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Optimal algorithms for comparing trees with labeled leaves. (English) Zbl 0589.62044
Summary: Let \(R_ n\) denote the set of rooted trees with n leaves in which: the leaves are labeled by the integer in \(\{\) 1,...,n\(\}\) ; and among interior vertices only the root may have degree two. Associated with each interior vertex v in such a tree is the subset, or cluster, of leaf labels in the subtree rooted at v. Cluster \(\{\) 1,...,n\(\}\) is called trivial.
Clusters are used in quantitative measures of similarity, dissimilarity and consensus among trees. For any k trees in \(R_ n\), the strict consensus tree \(C(T_ 1,...,T_ k)\) is that tree in \(R_ n\) containing exactly those clusters common to every one of the k trees. Similarity between trees \(T_ 1\) and \(T_ 2\) in \(R_ n\) is measured by the number \(S(T_ 1,T_ 2)\) of nontrivial clusters in both \(T_ 1\) and \(T_ 2\); dissimilarity, by the number \(D(T_ 1,T_ 2)\) of clusters in \(T_ 1\) or \(T_ 2\) but not in both. Algorithms are known to compute \(C(T_ 1,...,T_ k)\) in \(O(kn^ 2)\) time, and \(S(T_ 1,T_ 2)\) and \(D(T_ 1,T_ 2)\) in \(O(n^ 2)\) time.
I propose a special representation of the clusters of any tree T in \(R_ n\), one that permits testing in constant time whether a given cluster exists in T. I describe algorithms that exploit this representation to compute \(C(T_ 1,...,T_ k)\) in O(kn) time, and \(S(T_ 1,T_ 2)\) and \(D(T_ 1,T_ 2)\) in O(n) time. These algorithms are optimal in a technical sense. They enable well-known indices of consensus between two trees to be computed in O(n) time. All these results apply as well to comparable problems involving unrooted trees with labeled leaves.

62H30 Classification and discrimination; cluster analysis (statistical aspects)
68Q25 Analysis of algorithms and problem complexity
62-04 Software, source code, etc. for problems pertaining to statistics
Full Text: DOI
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