computing the quartet distance between general trees

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54 CHAPTER 7. SUB-CUBIC TIME ALGORITHMFrom the table it is evident that padding the matrices is not speeding up the multiplication.Only in three experiments it seems faster in some cases and only in a very smallpercentage of the multiplications. Furthermore, from the sizes of the internal nodes involvedwe observe that it is only when dealing with very small matrices (#rows/#columns< 10), that we “benefit” from padding. More likely, it is a coincidence rather than an actualtendency. My conclusion is that it is not beneficial to pad the matrices and consequently,there is no reason to separately take care of the case where max(d v ,d v ′) ω issmallest and I will continue to distinguish only between dv 2d v ′ and d v d 2 , as done in thev ′prototype.Expectations The implementation still does not meet the formal description of the article,which means that we can not count on the analysis and thus, the implementationmight brake the sub-cubic time bound and behave as a cubic algorithm. It seems reasonableto keep the assumptions from the prototype. Nevertheless, I hope to see a generalimprovement which of course is dependent on the proportion of time actually spent onmatrix multiplication. This should be more significant to the experiments involving thelargest multiplications, namely the star trees and the worst case trees which seemed tobe suffering because of the larger matrices encountered.Result The result of applying the two final implementations on the usual array of testdata is shown in Fig. 7.10 and Fig. 7.11. We observe that there is no significant improvementin running time. The general picture is much the same as for the prototype; thefinal implementations perform in the same range of time consumption as the prototypeimplementations with a few exceptions. The reason might very likely be the relativelylow amount of time spent doing matrix multiplication and the relatively high amount oftime spent processing the pairs of internal nodes.There are a couple of cases, however, where a change in the slope of the plot is clearlyobservable. That is, not surprisingly, the trees with few inner nodes, resulting in largematrices; star trees and wc trees. And only for the C++ implementation. Especially thestar tree yields a dramatic improvement. The plot of the C++ prototype, that was almostparallel to the O(n 3 ) line, has now improved to being almost parallel to the O(n 2 ) line.However, the plot is not making an exact straight line which I suspect might be becausethe overhead is gradually equalized by the complexity of the matrix multiplication as thesize of the matrices grow. Eventually this will result in the plot showing the complexityof the matrix multiplication only. The processing time of an 800-leaf star tree is loweredfrom over 100 seconds to below 10 seconds meaning that the change of matrix librarywas indeed a great improvement.
7.2. IMPLEMENTATION 55Performance of the Python implementation of the sub-cubic algorithm.time in seconds - t(n)1000100101bin, best exp= 2.04ran, best exp= 2.07sqrt, best exp= 1.79star, best exp= 2.00wc, best exp= 2.00nn 2n 30.10.0150 100 200 500 800number of leaves - nFigure 7.10: Performance of the sub-cubic algorithm in Python.time in seconds - t(n)10001001010.1Performance of the C++ implementation of the sub-cubic algorithm.bin, best exp= 2.05ran, best exp= 2.07sqrt, best exp= 1.81star, best exp= 2.04wc, best exp= 2.02nn 2n 30.0150 100 200 500 800number of leaves - nFigure 7.11: Performance of the sub-cubic algorithm in C++.
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Master’s thesisCOMPUTING THE QUAR
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AcknowledgementsFirst of all I woul
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viiiCONTENTS5.3 Implementing leaf s
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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: 7.2. IMPLEMENTATION 53well. My impl
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
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