computing the quartet distance between general trees

7.1. THE ALGORITHM 45More interesting are the steps needed to prepare each of the tables used for look-up– because the actual calculation of these is part of the algorithm and a thorough understandingis therefore essential. All tables are prepared during the preprocessing of eachpair of internal nodes, along with the table I , presented in Sec. 7.1.1. They are all resultsof further processing of I and necessary to make the constant time calculation. With theexception of the tables I1 ′′′ ′′′and I2, none of the tables are time-consuming to deal withand no worse than O(d v d v ′) which, for all pairs of inner nodes, leads to a total time of∑v∈T∑v ′ ∈T ′ d v d v′ = ( ∑ v∈T d v )( ∑ v ′ ∈T ′ d v ′) ≤ (2|E|)(2|E|) = O(n2 ).These two exceptions, I1 ′′′ ′′′and I2, are actually identical and only differ in the waythey are calculated. With one term, they shall simply be known as I ′′′ and the calculationof this table will be explained thoroughly in the following Sec. 7.1.4.1.7.1.4.1 The calculation of I ′′′The table named I ′′′ plays a part in the calculation of the number of different butterfliesand in order to keep the entire running time of the algorithm sub-cubic, special care isneeded in the calculation of the table. It is defined as follows:I ′′′ [i , j ] =∑d vd v ′∑k=1,k≠i l=1,l≠jI [i ,l]I [k, j ]I [k,l] (7.11)Filling the table naively, in accordance with the formula, takes time O(n 4 ) and thesub-cubic time bound promised will be broken. Hence, the calculation constitutes aserious barrier and demands separate attention. The solution is instead to calculate eitherI ′′′1 = (I I T )I or otherwise I ′′′2 = I (I T I ) which are both described in more detail in theappendix. At first sight, this does not seem to solve the problem, since the solution is relyingon matrix multiplication. The complexity of matrix multiplication, if done naively,is O(n 3 ). As explained in the article [14] choosing either the first or second solution willresult in an explicit running time of either O(d 2 v d v ′) or O(d v d 2 v ′ ) for processing a pair ofinternal nodes (v, v ′ ) with degrees d v and d v′ respectively. However, other methods forcalculating the matrix product may be utilized and this is essential to the algorithm. Applyinga matrix multiplication method with a time complexity of O(n ω ) on square matrices,one can make the calculation in time O(max(d v ,d v ′) ω ). This value might be smallerfor some matrices that are nearly square, but the approach requires that the matrices arepadded with zeroes to become square – i.e. extending the matrix to fit the requirementsof the matrix multiplication method used.It is difficult to predict the impact of the matrix multiplication on the entire algorithm;since it is on an internal node basis the complete running time is not identical tothe one of the multiplication method. In the article [14], Section 4 gives a thorough case