Search for: All records

We present the first near-linear-time algorithm that computes a (1+ε)-approximation of the diameter of a weighted unit-disk graph of n vertices. Our algorithm requires O(n log^2 n) time for any constant ε>0, so we considerably improve upon the near-O(n^{3/2})-time algorithm of Gao and Zhang (2005). Using similar ideas we develop (1+ε)-approximate \emph{distance oracles} of O(1) query time with a likewise improvement in the preprocessing time, specifically from near O(n^{3/2}) to O(n log^3 n). We also obtain similar new results for a number of related problems in the weighted unit-disk graph metric such as the radius and the bichromatic closest pair. As a further application we employ our distance oracle, along with additional ideas, to solve the (1+ε)-approximate \emph{all-pairs bounded-leg shortest paths\/} (apBLSP) problem for a set of n planar points. Our data structure requires O(n^2 log n) space, O(loglog n) query time, and nearly O(n^{2.579}) preprocessing time for any constant ε>0, and is the first that breaks the near-cubic preprocessing time bound given by Roditty and Segal (2011).