computing the quartet distance between general trees

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64 CHAPTER 9. CONCLUSIONenough to reflect the actual running time. The algorithm performs better than predictedon most input but given two star trees it should perform with exactly the same time complexityas the matrix multiplication method in use, namely in (O ω ) time.The resulting implementation of the sub-cubic algorithm is not only behaving wellwith regards to time development – it also performs better than the two reference algorithmsand thus, the algorithm does not suffer from being the product of a seeminglycomplex idea.Finally, I have verified the correctness of each algorithm and left a few contributionsin this regard. These are highlighted as Contribution 6.1, 7.1 and 7.2.In summary I have verified the correctness and time complexity of three algorithms,with focus on the sub-cubic algorithm. Hopefully I have left the reader with enoughinsight and the necessary tools to continue the work in a direction towards an even fasteralgorithm.9.1 Future workI find it natural that this work should be extended algorithmically in the attempt of findinga purely quadratic algorithm. I think the ideas used by Mailund et al. [14] are interestingand promising. In any case, the sub-cubic algorithm may be usable in practice,but it is not superior to the other existing algorithms for general trees and hence, I thinkalgorithmic improvements would be far more valuable than practical optimizations.I think focus should be on finding a different way to calculate diff B (T,T ′ ) as to getrid of the matrix multiplication. Alternatively, it might be possible to calculate the totalamount of quartets where both inherit a butterfly topology, call this quantity total B (T,T ′ ),and then calculate diff B (T,T ′ ) = total B (T,T ′ ) − shared B (T,T ′ ).
Bibliography[1] Mukul S. Bansal, Jianrong Dong, and David Fernández-Baca. Comparing and aggregatingpartially resolved trees. In LATIN’08: Proceedings of the 8th Latin Americanconference on Theoretical informatics, pages 72–83, Berlin, Heidelberg, 2008.Springer-Verlag. ISBN 3-540-78772-0, 978-3-540-78772-3.[2] Gerth Stølting Brodal, Rolf Fagerberg, and Christian N. S. Pedersen. Computing thequartet distance between evolutionary trees in time O(n log 2 n). Proceedings of the12th International Symposium for Algorithms and Computation (ISAAC), 2223:731–742, 2001.[3] Gerth Stølting Brodal, Rolf Fagerberg, and Christian N. S. Pedersen. Computing thequartet distance between evolutionary trees in time O(n logn). Algorithmica, 38:377–395, 2003.[4] David Bryant, John Tsang, Paul E. Kearney, and Ming Li. Computing the quartetdistance between evolutionary trees. In SODA, pages 285–286, 2000.[5] Chris Christiansen and Martin Randers. Computing the quartet distance betweentrees of arbitrary degrees. Master’s thesis, University of Aarhus, January 2006.[6] Chris Christiansen, Thomas Mailund, Christian N. S. Pedersen, and Martin Randers.Computing the quartet distance between trees of arbitrary degree. In WABI, pages77–88, 2005.[7] Don Coppersmith and Shmuel Winograd. Matrix multiplication via arithmetic progressions.Journal of Symbolic Computation, 9(3):251–280, 1990. ISSN 0747-7171.Computational algebraic complexity editorial.[8] Bhaskar DasGupta, Xin He, Tao Jiang, Ming Li, John Tromp, and Louxin Zhang. Oncomputing the nearest neighbor interchange distance. 1997.65
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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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