computing the quartet distance between general trees

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20 CHAPTER 4. QUARTIC TIME ALGORITHMbT baT aCT ccFigure 4.1: A center node C of the leaves a,b,c. The white subtrees make up the setof subtrees denoted T rest.the topologies found in the two trees, an array is computed, holding, at position i , thetopology of the quartet containing a, b, c and the i ’th leaf. Given two of those arrays,one for T and one for T ′ , the topologies can be compared in linear time and the numberof different topologies associated with the triplet (a,b,c) counted. This gives an overallrunning time of O(n 4 ).A number of different quartet topologies is thus computed by processing all triplets(a,b,c). However, each quartet (a,b,c,d) is actually considered four times; once for eachpossible triplet composed from the four leaves – in other words, each of the four leaveswill eventually act as the leaf x, described above. Consequently, the total number of differentquartets counted has to be divided by four to get the quartet distance. See Alg. 4.1for an outline of the algorithm.Regarding space consumption, this approach only needs memory for the tree datastructure and the two arrays, which are all linear in the number of leaves. Thus, thealgorithm uses O(n) space.4.1 ImplementationThe quartic algorithm is simple and easy to understand and this counts for the implementationof it as well. The algorithm is a bit clumsy by nature, making a lot of traversalsof the tree, but since it is the starting point for my investigation of quartet distance calculation,my focus will be on correctness rather than efficiency, and thus, I will make noattempt to optimize the algorithm in any way. Having a solid foundation and referencepoint for comparison with the other implementations is important to gain confidence inthe final results.The main loop, line 2 of Algorithm 4.1, goes through all distinct triplets, which can begenerated in various ways. The centers of a tree are found by first making three traversals
4.1. IMPLEMENTATION 21Algorithm 4.1 Quartic algorithm for quartet distance1: diffQ = 02: for all triplets (a,b,c) do ⊲ O(n 3 )3: Find center C in T and C ′ in T ′ ⊲ O(n)4: Collect leaf sets T a , T b , T c and T rest ⊲ O(n)5: and leaf sets T a ′ , T ′ b , T c ′ and T rest ′ ⊲ O(n)6: Create arrays A and A ′7: for all leaves x ∉ {a,b,c} do ⊲ O(n)8: Topology can be decided in constant time based on leaf set membership9: A[x] ← topology(a,b,c, x) in T ⊲ O(1)10: A ′ [x] ← topology(a,b,c, x) in T ′11: end for12: for all topologies t ∈ A and t ′ ∈ A ′ do13: if t ≠ t ′ then14: diffQ = diffQ + 115: end if16: end for17: end for18: return diffQ / 4to find the path between each pair of nodes (a,b), (a,c) and (b,c) and then examiningthese paths to find the single common internal node that is the center. The leaf sets T a ,T b and T c are easily found by traversing the tree with origin in the center node and T rest isnot needed, as this set is just every leaf that is not in one of the other sets. Now, to processeach iteration of the loop in line 7 in constant time, it is necessary to go through the leafsets and deal with each leaf contained in these, rather than to just go through all leaves,which would require repeated look-ups in each leaf set, breaking the time bound. Thefour topologies are represented by integers 0,1,2,3 and comparison of the two arrays isdone by traversing them in parallel and comparing elements.4.1.1 ResultThe algorithm has been implemented in Python and C++ and tested on the five groupsof artificial test trees described in Sec. 3.1. The result is displayed in Fig. 4.2 and Fig. 4.3.Because the algorithm is slow, the size of the trees used is limited to around 140 leaves.Larger trees would induce unacceptable running times and 140 leaves are enough toobserve any tendencies of the development in performance. The experiments have beenrepeated five times.Naturally, the expectation is to see a O(n 4 ) development on all trees and this is indeedthe case; all plots seem to be parallel to or less steep than the line indicating a quartic development.Furthermore, the estimated exponents of the lines plotted are below four.
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
Page 19: 2.2. CHOICE OF LANGUAGE AND TEST EN
Page 22 and 23: 14 CHAPTER 3. EXPERIMENTAL APPROACH
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
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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
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70 APPENDIX A. PREPROCESSING FOR TH
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Appendix CReal-life application of
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