A nearly logarithmic lower bound on the randomized competitive ratio for the metrical task systems problem is presented. This implies a similar lower bound for the extensively studied K-server problem. The proof is based on Ramsey-type theorems for metric spaces, that state that every metric space contains a large subspace which is approximately a hierarchically well-separated tree (and in part...

We introduce a new class of abstract structures, which we call generalized ultrametric semilattices, and in which the meet operation of the semilattice coexists with a generalized distance function in a tightly coordinated way. We prove a constructive fixed-point theorem for strictly contracting functions on directed-complete generalized ultrametric semilattices, and introduce a corresponding i...

Sub-dominant theory provides efficient tools for clustering. However it classically works only for ultrametrics and ad hoc extensions like Jardine and Sibson’s 2ultrametrics. In this paper we study the extension of the notion of sub-dominant to other distance models in classification accounting for overlapping clusters. We prove that a given dissimilarity admits one and only one lower-maximal q...

We review the solution and properties of the diffusion equation in a hierarchical or ultrametric space [ 11. One of the oldest equations in physics is the equation of diffusion. In its most general form it reads: dPi N E..p. dt = j=l rJ ’ c where Pi is the probability of finding a particle at site i (i = 1, . . . . N) of some arbitrary discretized space M, and cij is the hopping or transition p...

-UTCE is a platform for ultrametric tree construction and tree evaluation. Phylogenetic tree can be used by biologists to observe the relationship between species, and the ultrametric tree is a popular model which is assumed that the rate of evolution is constant. UPGMA is one of well-know ultrametric tree constructing algorithm. However, UPGMA is a heuristic algorithm and can not guarantee the...

High dimensional, sparsely populated data spaces have been characterized in terms of ultrametric topology. This implies that there are natural, not necessarily unique, tree or hierarchy structures de ned by the ultrametric topology. In this note we study the extent of local ultrametric topology in texts, with the aim of nding unique \ ngerprints" for a text or corpus, discriminating between tex...