squared edge lengths of the minimum spanning tree is . This bound follows from the observation that the edges have disjoint rhombi, with area proportional to the edge lengths squared. The bound on total length follows by application of the Cauchy–Schwarz inequality. Another interpretation of these results is that the average edge length for any set of points in a unit square is , at most proportional to the spacing of points in a regular grid; and that for random points in a unit square the average length is proportional to . However, in the random case, with high probability the longest edge has length approximately longer than the average by a non-constant factor. With high probability, the longest edge forms a leaf of the spanning tree, and connects a point far from all the other points to its nearest neighbor. For large numbers of points, the distribution of the longest edge length around its expected value converges to a Laplace distribution. Any geometric spanner, a subgraph of a complete geometric graph whose shortest paths approximate the Euclidean distance, must have total edge length at least as large as the minimum spanning tree, and one of the standard quality measures for a geometric spanner is the ratio between its total length and of the minimum spanning tree for the same points. Several methods for constructing spanners, such as the greedy geometric spanner, achieve a constant bound for this ratio. It has been conjectured that the Steiner ratio, the largest possible ratio between the total length of a minimum spanning tree and Steiner tree for the same set of points in the plane, is , the ratio for three points in an equilateral triangle. Subdivision If every edge of a Euclidean minimum spanning tree is subdivided, by adding a new point at its midpoint, then the resulting tree is still a minimum spanning tree of the augmented point set. Repeating this subdivision process allows a Euclidean minimum spanning tree to be subdivided arbitrarily finely. However, subdividing only some of the edges, or subdividing the edges at points other than the midpoint, may produce a point set for which the subdivided tree is not the minimum spanning tree. Computational complexity For points in any dimension, the minimum spanning tree can be constructed in time by constructing a complete graph with an edge between every pair of points, weighted by Euclidean distance, and