tqDist can be installed on Windows, Linux and Mac OS X. Doing this will install a set of command-line tools together with a Python module and an R package for scripting in Python or R. The software package is freely available under the GNU LGPL licence at http://birc.au.dk/software/tqDist.

Distance measures between trees are useful for comparing trees in a systematic manner, and several different distance measures have been proposed. The triplet and quartet distances, for rooted and unrooted trees, respectively, are defined as the number of subsets of three or four leaves, respectively, where the topologies of the induced subtrees differ. These distances can trivially be computed by explicitly enumerating all sets of three or four leaves and testing if the topologies are different, but this leads to time complexities at least of the order n3 or n4 just for enumerating the sets. The different topologies can be counted implicitly, however, and in this paper, we review a series of algorithmic improvements that have been used during the last decade to develop more efficient algorithms by exploiting two different strategies for this; one based on dynamic programming and another based on coloring leaves in one tree and updating a hierarchical decomposition of the other.

In this paper we present an experimental evaluation of the algorithms by Brodal et al. [SODA 2013] for computing the triplet and quartet distance measures between two leaf labelled rooted and unrooted trees of arbitrary degree, respectively. The algorithms count the number of rooted tree topologies over sets of three leaves (triplets) and unrooted tree topologies over four leaves (quartets), respectively, that have different topologies in the two trees.The algorithms by Brodal et al. maintain a long sequence of variables (hundreds for quartets) for counting different cases to be considered by the algorithm, making it unclear if the algorithms would be of theoretical interest only. In our experimental evaluation of the algorithms the typical overhead per node is about 2 KB and 10 KB per node in the input trees for triplet and quartet computations, respectively. This allows us to compute the distance measures for trees with up to millions of nodes. The limiting factor is the amount of memory available. With 31 GB of memory all our input instances can be solved within a few minutes.In the algorithm by Brodal et al. a few choices were made, where alternative solutions possibly could improve the algorithm, in particular for quartet distance computations. For quartet computations we expand the algorithm to also consider alternative computations, and make two observations: First we observe that the running time can be improved from O(max(d1, d2)where n is the number of leaves in the two trees, and d1 and d2 are the maximum degrees of the nodes in the two trees, respectively. Secondly, by taking a different approach to counting the number of disagreeing quartets we can reduce the number of calculations needed to calculate the quartet distance, improving both the running time and the space requirement by our algorithm by a constant factor.