An algorithm for finding a representation of a subtree distance

Kazutoshi Ando () and Koki Sato
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Kazutoshi Ando: Shizuoka University
Koki Sato: DAITO GIKEN, INC.

Journal of Combinatorial Optimization, 2018, vol. 36, issue 3, No 4, 742-762

Abstract: Abstract Generalizing the concept of tree metric, Hirai (Ann Combinatorics 10:111–128, 2006) introduced the concept of subtree distance. A nonnegative-valued mapping $$d:X\times X \rightarrow \mathbb {R}_+$$ d : X × X → R + is called a subtree distance if there exist a weighted tree T and a family $$\{T_x\mid x \in X\}$$ { T x ∣ x ∈ X } of subtrees of T indexed by the elements in X such that $$d(x,y)=d_T(T_x,T_y)$$ d ( x , y ) = d T ( T x , T y ) , where $$d_T(T_x,T_y)\ge 0$$ d T ( T x , T y ) ≥ 0 is the distance between $$T_x$$ T x and $$T_y$$ T y in T. Hirai (2006) provided a characterization of subtree distances that corresponds to Buneman’s (J Comb Theory, Series B 17:48–50, 1974) four-point condition for tree metrics. Using this characterization, we can decide whether or not a given mapping is a subtree distance in O $$(n^4)$$ ( n 4 ) time. In this paper, we show an O $$(n^3)$$ ( n 3 ) time algorithm that finds a representation of a given subtree distance. This results in an O $$(n^3)$$ ( n 3 ) time algorithm for deciding whether a given mapping is a subtree distance.

Keywords: Tree metric; Phylogeny; Realization algorithm (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s10878-017-0145-x

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