An E cient Approximation Algorithm for Weighted Region Shortest Path

In this paper we present an approximation algorithm for solving the following optimal motion planning problem: Given a planar space composed of triangular regions , each of which is associated with a positive unit weight, and two points s and t, nd a path from s to t with the minimum weight, where the weight of a path is deened to be the weighted sum of the lengths of the sub-paths within each region. Some previous algorithms (Lanthier et al 9] and Aleksandrov et al 1]) took a discretization approach by introducing m Steiner points on each edge. A discrete graph is constructed by adding edges connecting Steiner points in the same triangular region and an optimal path is computed in the resulting discrete graph using Dijkstra's algorithm. To avoid high time complexity , both 9] and 1] use a subgraph of the complete graph in each triangular region. As a result, in the discrete graph only an approximate optimal path can be achieved, whose error is proportional to the weight of the optimal path. This approximate optimal path then is used to approximate the optimal path in the original problem. In this paper we introduce a geometric structure called interval. Our new algorithm based on interval extends the previous algorithms by dynamically maintaining O(m) edges in each region to compute the exact shortest path in the discrete graph eeciently. The running time of this algorithm is O(mn log mn), where n is the number of triangular regions. Our algorithm provides an improvement over previous algorithms in the case when approximation error is to be bounded by an additive constant. Besides (additive) constant error bound, the interval-based algorithm also has the following three advantages: (a) can compute exact solutions for a discrete case of this problem; (b) can be applied to any dis-cretization; and (c) can be applied to a more general class of problems than the previous algorithms.