Suppose it is required to minimize an objective function. Suppose that we have a method for getting a lower bound on the cost of any solution among those in the set of solutions represented by some subset. If the best solution found so far costs less than the lower bound for this subset, we need not explore this subset at all.
The files of the form city<n>.ll hold the longitude and latitude of the largest n cities in the United States (excluding Alaska and Hawaii) in millionths of a degree.