Refined Buneman Trees

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A 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 B C1 C2 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 00 11 00 11 00 C 0 1 8 3 411 5 6 7 00 11 000000000 111111111 000000000 111111111 0 000000000 111111111 000000000 111111111 1 000000000 111111111 000000000 111111111 000 111 000000000 111111111 C’ 8 000000000 111111111 000000000 111111111 3 000000000 111111111 000000000 111111111 000000000 111111111 4 000000000 111111111 000000000 111111111 5 000000000 111111111 000000000 111111111 11000000000 111111111 00 116 000000000 111111111 11000000000 111111111 7 000000000 111111111 000000000 111111111 Figure 7.1: Manipulating the matrix in the single linkage clustering tree. A) we find the two rows and two columns with minimum entries (recall the matrix is symmetric). B) We reuse one set of row and column, and discard the other set or row and column. C) The new matrix is no longer able to index the row/ column that was discarded, and the reused row and column has been re-indexed. 53
Algorithm 6 The algorithm that calculates the single linkage clustering tree for x Require: δ is a distance measure on X Ensure: T is root in the single linkage clustering tree for x ∈ X 1: δ ′ = F ARRIST RANSF ORM(δ, x) 2: Q is a quad tree overlaid δ ′ 3: C is a list of clusters corresponding to δ ′ 4: while |C| > 1 do 5: (c 1 ,c 2 )=MINIMUM(Q) 6: c ′ = c 1 ∪ c 2 7: for c ′′ ∈ C, c ′′ ≠ c 1 ,c 2 do 8: δ c ′ ′ ,c = ′′ min{δ′ c 1,c ′′,δ′ c 2,c ′′} 9: end for 10: C = C ∪{c ′ }\{c 1 ,c 2 } 11: UPDATE(Q) 12: end while 13: T = C Thus, the algorithm for computing the single linkage clustering tree for x becomes the algorithm found in Algorithm 6. Analysing Algorithm 6 is straightforward: initialising δ ′ and Q takes time O(n 2 ). We iterate n times in the while loop, performing constant time search for the minimum entry in the distance matrix, linear time update of the new row/ column for C ′ with regards to the remaining clusters in the matrix, linear time removal of C 1 and C 2 , and linear time updating the quad tree data structure. All in all we have a O(n 2 ) implementation of the single linkage clustering tree algorithm. Thus, we can safely replace anchored Buneman trees with single linkage clustering trees in the main algorithm, since is has the same running time complexity. Also, we have argued that the anchored Buneman tree is contained in the single linkage clustering tree. And since we are building overapproximations of the refined Buneman tree, the extra splits from the single linkage clustering tree will not affect the algorithm adversely. 54
Page 1 and 2:
Refined Buneman Trees Lasse Westh-N
Page 3 and 4: This thesis is dedicated to my fami
Page 5 and 6: Contents 1 Introduction 7 1.1 Docum
Page 7 and 8: 13.3 Correctness of the reference i
Page 9 and 10: The theory of evolution has also be
Page 11 and 12: Chapter 2 Definitions This chapter
Page 13 and 14: A C B Figure 2.1: An evolutionary t
Page 15 and 16: 2.4 Quartets To every set of four s
Page 17 and 18: 2.6 Splits The partition of a finit
Page 19 and 20: evolutionary tree gives an invaluab
Page 21 and 22: time in the algorithm. In [BFÖ+ 03
Page 23 and 24: Figure 3.1: A tree of life. 22
Page 25 and 26: knowing its origins, but how does h
Page 27 and 28: anging from huge time complexity to
Page 29 and 30: Algorithm 2 The Neighbor-Joining al
Page 31 and 32: A, C, T 1 2 3 T A, C, T T C, T A, T
Page 33 and 34: Part II Implementing Refined Bunema
Page 35 and 36: Algorithm 3 Overapproximating the r
Page 37 and 38: the pseudo-code for the algorithm i
Page 39 and 40: AE DE D B e E BC A C BE AC Figure 5
Page 41 and 42: construction, but we still have to
Page 43 and 44: Chapter 6 TheTreeDataStructure This
Page 45 and 46: incidentedge Figure 6.2: The world
Page 47 and 48: interface EdgeIterator { boolean ha
Page 49 and 50: Figure 6.7: An example a node which
Page 51 and 52: So how do we find σ ′ We start
Page 53: Algorithm 5. Offhand, the algorithm
Page 57 and 58: a b c d root ab cd Figure 8.2: Upda
Page 59 and 60: 6000 Quad Tree performance characte
Page 61 and 62: 30000 Quad Tree performance charact
Page 63 and 64: sets A, B, C and D by scanning the
Page 65 and 66: Chapter 10 The Selection Algorithm
Page 67 and 68: is O(n 2 ). The algorithm uses a di
Page 69 and 70: Chapter 11 JSplits Figure 11.1: A s
Page 71 and 72: implementing an algorithm with a hi
Page 73 and 74: Chapter 12 Source Code The source c
Page 75 and 76: Chapter 13 The Reference Implementa
Page 77 and 78: • the splits that are generated.
Page 79 and 80: Chapter 14 Correctness This chapter
Page 81 and 82: The best possible way of testing wo
Page 83 and 84: 100000 90000 Performance of the ref
Page 85 and 86: of the size of the heap during the
Page 87 and 88: 140000 Space complexity best fit: x
Page 89 and 90: Chapter 16 Comparing Evolutionary T
Page 91 and 92: Figure 16.1: The size of the set B(
Page 93 and 94: Figure 16.2: The total number of sp
Page 95 and 96: that it over-induces splits, and th
Page 97 and 98: efined Buneman therefore suffers a
Page 99 and 100: Speedups might be achieved using op
Page 101 and 102: Appendix A Correctness of the Refer
Page 103 and 104: Quartet: 0 0 | 1 4 -0.1263501163396
Page 105 and 106:
Appendix B Garbage Collector Log 0.
Page 107 and 108:
Bibliography [AJL + 02] Bruce Alber
Page 109:
[Kim80] M. Kimura. A simple model f
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