computing the quartet distance between general trees

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70 APPENDIX A. PREPROCESSING FOR THE SUB-CUBIC ALGORITHMAnd two more arrays:R ′′ [i ] =C ′ ∑d v[j ] = I ′ [i , j ]M ′ =∑d vj =1i=1∑d v d v ′∑i=1 j =1I ′ [i , j ]I [i , j ](C [j ] − I [i , j ])(A.6)(A.7)(A.8)C ′′ ∑d v[j ] = I [i , j ](R[i ] − I [i , j ])(A.9)i=1Only one of the two following arrays will need calculation, depending on which of thetwo expressions for counting different butterflies is used, Eq. (7.9) or Eq. (7.10):R ′′′ [i ] =∑d vj =1I [i , j ] 2C ′′′ ∑d v[j ] = I [i , j ] 2With the approach taken in the article [14], the following table needs calculation.i=1(A.10)(A.11)I ′′′ [i , j ] =∑d vd v ′∑k=1,k≠i l=1,l≠jI [i ,l]I [k, j ]I [k,l]However, if calculated naively, the sub-cubic time bound promised will be broken. Thesolution is instead to calculate either I1 ′′ = I I T and then I1 ′′′ = I ′′ I , which is basically1dI 1 ′′v ′[i ,k] = ∑j =1I [i , j ]I [k, j ](A.12)I 1 ′′′d v[i , j ] = ∑k=1I [k, j ]I 1 ′′ [i ,k], (A.13)or otherwise calculate I2 ′′ = I T I and then I2 ′′′ = I I ′′ , which is basically2I 2 ′′d v[j,l] = ∑i=1I [i , j ]I [i ,l](A.14)dI 2 ′′′v ′[i , j ] = ∑I [i ,l]I 2 ′′ [l,k]. (A.15)l=1The choice is essential to the algorithm and is explained in further detail in Sec. 7.1.4.1.
Appendix BObtaining and running the programsA small website has been dedicated to this thesis. It gives information on how to download,compile and run the programs:link: http://www.cs.au.dk/~dalko/thesis/Otherwise, request the code at:e-mail: anders.kabell.kristensen@gmail.com71
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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
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4 CHAPTER 1. INTRODUCTIONIt is evid
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6 CHAPTER 1. INTRODUCTIONsub-cubic
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Chapter 2PrerequisitesFirst, this c
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2.2. CHOICE OF LANGUAGE AND TEST EN
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14 CHAPTER 3. EXPERIMENTAL APPROACH
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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 81 and 82: Appendix CReal-life application of
Page 83: 75t29t25t54t20t27 t13 t1t3t17t24t41
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