Touring a Sequence of Polygons in Weighted Regions

Given a subdivision of plane into convex polygon regions, a sequence of polygons to meet, a start point s, and a target point t, we are interested in determining the shortest weighted path on this plane which starts at s, visits each of the polygons in the given order, and ends at t. The length of a path in weighted regions is de¯ned as the sum of the lengths of the sub-paths within each region. We will present an approximation algorithm with maximum ± cost additive. Our algorithm is based on the shortest weighted path algorithm proposed by Mata and Mitchel [2]. The algorithm runs in O(((n3LW +RW) k ± )3) time, where n is the number of vertices of the region boundaries, L is the longest boundary, W is the maximum weight in the region, R is the sum of the perimeters of the regions, and k is the number of polygons. The main idea in the algorithm is to add Steiner points on the region boundaries and polygon edges. In addition, we will also present a solution to the query version of this problem. We will extend our result in unweighted version of the Touring a Sequence of Polygons" problem [3]. We will give an approximation algorithm to solve the general case of the problem (with non-convex intersecting polygons).

Keywords:

computational geometry , shortest path , weighted region , polygons