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.

The rooted triplet distance measures the structural dissimilarity of two phylogenetic trees or phylogenetic networks by counting the number of rooted phylogenetic trees with exactly three leaf labels (called rooted triplets, or triplets for short) that occur as embedded subtrees in one, but not both, of them. Suppose that N 1 = (V 1 , E 1 ) and N 2 = (V 2 , E 2 ) are phylogenetic networks over a common leaf label set of size n, that N i has level k i and maximum in-degree d i for i ∈ {1, 2} , and that the networks' out-degrees are unbounded. Write. Previous work has shown how to compute the rooted triplet distance between N 1 and N 2 in O(n log n) time in the special case k ≤ 1 . For k > 1 , no efficient algorithms are known; applying a classic method from 1980 by Fortune et al. in a direct way leads to a running time of Ω(N 6 n 3 ) and the only existing non-trivial algorithm imposes restrictions on the networks' in-and outdegrees (in particular, it does not work when non-binary vertices are allowed). In this article, we develop two new algorithms with no such restrictions. Their running times are O(N 2 M + n 3 ) and O(M + Nk 2 d 2 + n 3 ) , respectively. We also provide implementations of our algorithms, evaluate their performance on simulated and real datasets, and make some observations on the limitations of the current definition of the rooted triplet distance in practice. Our prototype implementations have been packaged into the first publicly available software for computing the rooted triplet distance between unrestricted networks of arbitrary levels.

The rooted triplet distance measures the structural dissimilarity of two phylogenetic trees or phylogenetic networks by counting the number of rooted phylogenetic trees with exactly three leaf labels (called rooted triplets, or triplets for short) that occur as embedded subtrees in one, but not both, of them. Suppose that $$N_1 = (V_1, E_1)$$ N 1 = ( V 1 , E 1 ) and $$N_2 = (V_2, E_2)$$ N 2 = ( V 2 , E 2 ) are phylogenetic networks over a common leaf label set of size n, that $$N_i$$ N i has level $$k_i$$ k i and maximum in-degree $$d_i$$ d i for $$i \in \{1,2\}$$ i ∈ { 1 , 2 } , and that the networks’ out-degrees are unbounded. Write $$N = \max (|V_1|, |V_2|)$$ N = max ( | V 1 | , | V 2 | ) , $$M = \max (|E_1|, |E_2|)$$ M = max ( | E 1 | , | E 2 | ) , $$k = \max (k_1, k_2)$$ k = max ( k 1 , k 2 ) , and $$d = \max (d_1, d_2)$$ d = max ( d 1 , d 2 ) . Previous work has shown how to compute the rooted triplet distance between $$N_1$$ N 1 and $$N_2$$ N 2 in $$\mathrm {O}(n \log n)$$ O ( n log n ) time in the special case $$k \le 1$$ k ≤ 1 . For $$k > 1$$ k > 1 , no efficient algorithms are known; applying a classic method from 1980 by Fortune et al. in a direct way leads to a running time of $${\Omega }(N^{6} n^{3})$$ Ω ( N 6 n 3 ) and the only existing non-trivial algorithm imposes restrictions on the networks’ in- and out-degrees (in particular, it does not work when non-binary vertices are allowed). In this article, we develop two new algorithms with no such restrictions. Their running times are $$\mathrm {O}(N^{2} M + n^{3})$$ O ( N 2 M + n 3 ) and $$\mathrm {O}(M + N k^{2} d^{2} + n^{3})$$ O ( M + N k 2 d 2 + n 3 ) , respectively. We also provide implementations of our algorithms, evaluate their performance on simulated and real datasets, and make some observations on the limitations of the current definition of the rooted triplet distance in practice. Our prototype implementations have been packaged into the first publicly available software for computing the rooted triplet distance between unrestricted networks of arbitrary levels.

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