Viewing the Rings of a Tree - Minimum Distortion Embeddings into Trees.

We describe a (1 + ε) approximation algorithm for finding the minimum distortion embedding of an n-point metric space, (X, dX), into a tree with vertex set X. The running time of our algorithm is n2. (Δ/ε)(O(δopt / ε))2λ + 1 parameterized with respect to the spread of X, denoted by Δ, the minimum possible distortion for embedding X into any tree, denoted by δopt, and the doubling dimension of X, denoted by λ. Hence we obtain a PTAS, provided δopt is a constant and X is a finite doubling metric space with polynomially bounded spread, for example, a point set with polynomially bounded spread in constant dimensional Euclidean space. Our algorithm implies a constant factor approximation with the same running time when Steiner vertices are allowed. Moreover, we describe a similar (1 + ε) approximation algorithm for finding a tree spanner of (X, dX) that minimizes the maximum stretch. The running time of our algorithm stays the same, except that δopt must be interpreted as the minimum stretch of any spanning tree of X. Finally, we generalize our tree spanner algorithm to a (1 + ε) approximation algorithm for computing a minimum stretch tree spanner of a weighted graph, where the running time is parameterized with respect to the maximum degree, in addition to the other parameters above. In particular, we obtain a PTAS for computing minimum stretch tree spanners of weighted graphs, with polynomially bounded spread, constant doubling dimension, and constant maximum degree, when a tree spanner with constant stretch exists.