Light orthogonal networks with constant geometric dilation

An orthogonal spanner network for a given set of n points in the plane is a plane straight line graph with axis-aligned edges that connects all input points. We show that for any set of n points in the plane, there is an orthogonal spanner network that (i) is short having a total edge length at most a constant times the length of a Euclidean minimum spanning tree for the point set; (ii) is small   having O(n)O(n) vertices and edges; and (iii) has constant geometric dilation, which means that for any two points u and v in the network, the shortest path in the network between u and v is at most a constant times longer than the Euclidean distance between u and v  . Such a network can be constructed in O(nlogn)O(nlogn) time.