lecture_18 - Introduction to Algorithms 6.046J/18.401...

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Introduction to Algorithms 6.046J/18.401 Lecture 1 7 Prof. Piotr Indyk
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Computational Geometry ctd. Segment intersection problem: – Given: a set of n distinct segments s 1 …s n , represented by coordinates of endpoints – Detection : detect if there is any pair s i s j that intersects – Reporting : report all pairs of intersecting segments © 2003 by Piotr Indyk Introduction to Algorithms November 15, 2004 L18.2
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Segment intersection Easy to solve in O(n 2 ) time Is it possible to get a better algorithm for the reporting problem ? NO (in the worst-case) However: – We will see we can do better for the detection problem – Moreover, the number of intersections P is usually small. Then, we would like an output sensitive algorithm, whose running time is low if P is small . © 2003 by Piotr Indyk Introduction to Algorithms November 15, 2004 L18.3
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Result We will show: – O(n log n) time for detection – O( (n +P) log n) time for reporting We will use … … (no, not divide and conquer) Binary Search Trees Specifically: Line sweep approach © 2003 by Piotr Indyk Introduction to Algorithms November 15, 2004 L18.4
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Orthogonal segments V-segment All segments are either horizontal or vertical H-segment Assumption: all coordinates are distinct Therefore, only vertical- horizontal intersections exist © 2003 by Piotr Indyk Introduction to Algorithms November 15, 2004 L18.5
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Orthogonal segments Sweep line: – A vertical line sweeps the plane from left to right – It “stops” at all “important” x-coordinates, i.e., when it hits a V-segment or endpoints of an H-segment
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