computing the quartet distance between general trees

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52 CHAPTER 7. SUB-CUBIC TIME ALGORITHMtegrations. The result of multiplying square matrices of increasing sizes is illustrated inthe plot of Fig. 7.9. As expected, the BLAS calls are significantly faster than the basic im-Performance of the different matrix multiplication methods used.10010python-scipypython-blascpp-boostcpp-blasn 3time in seconds - t(n)10.10.0150 100 200 500 800size of matrix - nFigure 7.9: Comparison of the matrix multiplication methods used.plementations. Nearly as much as a factor of one thousand. The plot is based on tenrepetitions of each matrix multiplication, but nevertheless, the points are a bit scatteredand not forming straight lines. I blame this on the methods used which may suffer fromstructural influences and this could also be indicated in the similar behaviour of the twoBLAS routines. Especially for the larger matrices the optimized calls are superior. It is,however, surprising to see that both Python calls are performing better than their respectiveC++ combatant.The only method that seems to be strict O(n 3 ) is the C++ Boost.uBlas call. In retrospectone might see this as the reason why the star tree was such a challenge to the C++prototype implementation.7.2.3 The final versionNow, had I been successful in finding an implementation of a theoretically good algorithm,the next step would be to estimate the exponent ω, and thus being able to choosethe smallest of max(d v ,d v ′) ω , dv 2d v ′ and d v d 2 . However, as discussed in Sec. 7.1.1, thev ′matrix multiplication routines hide away different coefficients and in addition, time isspent padding the matrices in some cases, all of which has to be taken into account as
7.2. IMPLEMENTATION 53well. My implementation will differ from the theory by making no difference between thenaive and some advanced method for matrix multiplication. I might as well use BLAS inevery case, since it is not specifically designed for square matrices. Anyway, even thoughI have no knowledge of the internals of the BLAS implementation I can treat it as a blackbox and test it as a piece in the algorithmic puzzle. With all of this in mind, I will makean array of experiments with the purpose of clarifying exactly when, if ever, it is feasibleto pad the matrices, making them square, before applying the BLAS routine.I have been executing an equivalent to the prototype algorithm, but substituting thematrix multiplication method by BLAS. In each case the matrix multiplication had to becarried out, I made both calculations, with and without padding of the matrices, comparingthe running time of that particular part of the execution. This has been donefor each combination of the different types of test trees, for larger trees of mainly 800leaves. The result is shown in Table 7.1. For each execution of the algorithm it is shownhow many times the multiplication was carried out and for how many of those paddingwas advantageous. This is also shown percentage-wise. For every case where paddingwas in fact advantageous the table shows the largest degree for an internal node and thelargest difference between the degree of two of the internal nodes being processed whichis identical to the difference between the number of rows and columns in the matrix.Trees Number of Number of times Largest Largest %multiplications padding was degree differencean advantagein degreebin vs bin 636804 0 - - 0.000ran vs ran 154449 30 9 4 0.019sqrt vs sqrt 841 0 - - 0.000star vs star 1 0 - - 0.000wc vs wc 160801 0 - - 0.000bin vs ran 313614 43 7 4 0.014bin vs sqrt 8358 0 - - 0.000bin vs star 798 0 - - 0.000bin vs wc 319998 0 - - 0.000ran vs sqrt 4221 0 - - 0.000ran vs star 393 0 - - 0.000ran vs wc 157593 130 7 4 0.082sqrt vs star 21 0 - - 0.000sqrt vs wc 4221 0 - - 0.000star vs wc 401 0 - - 0.000Table 7.1: Investigating when it is advantageous to multiply square matrices.
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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
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: 7.2. IMPLEMENTATION 51oretic approa
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
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computing the quartet distance between general trees

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