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

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Presentation transcript:

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

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

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

March Randomized Incremental Algorithm

March Inner nodes have degree 2 x-node y-node trapezoid

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

March Randomized Construction Algorithm

March 20118

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March New segment insertion

March 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.

March Intersection with more than one trapezoid

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March 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)