Lost in a Forest

Call a path an escape path if it eventually leads out of the forest no matter what the initial starting point or the relative orientations of the path and forest. To solve the "lost in a forest" problem we must find the "best" escape path. Bellman proposed two different interpretations of "best," one in which the maximum time to escape is minimized, and one in which the expected time to escape is minimized. A third interpretation (see Croft, Falconer, and Guy [12, p. 40]) involves maximizing the probability of escape within a specified time period. Bellman asked about two situations in particular: (1) the case in which the region is the infinite strip between two parallel lines a known distance apart, and (2) the case in which the region is a half-plane and the hiker's distance from the boundary is known. If "best" is taken to mean that the maximum time to escape is minimized, both of Bellman's particular situations have been investigated: for the first the minimax (i.e., shortest) escape path was found by Zalgaller [45] in 1961, and for the second the minimax escape path was described by Isbell [21] in 1957. In each of these two specific situations the shortest escape path is unique up to congruence. Little is known in either situation for other interpretations of "best."