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A path from s to t with seven breakpoints.

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Conference Paper
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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 finding a minimum cost path under the Manhattan metric for two given start and destination points. We propose an O(...

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... under the weighted Manhattan metric -the length of a path is the weighted sum of Manhattan lengths of the sub-paths within each region. Let π st denote a path between s and t which consists of some sub-paths between consecutive breakpoints. A breakpoint is a point on the path in which the path turns. We also consider s and t as breakpoints (see Fig. 1). Let ρ 1 , ρ 2 , ..., ρ k be sub-paths between consecutive breakpoints of a path π st in which each ρ i , for i = 1, 2, . . . , k lies completely within one region. If a part of a path π st does not lie totally in one of the regions, we decompose it to some sub-paths. We denote d(ρ i ) as the Manhattan distance between two endpoints ...
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... of G which lie on an edge of a triangle are oblique. These edges are useful when two triangular regions are close to each other and the region among them has a lower weight than these triangles. A path which passes between these two triangles cannot be completely horizontal or vertical since it will enter s t Figure 2: The constructed graph of Fig. 1. the triangles. So it will be oblique and lie on one of the edges of the triangles (see the sub-path between b 4 and b 5 on Fig. ...
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... and the region among them has a lower weight than these triangles. A path which passes between these two triangles cannot be completely horizontal or vertical since it will enter s t Figure 2: The constructed graph of Fig. 1. the triangles. So it will be oblique and lie on one of the edges of the triangles (see the sub-path between b 4 and b 5 on Fig. ...
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... computes all k intersections among n line segments in the plane in O(n log n + k) time. These intersection points are vertices of G. After specifying the set of vertices of G, the set of edges of G can be specified. It takes O(n 2 ) time to construct G since the graph has O(n 2 ) vertices and edges. The constructed graph of the work space on Fig. 1 is shown on Fig. 2. For simplicity, we do not triangulate the white regions with weight 1 in these figures. Precisely, we can apply the proposed algorithm in a polygonal subdivision in which the regions are not triangular. The triangulation of the regions just helps us for showing that G contains the minimum cost path between s and ...
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... than these triangles, by applying this conversion, some parts of the horizontal and vertical line segments may lie in the triangular region with higher weight. In this case, we can replace the part which lies in a triangular region with higher cost with a line segment which lies on an edge of the triangles (see the sub-path between b 4 and b 5 on Fig. 1). Since the weight of each of the edges of the work space is equal to the minimum weight of the regions that are incident to that edge, the cost of movement between two breakpoints on the replaced line segments is equal to the cost of movement along the oblique line segment. Therefore, a path between s and t can only consist of ...