Efficient algorithms for globally optimal trajectories

Presents serial and parallel algorithms for solving a system of equations that arises from the discretization of the Hamilton-Jacobi equation associated to a trajectory optimization problem of the following type. A vehicle starts at a prespecified point x₀ and follows a unit speed trajectory x(t) inside a region in ℜ^m , until an unspecified time T that the region is excited. A trajectory minimising a cost function of the form ∫₀^T r(x(t))dt+q(x(T)) is sought. The discretized Hamilton-Jacobi equation corresponding to this problem is usually served using iterative methods. Nevertheless, assuming that the function r is positive, one is able to exploit the problem structure and develop one-pass algorithms for the discretized problem. The first m resembles Dijkstra´s shortest path algorithm and runs in time O(n log n), where n is the number of grid points. The second algorithm uses a somewhat different discretization and borrows some ideas from Dial´s shortest path algorithm; it runs in time O(n), which is the best possible, under some fairly mild assumptions. Finally, the author shows that the latter algorithm can be efficiently parallelized: for two-dimensional problems and with p processors, its running time becomes O(n/p), provided that p=O(√n/log n)