Minimum dilation stars

The dilation of a Euclidean graph is defined as the ratio of distance in the graph divided by distance in Rd. In this paper we consider the problem of positioning the root of a star such that the dilation of the resulting star is minimal. We present a deterministic O(n log n)-time algorithm for evaluating the dilation of a given star; a randomized O(n log n) expected-time algorithm for finding an optimal center in Rd; and for the case d = 2, a randomized O(n2α(n) log2n) expected-time algorithm for finding an optimal center among the input points.