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Figure from:Memory-constrained algorithms for simple polygons

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Figure 5: The standard situation. In the standard situation, where everything is confined within one subpolygon Pour, we can apply the search strategy of the constant-workspace algorithm almost without change: pick a search direction, according to the same rules as in [2], and shoot a ray R’ partitioning the subpolygon into two. The decision which half contains the target is based on the exit chord of Pury. The only problem arises when R’ hits the boundary in the exit chord.

Figure 5 The standard situation. In the standard situation, where everything is confined within one subpolygon Pour, we can apply the search strategy of the constant-workspace algorithm almost without change: pick a search direction, according to the same rules as in [2], and shoot a ray R’ partitioning the subpolygon into two. The decision which half contains the target is based on the exit chord of Pury. The only problem arises when R’ hits the boundary in the exit chord.

Related Figures (6)

Figure 1: The shortest path tree SPT from ag to all other vertices, and the resulting triangu- lation. The SPT edges are purple. The additional triangulation edges are dotted. The figure illustrates two types of steps of the triangulation algorithm: a forward step in the right part, and a backward step in the middle part. The angular sweeps are indicated by shaded areas.
Figure 1: The shortest path tree SPT from ag to all other vertices, and the resulting triangu- lation. The SPT edges are purple. The additional triangulation edges are dotted. The figure illustrates two types of steps of the triangulation algorithm: a forward step in the right part, and a backward step in the middle part. The angular sweeps are indicated by shaded areas.
Figure 2: The three cases for the Eulerian traversal of SPT. The triangles that are reported while performing a step are shaded. Edges in the triangulation that are not contained in SPT are drawn dashed.
Figure 2: The three cases for the Eulerian traversal of SPT. The triangles that are reported while performing a step are shaded. Edges in the triangulation that are not contained in SPT are drawn dashed.
Figure 3: Decomposition of a polygon into mountains.
Figure 3: Decomposition of a polygon into mountains.
Figure 4: A decomposition of a simple polygon into 7 disjoint subpolygons using 6 chords and its associated tree decomposition. Given a polygon P and a parameter s representing the amount of workspace that we are allowed to use, we build a data structure of size O(s) that will speed up the computation of shortest paths between any two arbitrarily specified points in P. After a preprocessing of O(n?) time (or O(n) if s = O(n)), our query algorithm will be able to solve any shortest path query in O(n?/s) time using O(s) workspace. In particular, this algorithm will run in O(n?) time when we have only constant work space, and linear time when s = O(n). Hence, our algorithm gives a continuous tradeoff (whose both extremes match the running time of the currently best known methods) between the running time and allowed space.
Figure 4: A decomposition of a simple polygon into 7 disjoint subpolygons using 6 chords and its associated tree decomposition. Given a polygon P and a parameter s representing the amount of workspace that we are allowed to use, we build a data structure of size O(s) that will speed up the computation of shortest paths between any two arbitrarily specified points in P. After a preprocessing of O(n?) time (or O(n) if s = O(n)), our query algorithm will be able to solve any shortest path query in O(n?/s) time using O(s) workspace. In particular, this algorithm will run in O(n?) time when we have only constant work space, and linear time when s = O(n). Hence, our algorithm gives a continuous tradeoff (whose both extremes match the running time of the currently best known methods) between the running time and allowed space.
Figure 6: Procedure Catch-up: Growing the shorter side of the funnel
Figure 6: Procedure Catch-up: Growing the shorter side of the funnel
Figure 7: Procedure Extend. The ray R’ splits the funnel into two funnels.
Figure 7: Procedure Extend. The ray R’ splits the funnel into two funnels.