Quartet distance

The quartet distance[1] is a way of measuring the distance between two phylogenetic trees. It is defined as the number of subsets of four leaves that are not related by the same topology in both trees.

Computing the quartet distance

The most straightforward computation of the quartet distance would require $O(N^{4})$ time, where $N$ is the number of leaves in the trees.

For binary trees, better algorithms have been found to compute the distance in

$O(N^{2})$ time[2]
$O(N\log ^{2}N)$ time[3]

and

$O(N\log N)$ time[4]

Gerth Stølting Brodal et al. found an algorithm that takes $O(DN\log N)$ time to compute the quartet distance between two multifurcating trees when $D$ is the maximum degree of the trees,[5] which is accessible in C, perl, and the R package Quartet.

References

Estabrook, George F.; McMorris, F. R.; Meacham, Christopher A. (1985). "Comparison of Undirected Phylogenetic Trees Based on Subtrees of Four Evolutionary Units". Systematic Zoology. 34 (2): 193–200. doi:10.2307/2413326. JSTOR 2413326.
Bryant, D.; J. Tsang; P. E. Kearney; M. Li. (11 Jan 2000). "Computing the quartet distance between evolutionary trees". Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms. N.Y.: ACM Press: 285–286.
Brodal, Gerth Stølting; Fagerberg, Rolf; Pedersen, Christian N. S. (2001). "Computing the Quartet Distance between Evolutionary Trees in Time $O(n\log ^{2}n)$ ". Algorithms and Computation. Lecture Notes in Computer Science. 2223. pp. 731–742. doi:10.1007/3-540-45678-3_62. ISBN 978-3-540-42985-2.
Brodal, Gerth Stølting; Rolf Fagerberg; Christian Nørgaard Storm Pedersen (2003). "Computing the Quartet Distance Between Evolutionary Trees in Time $O(n\log n)$ ". Algorithmica. 38 (2): 377–395. doi:10.1007/s00453-003-1065-y.
Brodal, Gerth Stølting; Rolf Fagerberg; T Mailund; Christian Nørgaard Storm Pedersen; A Sand (2013). "Efficient algorithms for computing the triplet and quartet distance between trees of arbitrary degree" (PDF). Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM: 1814–1832. doi:10.1137/1.9781611973105.130. ISBN 978-1-61197-251-1.

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[eastbrook1985-1] Estabrook, George F.; McMorris, F. R.; Meacham, Christopher A. (1985). "Comparison of Undirected Phylogenetic Trees Based on Subtrees of Four Evolutionary Units". Systematic Zoology. 34 (2): 193–200. doi:10.2307/2413326. JSTOR 2413326.

[2] Bryant, D.; J. Tsang; P. E. Kearney; M. Li. (11 Jan 2000). "Computing the quartet distance between evolutionary trees". Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms. N.Y.: ACM Press: 285–286.

[3] Brodal, Gerth Stølting; Fagerberg, Rolf; Pedersen, Christian N. S. (2001). "Computing the Quartet Distance between Evolutionary Trees in Time $O(n\log ^{2}n)$ ". Algorithms and Computation. Lecture Notes in Computer Science. 2223. pp. 731–742. doi:10.1007/3-540-45678-3_62. ISBN 978-3-540-42985-2.

[4] Brodal, Gerth Stølting; Rolf Fagerberg; Christian Nørgaard Storm Pedersen (2003). "Computing the Quartet Distance Between Evolutionary Trees in Time $O(n\log n)$ ". Algorithmica. 38 (2): 377–395. doi:10.1007/s00453-003-1065-y.

[5] Brodal, Gerth Stølting; Rolf Fagerberg; T Mailund; Christian Nørgaard Storm Pedersen; A Sand (2013). "Efficient algorithms for computing the triplet and quartet distance between trees of arbitrary degree" (PDF). Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM: 1814–1832. doi:10.1137/1.9781611973105.130. ISBN 978-1-61197-251-1.