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This paper proposes an efficient method for the weighted region problem (WRP) on the surface of three-dimensional terrains. WRP is a classical path planning problem, asking for the minimum cost path between two given points crossing different regions in which each region is assigned a traversal cost per unit distance. Although WRP has been studied for decades, the exact solution for WRP, even in a two-dimensional environment, is unknown. Thus, the existing solutions for WRP are all approximations with decomposition-based and heuristic methods being the most widely-used in practice. However, when a very-close to optimal path is required, especially on real terrains with many regions, these approaches are not guaranteed or cannot return a satisfactory result in reasonable time. In this paper, we first present a new algorithm of finding a very-close optimal path, based on a user-defined parameter , between two points, crossing the surface of a sequence of regions in 3D, using Snell's law of physical refraction. We then show how to combine this algorithm with one existing decomposition-based method to compute a close optimal path over the whole terrain. In addition to a theoretical analysis, with an extensive set of test cases, the practicality and feasibility of our method are confirmed by that, our method always runs faster and returns closer to optimal paths in comparison with the existing ones.

Given k skew segments in an ordered sequence E and two points s and t in a three-dimensional environment, for any , we study a classical geometric problem of finding a -approximation Euclidean shortest path between s and t , crossing the segments in E in order. Let L be the maximum Euclidean length of the segments in E and h be the minimum distance between two consecutive segments in E . The running time of our algorithm is . Currently, the running time of finding the exact shortest path for this problem is exponential. Thus, most practical algorithms of this problem are approximations. Among these practical algorithms, placing discrete points, named Steiner points, on every segment in E , then constructing a graph to find an approximate path between s and t , is most widely used in practice. However, using Steiner points will cause the running time of this approach to always depend on a polynomial function of the term , which is not a close optimal solution. Differently, in this paper, we solve the problem directly in a continuous environment, without using Steiner points, in terms of the running time depending on a logarithmic function of the term , which we call a close optimal solution.

No abstract

This paper proposes a new practical method for the weighted region problem (WRP). The objective of WRP is to find a minimum cost path between two vertices among different regions where each region incurs a traversal cost per unit distance. Currently, there is no practical algorithm that solves this problem exactly. Among the approximation methods that solve instances of WRP, there is a limited number of algorithms that compute paths whose lengths are close to optimal, which we call very-close optimum paths. However, they are considered as theoretical methods. On the other hand, algorithms for solving WRP that can be applied to practical data sets (using decomposition ideas or heuristics) are not guaranteed to find a very-close optimum path within an acceptable amount of time. In this paper, we consider an alternative method for solving WRP that exploits Snell's law of physical refraction. We compare the performance of our new algorithm with that of two existing algorithms, using at least 500 test cases for each such comparison. The experimental results show that our algorithm returns a very-close optimum weighted shortest path in reasonable time.