We present 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 x(0) and follows a unit speed trajectory x(t) inside a region in R(m), until an unspecified time T that the region is exited, A trajectory minimizing a cost function of the form integral(0)(T) r(x(t)) dt+q(x(T)) is sought, The discretized Hamilton-Jacobi equation corresponding to this problem is usually solved using iterative methods, Nevertheless, assuming that the function r is positive, we are able to exploit the problem structure and develop one-pass algorithms for the discretized problem, The first algorithm 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 a variation of Dial’s shortest path algorithm that we develop here; it runs in time O(n), which is the best possible, under some fairly mild assumptions, Finally, we show 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(root n/log n).