computing the quartet distance between general trees

7.1. THE ALGORITHM 39CcabADFigure 7.2: Example of a claim. The directed edge is a unique identifierof the directed quartet ab → cd.dedges, count the quartet as a shared butterfly if the two claims give rise to the same topology,and otherwise count the quartet as a different butterfly. Making heavy use of preprocessing,one such pair of edges can be handled in constant time, see Sec. 7.1.4. Since|E| = O(n) and there are O(n 2 ) pairs of edges, the process of counting the butterflies canbe done in O(n 2 ) time. Thus, the preprocessing step is crucial to the running time.7.1.1 Basic preprocessingHere I will describe only a fundamental part of the preprocessing that is necessary tounderstand the intuition behind the algorithm and how to count shared and differentbutterflies. More preprocessing is needed to make it possible to process a pair of directededges in constant time as mentioned. Unfortunately that part of the preprocessing posesa threat to the sub-cubic complexity of the entire algorithm and has direct influence onthe running time and must be handled with care. For now it is merely confusing and Iwill postpone the introduction of this until the appropriate section.As we shall see, it comes in handy that the notion of claims and the concept of sharedleaf set sizes both deal with subtrees. The first preprocessing step is to calculate theshared leaf set sizes, as explained in Sec. 5.2, which has quadratic time and space consumption.The next step is to calculate, for each pair of internal nodes v ∈ T and v ′ ∈ T ′ , withsubtrees F 1 ,...,F dv and G 1 ,...,G d′v, a matrix I where I [i , j ] = |F i ∩G j |. When processingpairs of edges as mentioned above, we will need this matrix, I , associated with the twonodes that the edges point to.This is enough information about the preprocessing step to complete the intuitiveexplanation for counting butterflies in Sec. 7.1.2 and 7.1.3.