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Unformatted text preview: and Pi+1. Then either: ai = ai+1 or ai+1 is the highest among the vertices adjacent to a i Proof Sketch: (by cases) ai is vertex of both Pi and Pi+1 ai = ai+1 else ai deleted in construction of Pi+1
Let bi+1 = highest vertex of Pi+1 among those adjacent to ai in Pi Claim: b is highest vertex of P i+1 But we still need bound on # i+1 vertices adjacent to ai+1... Extremal Polytope Queries Details (continued) Lemma 7.10.3: Let ai and ai+1 be uniquely highest vertices of Pi and Pi+1. Then either ai = ai+1 or ai is the highest among the parents of the extreme edges Li+1 and Ri+1 Proof Sketch: Project Pi+1 onto plane orthogonal to (e.g. xz plane) Let Li+1, Ri+1 = 2 edges of Pi+1 projecting to 2 edges of P'i+1 incident to a'i+1 L'i+1 = R'i+1 Parents of edge of are vertices of Pi from ' which they derive. tetrahedron projected New extreme edges are "close" to the old. See p. 281-283 for remaining details. P'i+1 onto xz plane Extremal Polytope Queries
Main Idea (again)
Algorithm: EXTREME POINT of a POLYTOPE Input: polytope P and direction vector u Output: vertex a of P extreme in u direction Construct nested polytope hierarchy P = P0, P1,..., Pk ak vertex of Pk extreme in u direction Compute Lk and Rk for i = k - 1, k - 2, ...,1, 0 do ai extreme vertex among ai+1 and parents of Li+1 and Ri+1 for all edges incident to ai do save extreme edges Li and Ri if ai = ai+1 then u Pk P0 polytope extreme-point queries can be answered in O(log n) time each else (ai = ai+1) compute and space preprocessing, After O(n) time Li from Li+1 etc... Planar Point Location Goal: Given a planar subdivision of n vertices, preprocess it so that point location query can be quickly answered. A polygonal planar subdivision can be preprocessed in O(n) time and space for O(log n) query. 2D version/variant of independent set/nested approach ( may need to triangulate polygonal faces) Monotone subdivision approach Randomized trapezoidal decomposition Planar Point Location (continued) Monotone subdivision approach: -Partition subdivision (e.g. Voronoi diagram) into "horizontal" strips. -Double binary search: -Vertical search on strips to locate query point between 2 separators -Horizontal search to locate it within 1 strip -O(log2n) query time Planar Point Location (continued) And now for something completely different... As foreshadowed in Ch. 2 (polygon partitioning) Seidel's randomized trapezoidal decomposition for non-crossing segments:
Assume no 2 points are at same height. O(nlogn) construction time: extend
horizontal line through each endpoint. O(logn) query time. Planar Point Location (continued) Lemma allows us to incrementally build binary search tree. 3 types of nodes:
1. 2. 3. internal X nodes, which branch left or right of a segment s i internal Y nodes, which branch above or below a segment endpoint Leaf trapezoid nodes. Planar Point Location (continued) Planar Point Location (continued) Planar Point Location (continued)...
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- Spring '10