computing the quartet distance between general trees

6.1. IMPLEMENTATION 33that we can look up in constant time:|T ′ rest ∩ T a| = |T a | − (|T a ∩ T ′ a | + |T a ∩ T ′ b | + |T a ∩ T ′ c |)|T ′ rest ∩ T b| = |T b | − (|T b ∩ T ′ a | + |T b ∩ T ′ b | + |T b ∩ T ′ c |)|T rest ′ ∩ T c| = |T c | − (|T c ∩ T a ′ | + |T c ∩ T ′ b | + |T c ∩ T c ′ |) (6.3)The last expression is derived directly from the number of leaves in T ′ :|T rest ′ | = n − (|T a ′ | + |T ′ b | + |T c ′ |) (6.4)We are now able to compute the number of shared star quartets and thus, the overallnumber of shared quartets of some triplet in constant time. Combining this with theapproach of finding all centers associated with some pair of leaves in linear time, yieldsa running time of the entire algorithm of O(n 3 ). Because of the table for shared leaf setsizes, the algorithm requires a space consumption of O(n 2 ).Just like with the quartic algorithm the shared quartets are counted too many times.Here, however, each quartet is counted twelve times. This is due to the fact that we dealwith pairs and that each quartet is considered once for each possible six pairs that canbe composed from the four leaves. In addition, each of those pairs will be used for constructionof two triplets; one for each of the two remaining leaves. Therefore, the numberof shared quartet topologies counted is divided by twelve.See Alg. 6.1 for an outline of the algorithm.Contribution 6.1 Despite of the description in Section 2.2 of Christiansen et al. [6]which states that each quartet is counted four times, the cubic algorithm is actuallycounting each quartet twelve times as explained in the text. This is a minor change,but of course necessary to keep in mind, as to get the correct result when implementingthe algorithm. I made the discovery while working on my implementation, seeing thatthe result was consistently triple of what was expected.6.1 ImplementationThe cubic algorithm has been implemented first in Python and later in C++. Since nounusual tricks or language features are required, the two implementations are very similar.The first step is the implementation of the algorithms for calculating the leaf sets –line 1 and 2 of Alg. 6.1. These have been described and tested in Sec. 5.3.