computing the quartet distance between general trees

7.2. IMPLEMENTATION 477.2 ImplementationThis section gives a description of my work on implementing the sub-cubic algorithmof Sec. 7.1. As mentioned, it comes down to implementing two algorithms, namelyshared B (T,T ′ ) and diff B (T,T ′ ), used for counting shared and different butterflies in thetwo trees T and T ′ respectively.To calculate the total number of butterflies in one tree the algorithm for countingshared butterflies is used. This means that the basic preprocessing, namely calculatingthe shared leaf set sizes, has to be done three times; once for counting shared and differentbutterflies in the two trees, once for counting butterflies in the tree T and oncefor counting butterflies in the tree T ′ . The implementation of the algorithm for sharedleaf set size calculation has been described in Sec. 5.3 and will comprise the most spaceconsuming part of the sub-cubic algorithm, resulting in O(n 2 ) memory usage.The paper [14] presents the algorithms as follows: do the preprocessing for each pairof inner nodes, then do the counting for each pair of directed edges. Instead, I have chosena strategy of processing each pair of inner nodes in turn, including preprocessing andcounting for each pair of directed edges adjacent to the nodes. Consequently, the onlyreal preprocessing that is done prior to processing the nodes is calculating the sharedleaf set sizes, the implementation of which is described in Sec. 5.3.The full amount of preprocessing information needed to deal with a pair of nodes islisted in App. A. Some of it is needed in both shared B (T,T ′ ) and diff B (T,T ′ ), and someis specific to the latter. Therefore, I implemented the two together in a single functioncapable of returning both counts. If both values are needed, the preprocessing specificto the counting of different butterflies is just an extension to the preprocessing done forshared butterflies.Implementing shared B (T,T ′ ) is straight forward following the recipe I have given inSec. 7.1.2, 7.1.4 and App. A. I loop through every pair of internal nodes and fill out allthe preprocessing tables needed for shared quartets only. Then I go through the pairs ofedges adjacent to the two nodes, apply Eq. (7.8) and add the result to the total sum ofshared butterflies. I make use of the Boost 1 library for binomial coefficient calculation.Before returning the total number of shared butterflies the result is divided by 4; by 2because of symmetry (see clarification in Contribution 7.1) and 2 because there are twiceas many directed edges as there are undirected edges. One thing to note is that it onlymakes sense to look at edges pointing to internal nodes since there has to be at least twoleaves behind the edge (e.g. in the subtree F i on Fig. 7.3) to form a butterfly.Implementing diff B (T,T ′ ) requires more attention. If the analysis given in the paper1 Boost: http://www.boost.org/