Fly Cheaply

In planning a flight, stops at intermediate airports are sometimes necessary to minimize fuel consumption, even if a direct flight is available. We investigate the problem of finding the cheapest path from one airport to another, given a set of n airports in R2 and a function l: R2 R2 R+ representing the cost of a direct flight between any pair. Given a source airport s, the cheapest-path map is a subdivision of R2 where two points lie in the same region iff their cheapest paths from s use the same sequence of intermediate airports. We show a quadratic lower bound on the combinatorial complexity of this map for a class of cost functions. Nevertheless, we are able to obtain subquadratic algorithms to find the cheapest path from s to all other airports for any well-behaved cost function l: our general algorithm runs in O(n4/3+ ) time, and a simpler, more practical variant runs in O(n3/2+ ) time, while a special class of cost functions requires just O(nlogn) time.