Conference Paper

Computing the Quartet Distance Between Evolutionary Trees of Bounded Degree.

DOI: 10.1142/9781860947995_0013 Conference: Proceedings of 5th Asia-Pacific Bioinformatics Conference, APBC 2007, 15-17 January 2007, Hong Kong, China
Source: DBLP


We present an algorithm for calculating the quartet distance between two evolutionary trees of bounded degree on a common set of n species. The previous best algorithm has running time O(d 2n 2) when considering trees, where no node is of more than degree d. The algorithm developed herein has running time O(d 9n log n)) which makes it the first algorithm for computing the quartet distance between non-binary trees which has a sub-quadratic worst case running time.

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    • "Two algorithms exist that can be directly applied to compute the parametric quartet distance (see also [11]). One runs in time O(n 2 min{d 1 , d 2 }), where, for i ∈ {1, 2}, d i is the maximum degree of a node in T i [12]; the other takes O(d 9 n log n) time, where d is the maximum degree of a node in T 1 and T 2 [34]. 4 Our faster O(n 2 ) algorithm offers a 2-approximate solution when an exact value of the parametric quartet distance is not required. "
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    ABSTRACT: We define, analyze, and give efficient algorithms for two kinds of distance measures for rooted and unrooted phylogenies. For rooted trees, our measures are based on the topologies the input trees induce on triplets; that is, on three-element subsets of the set of species. For unrooted trees, the measures are based on quartets (four-element subsets). Triplet and quartet-based distances provide a robust and fine-grained measure of the similarities between trees. The distinguishing feature of our distance measures relative to traditional quartet and triplet distances is their ability to deal cleanly with the presence of unresolved nodes, also called polytomies. For rooted trees, these are nodes with more than two children; for unrooted trees, they are nodes of degree greater than three. Our first class of measures are parametric distances, where there is parameter that weighs the difference between an unresolved triplet/quartet topology and a resolved one. Our second class of measures are based on Hausdorff distance. Each tree is viewed as a set of all possible ways in which the tree could be refined to eliminate unresolved nodes. The distance between the original (unresolved) trees is then taken to be the Hausdorff distance between the associated sets of fully resolved trees, where the distance between trees in the sets is the triplet or quartet distance, as appropriate.
    Full-text · Article · Jun 2009 · Theoretical Computer Science
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    • "In [4] we developed two algorithms: the first algorithm runs in time O(n 3 ) and space O(n 2 )—and is thus independent of the degree of the inner nodes—the second in time O(n 2 d 2 ) and space O(n 2 ), where d is the maximal degree of inner nodes in the trees— and thus depend on the degree of the nodes. The O(n 2 d 2 ) was later improved to O(n 2 d) [5] and by taking an approach similar to the Brodal et al. [2] O(n log n) we developed a sub-quadratic algorithm in terms of n but at a significant cost in terms of d: O(d 9 n log n) [10]. "
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    ABSTRACT: We derive a quadratic time and space algorithm for computing the quartet distance between a pair of general trees, i.e. trees where inner nodes can have any degree 3. The time and space complexity of our algorithm is quadratic in the number of leaves and does not depend on the degree of the inner nodes. This makes it the fastest algorithm for computing the quartet distance between general trees independent of the degree of the inner nodes.
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