Maybe not. In fact, the algorithm cannot know the actual values of the workfunction w on the space M.

More Detailed Explanation

The algorithm may as well assume that M is a universal space. If the algorithm remembers everything it can, it knows that the true workfunction w lies within some set of possible workfunctions, say W = {w_1, w_2, ... }

After t steps, the algorithm's knowledge of the past consists of a partial distance function on pair of points drawn from the set R = {s(0), r(0), r(1), .... r(t)}. That is the set of all points on which the algorithm has had servers. That set has C(t+1,2) = (t^2+t)/2 pairs, but the algorithm knows only 2t distances. In principle, the algorithm can compute the trackless work function as follows:

1. Loop
2. Choose a metric on R consistent with the algorithm's knowledge
3. Using that metric, compute the values of the workfunction on R
4. Project those values to the cells of the diagram.
5. If all possible metrics have been chosen Exit
6. End Loop
7. For each cell, pick the minimum of all the projected values. (Default value is infinity)
8. For any cell(x,y), define f(x,y) to be the minimum of the quantity value(e,f) + max{|x-e|,|y-f|}, minimized over all choices of cell(e,f) which got a value in the previous step.