Faster algorithms for some geometric graph problems in higher dimensions

We show how to apply the well-separated pair decomposition of a point-set P in ?X+ to significantly improve known time bounds on several geometric graph problems. We first present an algorithm to find an approximate Euclidean minimum spanning tree of P whose weight is at most 1 + e times the exact minimum. We achieve a time complexity of O(nlogn + (cd/2 log ‘)n), improving the best known bound of O(cmdn16gn). We then show how to construct a graph with O(cmd+rn) edges in which the shortest path between any pair of points is within 1 + c of the Euclidean distance. Our time complexity is O(nlogn+(c-d log $)n), a significant improvement over the best previous algorithm that produces a graph of this size. Finally, we show how to compute the exact Euclidean minimum spanning tree in time o(Td(n, n) lOgn), where Td(m, n) iS the time to find the bichromatic closest pair between m red points and n blue points. The previous bound was O(Td(n, n)logd n). As with the previous algorithm, our complexity reduces to O(Td(n,n)) if ??d(n,n) = i-@ 1+a) for some (Y > 0.

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