Refined Buneman Trees

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0.972: [GC 1675K->1182K(1984K), 0.0011610 secs] 0.989: [GC 1694K->1202K(1984K), 0.0014850 secs] 1.004: [GC 1714K->1222K(1984K), 0.0015970 secs] 1.028: [GC 1734K->1243K(1984K), 0.0016290 secs] 1.038: [GC 1755K->1264K(1984K), 0.0015250 secs] 1.048: [GC 1776K->1284K(1984K), 0.0015470 secs] 1.058: [GC 1796K->1306K(1984K), 0.0015560 secs] 1.070: [GC 1818K->1328K(1984K), 0.0019730 secs] 1.092: [GC 1840K->1355K(1984K), 0.0022360 secs] 1.104: [GC 1867K->1385K(1984K), 0.0025080 secs] 1.115: [GC 1897K->1406K(1984K), 0.0023980 secs] 1.126: [GC 1918K->1417K(1984K), 0.0017430 secs] 1.136: [GC 1929K->1441K(1984K), 0.0022290 secs] 1.147: [GC 1953K->1455K(1984K), 0.0018660 secs] 1.149: [Full GC 1455K->356K(1984K), 0.0411630 secs] 1.198: [GC 868K->384K(1984K), 0.0014270 secs] 1.208: [GC 896K->409K(1984K), 0.0015440 secs] 1.218: [GC 921K->428K(1984K), 0.0019500 secs] 1.228: [GC 940K->447K(1984K), 0.0017300 secs] 1.238: [GC 959K->465K(1984K), 0.0021660 secs] 1.249: [GC 977K->483K(1984K), 0.0019060 secs] 1.259: [GC 995K->495K(1984K), 0.0020870 secs] 1.270: [GC 1007K->516K(1984K), 0.0017420 secs] 1.280: [GC 1028K->528K(1984K), 0.0016770 secs] 1.290: [GC 1040K->541K(1984K), 0.0016130 secs] 1.300: [GC 1053K->554K(1984K), 0.0016020 secs] 1.310: [GC 1066K->566K(1984K), 0.0014070 secs] 1.320: [GC 1078K->582K(1984K), 0.0017910 secs] 1.330: [GC 1094K->597K(1984K), 0.0015940 secs] 1.340: [GC 1109K->613K(1984K), 0.0015960 secs] 1.350: [GC 1125K->638K(1984K), 0.0024570 secs] 1.361: [GC 1150K->652K(1984K), 0.0019230 secs] 1.371: [GC 1164K->672K(1984K), 0.0020700 secs] 1.381: [GC 1184K->689K(1984K), 0.0018670 secs] 1.391: [GC 1201K->713K(1984K), 0.0021350 secs] 1.401: [GC 1225K->734K(1984K), 0.0020890 secs] 1.412: [GC 1246K->758K(1984K), 0.0024130 secs] 1.422: [GC 1270K->772K(1984K), 0.0019690 secs] 1.433: [GC 1284K->793K(1984K), 0.0020540 secs] 1.466: [GC 1305K->800K(1984K), 0.0018280 secs] 1.481: [GC 1312K->809K(1984K), 0.0016140 secs] 1.495: [GC 1321K->815K(1984K), 0.0008950 secs] 1.512: [GC 1327K->823K(1984K), 0.0010560 secs] 1.529: [GC 1335K->830K(1984K), 0.0010790 secs] 1.542: [GC 1342K->837K(1984K), 0.0012930 secs] 1.556: [GC 1349K->846K(1984K), 0.0015160 secs] 1.568: [GC 1358K->856K(1984K), 0.0012840 secs] 1.581: [GC 1368K->864K(1984K), 0.0013920 secs] 1.593: [GC 1376K->872K(1984K), 0.0014740 secs] 1.606: [GC 1384K->881K(1984K), 0.0013130 secs] 1.619: [GC 1393K->889K(1984K), 0.0013220 secs] 1.632: [GC 1401K->897K(1984K), 0.0012040 secs] 1.647: [GC 1409K->904K(1984K), 0.0012350 secs] 1.659: [GC 1416K->912K(1984K), 0.0012510 secs] 1.673: [GC 1424K->922K(1984K), 0.0013530 secs] 1.685: [GC 1434K->931K(1984K), 0.0014230 secs] 1.698: [GC 1443K->941K(1984K), 0.0016090 secs] 1.712: [GC 1453K->950K(1984K), 0.0013770 secs] 1.724: [GC 1462K->956K(1984K), 0.0013470 secs] 1.737: [GC 1468K->965K(1984K), 0.0014090 secs] 1.750: [GC 1477K->974K(1984K), 0.0013430 secs] 1.763: [GC 1486K->983K(1984K), 0.0013740 secs] 1.775: [GC 1495K->991K(1984K), 0.0012020 secs] 1.787: [GC 1503K->996K(1984K), 0.0012050 secs] 1.800: [GC 1508K->1005K(1984K), 0.0012510 secs] 1.813: [GC 1517K->1013K(1984K), 0.0013920 secs] 1.825: [GC 1525K->1020K(1984K), 0.0013730 secs] 1.837: [GC 1532K->1029K(1984K), 0.0013140 secs] 1.850: [GC 1541K->1036K(1984K), 0.0013250 secs] 1.863: [GC 1548K->1044K(1984K), 0.0012460 secs] 1.875: [GC 1556K->1052K(1984K), 0.0013450 secs] 1.887: [GC 1564K->1059K(1984K), 0.0012150 secs] 1.899: [GC 1571K->1065K(1984K), 0.0010660 secs] 1.911: [GC 1577K->1074K(1984K), 0.0013640 secs] 1.923: [GC 1586K->1083K(1984K), 0.0012460 secs] 105
Bibliography [AJL + 02] Bruce Alberts, Alexander Johnson, Julian Lewis, Martin Raff, Keith Roberts, and Peter Walter. Molecular Biology of the Cell, Fourth Edition. Garland Science, Taylor & Francis Group, 29 West 35th Street, New York, 2002. [BB99] Vincent Berry and David Bryant. Faster reliable phylogenetic analysis. In Proceedings of the third annual international conference on Computational molecular biology, pages 59–68. ACM Press, 1999. [BFM + 03] [BFÖ+ 03] [BG91] [BG00] [BM99] [Bun71] Gerth Stølting Brodal, Rolf Fagerberg, Thomas Mailund, Christian N. S. Pedersen, and Derek Phillips. Speeding up neighbourjoining tree construction. Technical report, ALCOM-FT, 2003. Gerth Stølting Brodal, Rolf Fagerberg, Anna Östlin, Christian N. S. Pedersen, and S. Srinivasa Rao. Computing Refined Buneman Trees in Cubic Time. In Proceedings of the 3rd Workshop on Algorithms in BioInformatics (WABI 2003), volume 2812 of Lecture Notes in Computer Science, pages 259–270. Springer Verlag, September 2003. Jean-Pierre Barthélemy and Alain Guénoche. Trees and Proximity Representations. John Wiley & Sons, 1991. Vincent Berry and Olivier Gascuel. Inferring evolutionary trees with strong combinatorial evidence. Theoretical Computer Science, 240(2):271–298, 2000. David Bryant and Vincent Moulton. A polynomial time algorithm for constructing the refined buneman tree. Applied Mathematics Letters, 12:51–56, 1999. Peter Buneman. The recovery of trees from measures of dissimilarity. In Mathematics in the Archeological and Historical Sciences, pages 387–395. Edinburgh University Press, 1971. [cal04] Cambridge advanced learner’s dictionary. Webpage, 2004. http://dictionary.cambridge.org/. 106
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Refined Buneman Trees Lasse Westh-N
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This thesis is dedicated to my fami
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Contents 1 Introduction 7 1.1 Docum
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13.3 Correctness of the reference i
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The theory of evolution has also be
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Chapter 2 Definitions This chapter
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A C B Figure 2.1: An evolutionary t
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2.4 Quartets To every set of four s
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2.6 Splits The partition of a finit
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evolutionary tree gives an invaluab
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time in the algorithm. In [BFÖ+ 03
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Figure 3.1: A tree of life. 22
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knowing its origins, but how does h
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anging from huge time complexity to
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Algorithm 2 The Neighbor-Joining al
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A, C, T 1 2 3 T A, C, T T C, T A, T
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Part II Implementing Refined Bunema
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Algorithm 3 Overapproximating the r
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the pseudo-code for the algorithm i
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AE DE D B e E BC A C BE AC Figure 5
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construction, but we still have to
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Chapter 6 TheTreeDataStructure This
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incidentedge Figure 6.2: The world
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interface EdgeIterator { boolean ha
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Figure 6.7: An example a node which
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So how do we find σ ′ We start
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Algorithm 5. Offhand, the algorithm
Page 55 and 56: Algorithm 6 The algorithm that calc
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: Appendix B Garbage Collector Log 0.
Page 109: [Kim80] M. Kimura. A simple model f
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