Minimum spanning trees of moving points in the plane

Consideration is given to the following problem. Preprocess n moving points in a plane, such that the Euclidean minimum spanning tree of these points at a given time t can be reported efficiently. In the result, if the moving points are in k-motion, after an O(kn⁴ log n ) time preprocessing step and using O(m) space to store the preprocessing result, the Euclidean minimum spanning tree at t can be reported in O(n) time, where m denotes the number of changes of the Euclidean minimum spanning tree of these points from time t=0 to time t=∞