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This paper introduces the concept of precisionsensitive algorithms, in analogy to the well-known output-sensitive algorithms. We exploit this idea in studying the complexity of the 3-dimensional Euclidean shortest path problem. Specifically, we analyze an incremental approximation approach based on ideas in [CSY], and show that this approach yields an asymptotic improvement of running time. By using an optimization technique to improve paths on fixed edge sequences, we modify this algorithm to guarantee a relative error of 0(2-’) in a time polynomial in r and 1/6, where 6 denotes the relative difference in path length between the shortest and the second shortest path.

Precision-sensitive Euclidean shortest path in 3-space (extended abstract)