Abstract
The triplet distance is a distance measure that compares two rooted trees on the same set of leaves by enumerating all sub-sets of three leaves and counting how often the induced topologies of the tree are equal or different. We present an algorithm that computes the triplet distance between two rooted binary trees in time O (n log2 n). The algorithm is related to an algorithm for computing the quartet distance between two unrooted binary trees in time O (n log n). While the quartet distance algorithm has a very severe overhead in the asymptotic time complexity that makes it impractical compared to O (n2) time algorithms, we show through experiments that the triplet distance algorithm can be implemented to give a competitive wall-time running time.
Highlights
Using trees to represent relationships is widespread in many scientific fields, in particular in biology where trees are used e.g. to represent species relationships, so called phylogenies, the relationship between genes in gene families or for hierarchical clustering of high-throughput experimental data
In this paper we focus on the triplet distance and develop an O (n log2 n) time algorithm for computing this distance between two rooted binary trees
In the following we show how the hierarchical decomposition tree (HDT) of T2 can be constructed in time O(n), and we prove that the height of the HDT is O
Summary
Using trees to represent relationships is widespread in many scientific fields, in particular in biology where trees are used e.g. to represent species relationships, so called phylogenies, the relationship between genes in gene families or for hierarchical clustering of high-throughput experimental data Common for these applications is that differences in the data used for constructing the trees, or differences in the computational approach for constructing the trees, can lead to slightly different trees on the same set of leaf IDs. Common for these applications is that differences in the data used for constructing the trees, or differences in the computational approach for constructing the trees, can lead to slightly different trees on the same set of leaf IDs To compare such trees, distance measures are often used. The triplet distance (for rooted trees) and quartet distance (for unrooted trees) enumerate all subsets of leaves of size three and four, respectively, and test if the induced topology of the leaves is the same in the two trees
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