Kevin A. Wortman

Let G be a graph that may be drawn in the plane in such a way that all internal faces are centrally symmetric convex polygons. We show how to find a drawing of this type that maximizes the angular resolution of the drawing, the minimum angle between any two incident edges, in polynomial time, by reducing the problem to one of finding parametric shortest paths in an auxiliary graph. The running time is at most O(t^3), where t is a parameter of the input graph that is at most O(n) but is more typically proportional to n^.5.

We provide an efficient reduction from the problem of querying approximate multiplicatively weighted farthest neighbors in a metric space to the unweighted problem. Combining our techniques with core-sets for approximate unweighted farthest neighbors, we show how to find (1+epsilon)-approximate farthest neighbors in time O(log n) per query in D-dimensional Euclidean space for any constants D and epsilon. As an application, we find an O(n log n) expected time algorithm for choosing the center of a star topology network connecting a given set of points, so as to approximately minimize the maximum dilation between any pair of points.

The dilation of a Euclidean graph is defined as the ratio of distance in the graph divided by distance in R^d. 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 R^d; and for the case d=2, a randomized O(n 2^(alpha(n)) log^2 n) expected-time algorithm for finding an optimal center among the input points.