We will find a way to quickly keep track of the current sum after each move of the robot.

Assume that robot moved to the right (east) from ~(a, b)~ to ~(a+1, b)~. Distance to any control point changed by ~1~ or ~-1~ after this move. To be more precise, distance increased by ~1~ for every control point with x-coordinate no greater than ~a~, and decreased by ~1~ for the rest of them. If we denote the number of points in the first group ~f(a)~, then the sum of these distances changed by ~f(a)-(N-f(a))~. We can do the same for the y-coordinate.

Next, we must figure out how to calculate ~f(a)~ efficiently. One of the ways to do this is to keep the control points in two arrays, one sorted by x-coordinate and the other sorted by y-coordinate. Now we can easily binary search ~f(a)~ for any value ~a~.

This solution has a complexity of ~\mathcal O(N \log N + M \log N)~. Better complexity can be achieved by precomputing all the ~f(a)~ values using dynamic programming.