Cache Oblivious Algorithms for Computing the Triplet Distance between Trees

Abstract

We consider the problem of computing the triplet distance between two rooted unordered trees with n labeled leaves. Introduced by Dobson in 1975, the triplet distance is the number of leaf triples that induce different topologies in the two trees. The current theoretically fastest algorithm is an O( n log n ) algorithm by Brodal et al. (SODA 2013). Recently, Jansson and Rajaby proposed a new algorithm that, while slower in theory, requiring O( n log 3 n ) time, in practice it outperforms the theoretically faster O( n log n ) algorithm. Both algorithms do not scale to external memory. We present two cache oblivious algorithms that combine the best of both worlds. The first algorithm is for the case when the two input trees are binary trees, and the second is a generalized algorithm for two input trees of arbitrary degree. Analyzed in the RAM model, both algorithms require O( n log n ) time, and in the cache oblivious model O( n / B log 2 n / M ) I/Os. Their relative simplicity and the fact that they scale to external memory makes them achieve the best practical performance. We note that these are the first algorithms that scale to external memory, both in theory and in practice, for this problem.

Building a small and informative phylogenetic supertree

“…The Algorithm. It is known how to compute D(T 1 , T 2 ) in O(n log n) time [4,5]. Below, we show how to compute D(T 1 , T 2 ) in O(qn) time, which is faster than [4,5] when q = o(log n).…”

“…The performance of q-MAXRTC in the experiments of this dataset is then defined by the ratio S(T, T q )/ n 3 , where S(T, T q ) = n 3 − D(T, T q ). To compute this ratio efficiently, we used the rooted triplet distance implementation in [5]. We measured the performance of q-MAXRTC for q ∈ {2, 3, 5, 7, 9, 11}.…”

“…Finally, in the design of phylogenetic tree comparison algorithms, trees with fewer internal nodes sometimes admit faster running times. For example, given two trees built on the same leaf label set of size n, the fastest known algorithms for computing the so-called rooted triplet distance between the two trees takes O(n log n) time [4,5], but if at least one of the input trees has O(1) internal nodes then the time complexity can be reduced to O(n); see Section 5. As the available published trees get larger and larger (the total number of species on Earth was recently estimated to be 1 trillion [17]), to make their comparison practical, it may become necessary to approximate them using trees with fewer internal nodes while keeping enough triplet branching structure to represent them accurately.…”

Computing the Rooted Triplet Distance Between Phylogenetic Networks

“…When k = 0 , both N 1 and N 2 are trees. This case has been extensively studied in the literature [4,[18][19][20][21][22][23][24], with the most efficient algorithms in theory and practice [19,20,24] running in O(n log n) time. For k = 1 , an O(n 2.687 )-time algorithm based on counting 3-cycles in an auxiliary graph was given in [17], and a faster, O(n log n)-time algorithm that transforms the input to a constant number of instances with k = 0 was given in [25].…”

“…All of these algorithms allow the vertices in the input networks to have arbitrary degrees. Moreover, software implementations of the fast algorithms for k = 0 and k = 1 are available [20,[23][24][25].…”