<P> The final result is an algorithm taking O (n log n) time and O (n) space . </P> <P> If the input coordinates are integers and can be used as array indices, faster algorithms are possible: the Delaunay triangulation can be constructed by a randomized algorithm in O (n log log n) expected time . Additionally, since the Delaunay triangulation is a planar graph, its minimum spanning tree can be found in linear time by a variant of Borůvka's algorithm that removes all but the cheapest edge between each pair of components after each stage of the algorithm . Therefore, the total expected time for this algorithm is O (n log log n). </P> <P> The problem can also be generalized to n points in the d - dimensional space R. In higher dimensions, the connectivity determined by the Delaunay triangulation (which, likewise, partitions the convex hull into d - dimensional simplices) contains the minimum spanning tree; however, the triangulation might contain the complete graph . Therefore, finding the Euclidean minimum spanning tree as a spanning tree of the complete graph or as a spanning tree of the Delaunay triangulation both take O (dn) time . For three dimensions it is possible to find the minimum spanning tree in time O ((n log n)), and in any dimension greater than three it is possible to solve it in a time that is faster than the quadratic time bound for the complete graph and Delaunay triangulation algorithms . For uniformly random point sets it is possible to compute minimum spanning trees as quickly as sorting . Using a well - separated pair decomposition, it is possible to produce a (1 + ε) - approximation in O (n log n) time . </P> <P> All edges of an EMST are edges of a relative neighborhood graph, which in turn are edges of a Gabriel graph, which are edges in a Delaunay triangulation of the points, as can be proven via the equivalent contrapositive statement: every edge not in a Delaunay triangulation is also not in any EMST . The proof is based on two properties of minimum spanning trees and Delaunay triangulations: </P>

Give a lower bound of ω(n log n) for constructing the cartesian tree of a tree