Path Planning in a Weighted Planar Subdivision Under the Manhattan Metric

In this paper, we consider the problem of path planning in a weighted polygonal planar subdivision. Each polygon has an associated positive weight which shows the cost of path per unit distance of movement in that polygon. The goal is to find a minimum cost path under the Manhattan metric for two given start and destination points. First, we propose an \(O(n^2)\) time and space algorithm to solve this problem, where n is the total number of vertices in the subdivision. Then, we improve the time and space complexity of the algorithm to \(O(n \log ^2 n)\) and \(O(n \log n)\), respectively, by applying a divide and conquer approach. We also study the case of rectilinear regions in three dimensions and show that the minimum cost path under the Manhattan metric is obtained in \( O(n^2 \log ^3 n) \) time and \( O(n^2 \log ^2 n) \) space.