc version 4.2.1 (Apple Inc. build 5664))libraries: Boost.uBlas (Boost libraries v. 1.42.0)Boost.NumericBindings (v. v1)BLAS (Apple Accelerate framework: v. vecLib-268.0)
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: Chapter 2PrerequisitesFirst, this c
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Page 33 and 34: Chapter 5Calculating leaf set sizes
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Page 39 and 40: Chapter 6Cubic time algorithmHere I
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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.
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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
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60 CHAPTER 8. RESULTS AND DISCUSSIO
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Chapter 9ConclusionThe focus of thi
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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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