computing the quartet distance between general trees

Chapter 9ConclusionThe focus of this thesis has been a thorough experimental study of the algorithm forquartet distance computation between general trees described by Mailund et al. [14]with the aim of eventually being able to either accept or reject the postulated sub-cubicrunning time as being a practically usable result.As a foundation I have given a gentle introduction to the subject of quartet distancecalculation followed by a detailed description of two reference algorithms – a quartic anda cubic – including implementations and experimental verification of those.Since my focus has been on experiments, I have given a careful review of a wide rangeof test data, arguing that the whole spectrum of challenging input has been covered.I have implemented the supposed sub-cubic time algorithm and my conclusion ispositive – I am certainly convinced that the algorithm performs in sub-cubic time in practice.However, my implementation is not strictly following the theoretical proposal andconsequently I do not have the support of the analytic result to rely on. Nevertheless Ican provide robust evidence to support my conclusion: exposing the algorithm to theentire set of test trees showed good practical behavior in every situation. As a matterof fact, the algorithm is showing a quadratic behavior on most input and especially inthe least artificial cases. The most challenging input introduced has been the star tree– a single internal node with n leaves attached – which forces the algorithm to make amultiplication of two n × n matrices, however, the situation is very artificial and the twomatrices multiplied contain zeroes in every entry except n entries that contain ones. Itis not within the scope of this thesis to gain knowledge about the internals of the libraryused for matrix multiplication and therefore, I do not know if such matrices are treateddifferently. The experiments on star trees show good results but the need for sub-cubicmatrix multiplication is inevitable.Another view on this conclusion is that the analysis of the algorithm is not tight63