computing the quartet distance between general trees

56 CHAPTER 7. SUB-CUBIC TIME ALGORITHMFurther verifications of the quality of the algorithm In order to establish completeconfidence in the quality of the algorithm two more experiments have been carried out.The first one is an extension of the experiments already done, but with the purpose ofexamining more challenging combinations of trees to compare. This is done to furtherensure that the algorithm really is behaving well in all imaginable scenarios which mightindicate that it is in fact sub-cubic, even though we do not have the proof of the analysisto back up such a claim.The trees having a single internal node of very high degree, namely the star tree andthe wc tree, were the ones showing the steepest growth because of the matrix multiplicationof larger matrices. I will combine these with the trees having a larger number ofinternal nodes and yielding a worse actual running time to see if the combination of thetwo properties will result in a longer running time than the two do separately.The result is clear. These new combinations of input are not a greater challenge tothe algorithm. Figure 7.12 shows that the resulting performance is no worse than whatwe have previously witnessed; the growth in running time seems close to quadratic andthe actual running time is better than the worst we have seen so far.time in seconds - t(n)1000100101Performance of the C++ implementation on different combinations of trees.star-bin, best exp= 2.03star-ran, best exp= 2.05star-wc, best exp= 2.01wc-bin, best exp= 2.04wc-ran, best exp= 2.04nn 2n 30.10.0150 100 200 500 800number of leaves - nFigure 7.12: Combining the trees with high-degree inner nodes with other trees.The purpose of the second experiment is to ensure that the arrangement, or order,of the input is not influencing the running time. Some algorithms perform significantlybetter on certain inputs. For example, some sorting algorithms, like quick sort and bub-