computing the quartet distance between general trees

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34 CHAPTER 6. CUBIC TIME ALGORITHMAlgorithm 6.1 Cubic algorithm for quartet distance1: Calculate subtree leaf set sizes ⊲ O(n)2: Calculate shared leaf set sizes ⊲ O(n 2 )3: sharedQ = 04: for all pairs (a,b) do ⊲ O(n 2 )5: Find paths p in T and p ′ in T ′ between a and b ⊲ O(n)6: Find all centers C on the path p and C ′ on p ′ ⊲ O(n)7: for all leaves c ∉ {a,b} do ⊲ O(n)8: tmpQ = count quartets (a,b,c,x) with same topology in the two trees ⊲ O(1)9: – this is done in constant time using C abc and shared leaf set sizes10: sharedQ = sharedQ + tmpQ11: end for12: end for13: diffQ = ( n4)− sharedQ/1214: return diffQGoing through each pair of leaves (a,b) is easy. The path between two leaves is foundby making one traversal of the tree. Every internal node on the path has a number ofsubtrees (besides the two containing a and b) and each of these are traversed in turn,registering which leaves are contained in that subtree. A center is simply storing the idsof the edges that identify the subtrees containing a, b and a third leaf. An array, call it Ais created to store each of the n −2 centers. Centers are stored in the array A on the indexcorresponding to the id of the third leaf known as c.Now only the counting remains. Looking at the pair of centers associated with everychoice of third leaf, the shared butterfly quartets and shared star quartets associated withexactly those centers are counted, cf. Eq. (6.1), Eq. (6.2) and subsequent expressions.6.1.1 ResultThe implementations have been exposed to the usual range of trees. The result is displayedin Fig. 6.2 and Fig. 6.3.As expected, both implementations seem to have a cubic behaviour on all trees, sincethey are parallel to the purely cubic line. This is further supported by the estimated exponentswhich are all very close to three.The experiments have been continued until the running times exceeded what is acceptable.Because of the fact that the Python implementation is significantly slower andsince a cubic development in running time is rather dramatic, the Python experimentshave only been continued to as much as 400 leaves. The C++ experiments to 800 leaves.The experiments have in general been repeated five times, however, the larger Pythonexperiments take more than an hour of running time each and have therefore only been
6.1. IMPLEMENTATION 35carried out once.
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
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Page 28 and 29: 20 CHAPTER 4. QUARTIC TIME ALGORITH
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Page 33 and 34: Chapter 5Calculating leaf set sizes
Page 35 and 36: 5.3. IMPLEMENTING LEAF SET ALGORITH
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Page 39 and 40: Chapter 6Cubic time algorithmHere I
Page 41: 6.1. IMPLEMENTATION 33that we can l
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 78 and 79: 70 APPENDIX A. PREPROCESSING FOR TH
Page 81 and 82: Appendix CReal-life application of
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