Approximate Shortest Descending Paths

We present an approximation algorithm for the shortest descending path problem. Given a source s and a destination t on a terrain, a shortest descending path from s to t is a path of minimum Euclidean length on the terrain subject to the constraint that the height decreases monotonically as we traverse that path from s to t. Given any ε ∈ (0, 1), our algorithm returns in O(n log(n/ε)) time a descending path of length at most 1 + ε times the optimum. This is the first algorithm whose running time is polynomial in n and log(1/ε) and independent of the terrain geometry.