Δ additive and Δ ultra-additive maps, Gromov's trees, and the Farris transform

In phylogenetic analysis, one searches for phylogenetic trees that reflect observed similarity between a collection of species in question. To this end, one often invokes two simple facts: (i) Any tree is completely determined by the metric it induces on its leaves (which represent the species). (ii) The resulting metrics are characterized by their property of being additive or, in the case of dated rooted trees, ultra-additive. Consequently, searching for additive or ultra-additive metrics A that best approximate the metric D encoding the observed similarities is a standard task in phylogenetic analysis. Remarkably, while there are efficient algorithms for constructing optimal ultra-additive approximations, the problem of finding optimal additive approximations in the l1 or l∞ sense is NP-hard. In the context of the theory of δ-hyperbolic groups, however, good additive approximations A of a metric D were found by Gromov already in 1988 and shown to satisfy the bound∥D-A∥∞⩽Δ(D)⌈log2(#X-1)⌉,where Δ(D), the hyperbolicity of D, i.e. the maximum of all expressions of the formD(u,v)+D(x,y)-max(D(u,x)+D(v,y),D(u,y)+D(v,x))(u,v,x,y∈X). Yet, besides some notable exceptions (e.g. Adv. Appl. Math. 27 (2001) 733-767), the potential of Gromov's concept of hyperbolicity is far from being fully explored within the context of phylogenetic analysis. In this paper, we provide the basis for a systematic theory of Δultra-additive and Δadditive approximations. In addition, we also explore the average and worst case behavior of Gromov's bound.