Some embodiments of the invention provide a method for costing an expansion to a two-dimensional state in a path search that searches for a path between two sets of states in a space. The method identifies a cost function that is defined over the two-dimensional state. The method computes a second cost function that is defined over the two-dimensional state. It also computes a third cost function that is defined over the two-dimensional state. It then adds the second and third cost functions to obtain the first cost function.