A More Practical Algorithm for the Rooted Triplet Distance

Jansson, Jesper; Rajaby, Ramesh

doi:10.1089/cmb.2016.0185

“…The rooted triplet distance measures the dissimilarity between two leaf-labeled trees with identical labels. It is given by the number of rooted triplets that induce different minimal topologies ( Figure 1) in the two trees over the total number of triplets [28]. As tumor progression trees are fully-labeled, such metric cannot be directly applied: in this section we propose a novel similarity measure, inspired by the triplet distance, specifically designed for these more general trees.…”

Section: Methodsmentioning

confidence: 99%

Triplet-based similarity score for fully multi-labeled trees with poly-occurring labels

Ciccolella

¹

,

Bernardini

²

,

Denti

³

et al. 2020

Preprint

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The latest advances in cancer sequencing, and the availability of a wide range of methods to infer the evolutionary history of tumors, have made it important to evaluate, reconcile and cluster different tumor phylogenies.Recently, several notions of distance or similarities have been proposed in the literature, but none of them has emerged as the golden standard. Moreover, none of the known similarity measures is able to manage mutations occurring multiple times in the tree, a circumstance often occurring in real cases.To overcome these limitations, in this paper we propose MP3, the first similarity measure for tumor phylogenies able to effectively manage cases where multiple mutations can occur at the same time and mutations can occur multiple times. Moreover, a comparison of MP3 with other measures shows that it is able to classify correctly similar and dissimilar trees, both on simulated and on real data.

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“…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].…”

Section: Previous Workmentioning

confidence: 99%

“…degrees. Moreover, software implementations of the fast algorithms for k = 0 and k = 1 are available [20,[23][24][25].…”

Section: Previous Workmentioning

confidence: 99%

Computing the Rooted Triplet Distance Between Phylogenetic Networks

Jansson

¹

,

Mampentzidis

²

,

Rajaby

³

et al. 2021

Self Cite

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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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“…Results. All related work can be found in [5,1,14,3,15,9,10,11]. Previous and new results are shown in the table below.…”

Section: Introductionmentioning

confidence: 99%

“…Previous and new results are shown in the table below. For the cache oblivious model [8], the papers [5,1,14,3,10,11] do not provide an analysis, so here we provide an upper bound. The common main bottleneck with all previous approaches is that the data structures used rely intensively on Ω(n log n) random memory accesses.…”

Section: Introductionmentioning

confidence: 99%

Cache Oblivious Algorithms for Computing the Triplet Distance between Trees

Brodal

¹

,

Mampentzidis

²

2021

ACM J. Exp. Algorithmics

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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.

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A More Practical Algorithm for the Rooted Triplet Distance

Cited by 8 publications

References 19 publications

Triplet-based similarity score for fully multi-labeled trees with poly-occurring labels

Triplet-based similarity score for fully multi-labeled trees with poly-occurring labels

Computing the Rooted Triplet Distance Between Phylogenetic Networks

Cache Oblivious Algorithms for Computing the Triplet Distance between Trees

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