Minimum spanning trees in d dimensions

Abstract

It is shown that a minimum spanning tree of n points in Rd under any fixed Lp-metric, with p = 1, 2,..., ∞, can be computed in optimal O(Td(n, n)) time in the algebraic computational tree model. Td(n, m) denotes the time to find a bichromatic closest pair between n red points and m blue points. The previous bound in the model was O(Td(n, n)log n) and it was proved only for the L2 (Euclidean) metric. Furthermore, for d = 3 it is shown that a minimum spanning tree can be found in O(n log n) time under the L1 and L∞-metrics. This is optimal in the algebraic computation tree model. The previous bound was O(n log n log log n).