computing the quartet distance between general trees

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 ′ ).