1. March 20111 Trapezoidal Maps Shmuel Wimer Bar Ilan Univ., School of Engineering

2. March 20112 Trapezoidal Map Planar subdivision Abscissas are all distinct n segments 6n+4 vertices at most 3n+1 trapezoids at most

3. March 20113 Trapezoidal map can be constructed in O(nlogn) time by a scan-line algorithm.

4. March 20114 Randomized Incremental Algorithm

5. March 20115 Inner nodes have degree 2 x-node y-node trapezoid

6. March 20116 Does q lie to the left or to the right ? Does q lie above or below? Querying a point location

7. March 20117 Randomized Construction Algorithm

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13. March 201113 New segment insertion

14. March 201114 The information attached to new trapezoids is their left and right neighbor trapezoids, top and bottom segment and points defining their left and right vertical segment. If the information in Δ is properly stored, above info can be determined in a constant time from s i and Δ. Assuming that a point is contained in Δ, the sub tree replacing its leaf is sufficient to determine whether the point is in A, B, C or D. If p i =leftPoint(Δ) and / or q i =rightPoint(Δ), Δ is divided into two or three trapezoids and sub-tree replacement is simpler.

15. March 201115 Intersection with more than one trapezoid

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26. March 201126 Given a set of segments, nothing is guaranteed on the maximal run time, which can be quadratic. Considering all possible problems of n segments, what is the expected maximal query time? O(logn)