Replacing iterative algorithms with single-pass algorithms

Consider the rather mundane question: what is the distance from point A to point B? Of course, in Euclidean geometry, the answer is trivial, but what if A and B are cities, and we are taking ground transportation? Seeking the optimal path is now more challenging because we must consider terrain conditions. In other words, the optimal path depends on the rate of speed that is possible at each point. This problem is considered isotropic because the rate of speed at each point is independent of direction. It can be solved by using existing single-pass algorithms such as those by Dijkstra (1), Tsitsiklis (2), and Sethian (3). Of course, a more realistic problem also would take into account that the possible speed at a given point also depends on the direction we are traveling (uphill/downhill, with/against traffic, etc.). This is the anisotropic version of the problem and is substantially more complicated. The algorithm of Sethian and Vladimirsky (4) in this issue of PNAS successfully generalizes Sethian's fast marching method (3) to make it the only single-pass algorithm able to solve the anisotropic path planning problem. But in principle, the notion that information is essentially starting from point A, emanating in some sense radially outward from A, and eventually striking the point B is apparent.