computing the quartet distance between general trees

54 CHAPTER 7. SUB-CUBIC TIME ALGORITHMFrom the table it is evident that padding the matrices is not speeding up the multiplication.Only in three experiments it seems faster in some cases and only in a very smallpercentage of the multiplications. Furthermore, from the sizes of the internal nodes involvedwe observe that it is only when dealing with very small matrices (#rows/#columns< 10), that we “benefit” from padding. More likely, it is a coincidence rather than an actualtendency. My conclusion is that it is not beneficial to pad the matrices and consequently,there is no reason to separately take care of the case where max(d v ,d v ′) ω issmallest and I will continue to distinguish only between dv 2d v ′ and d v d 2 , as done in thev ′prototype.Expectations The implementation still does not meet the formal description of the article,which means that we can not count on the analysis and thus, the implementationmight brake the sub-cubic time bound and behave as a cubic algorithm. It seems reasonableto keep the assumptions from the prototype. Nevertheless, I hope to see a generalimprovement which of course is dependent on the proportion of time actually spent onmatrix multiplication. This should be more significant to the experiments involving thelargest multiplications, namely the star trees and the worst case trees which seemed tobe suffering because of the larger matrices encountered.Result The result of applying the two final implementations on the usual array of testdata is shown in Fig. 7.10 and Fig. 7.11. We observe that there is no significant improvementin running time. The general picture is much the same as for the prototype; thefinal implementations perform in the same range of time consumption as the prototypeimplementations with a few exceptions. The reason might very likely be the relativelylow amount of time spent doing matrix multiplication and the relatively high amount oftime spent processing the pairs of internal nodes.There are a couple of cases, however, where a change in the slope of the plot is clearlyobservable. That is, not surprisingly, the trees with few inner nodes, resulting in largematrices; star trees and wc trees. And only for the C++ implementation. Especially thestar tree yields a dramatic improvement. The plot of the C++ prototype, that was almostparallel to the O(n 3 ) line, has now improved to being almost parallel to the O(n 2 ) line.However, the plot is not making an exact straight line which I suspect might be becausethe overhead is gradually equalized by the complexity of the matrix multiplication as thesize of the matrices grow. Eventually this will result in the plot showing the complexityof the matrix multiplication only. The processing time of an 800-leaf star tree is loweredfrom over 100 seconds to below 10 seconds meaning that the change of matrix librarywas indeed a great improvement.