Path Planning in a Weighted Planar Subdivision Under the Manhattan Metric

Abstract

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 $$O(n^2)$$ 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 $$O(n \log ^2 n)$$ O ( n log 2 n ) and $$O(n \log n)$$ 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 $$ O(n^2 \log ^3 n) $$ O ( n 2 log 3 n ) time and $$ O(n^2 \log ^2 n) $$ O ( n 2 log 2 n ) space.