computing the quartet distance between general trees

Chapter 6Cubic time algorithmHere I will describe another algorithm for quartet distance computation by Christiansenet al. [6] that improves the solution to a cubic running time at the expense of an increasedspace consumption. The algorithm is based on the concept of shared leaf set sizes, introducedin Section 5.2, and further extends the idea of using centers, as introduced alongwith the quartic algorithm described in Chapter 4.We are interested in the number of quartets for which the topology differ betweenthe two trees. Therefore, we might as well count or calculate the number of quartetsthat share the same topology and then subtract this number from the overall number ofquartets, ( n4).Having calculated the shared leaf set sizes, the number of leaves common to twosubtrees, |T x ∩ T x ′ |, can be found with a constant time look-up. This will be used extensivelyto calculate the number of shared quartets containing some triplet of leaves,(a,b,c), in constant time. However, the main idea of the algorithm is to process pairs ofleaves, (a,b), of which there are O(n 2 ), and then, in linear time, to calculate the numberof shared quartets that include a given pair. This is done by considering all internalnodes on the path from a to b as centers. These centers can clearly be found in lineartime. Every leaf c, different from a and b, can be reached from exactly one of the centersC , by following an outgoing edge from C that is not part of the path between a and b. SeeFig. 6.1.In linear time, a path between a and b is found and an array computed, storing inentry i , the center of the triplet (a,b,i ). This is done for both trees. The arrays are linearin size and processing each pair of centers of leaves (a,b,i ) in constant time gives anoverall running time of the algorithm of O(n 3 ). The following and remaining part of thealgorithm is a recipe for constant time computation of the number of shared quartetscontaining a triplet (a,b,c), given two centers, C in T and C ′ in T ′ , of the triplet.31