Maximum gradient embeddings and monotone clustering

Abstract

Let (X, d X ) be an n-point metric space. We show that there exists a distribution D over non-contractive embeddings into trees f : X → T such that for every x ∈ X,where C is a universal constant. Conversely we show that the above quadratic dependence on log n cannot be improved in general. Such embeddings, which we call maximum gradient embeddings, yield a framework for the design of approximation algorithms for a wide range of clustering problems with monotone costs, including fault-tolerant versions of k-median and facility location. IntroductionMetric embeddings are an invaluable tool in analysis, Riemannian geometry, group theory, graph theory, and the design of approximation algorithms. In most cases embeddings are used to simplify a geometric object that we wish to understand, or on which we need to preform certain algorithmic tasks. Thus one tries to faithfully represent a metric space as a subset of another space with controlled geometry, whose structure is well enough understood to successfully address the problem at hand. There is some obvious exibility in this approach: Both the choice of target space and the notion of faithfulness of an embedding can be adapted to the problem we wish to solve. Of course, once these choices are made, the main difficulty is the construction of the required embedding, and in the algorithmic context we have the additional burden of making sure that the embedding can be computed efficiently.The present paper is inspired by problems from mathematical analysis, but its main motivation is algorithmic. We introduce a new notion of embedding, called maximum gradient embeddings, which is just right for approximating a wide range of clustering problems. We will then provide optimal maximum gradient embeddings of general nite metric spaces, and use them to design the best known approximation algorithms for several natural clustering problems. We do not attempt to explore here all the possible applications of maximum gradient embeddings, and we suspect that there are many more situations in which our method is applicable. Indeed, rather than being encyclopedic, the main emphasis of the present paper is that these embeddings yield a generic approach to many problems, and we give some example that illustrate this fact. In addition, our work raises interesting algorithmic questions which deserve further investigation.The exibility that was described above in the choice of notions of embedding has been exploited to great success by numerous authors in the past four decades. Due to the vast amount of work on this topic, we will *

Random martingales and localization of maximal inequalities

“…Important contributions of Calinescu, Karloff and Rabani [12] and Fakcharoenphol, Rao and Talwar [22] resulted in a sharp form of "Bartal's random tree method", and our work builds on these ideas. In [36,37] such random ultrametrics were used in order to prove maximal-type inequalities of a very different nature (motivated by embedding problems, as ultrametrics are isometric to subsets of Hilbert space [29]); these results also served as some inspiration for our work.…”