lec19 - Introduction to Algorithms 6.046J/18.401 Lecture 19...

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Introduction to Algorithms 6.046J/18.401 Lecture 19 Prof. Piotr Indyk
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Introduction to Algorithms April 29, 2003 L19.2 © 2003 by Piotr Indyk Computational Geometry Algorithms for geometric problems Applications: CAD, GIS, computer vision,……. E.g., the closest pair problem: – Given: a set of points P={p 1 …p n } in the plane, such that p i =(x i ,y i ) – Goal: find a pair p i p j that minimizes ||p i – p j || We will see more examples in the next lecture
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Introduction to Algorithms April 29, 2003 L19.3 © 2003 by Piotr Indyk Computational Model In the next two lectures, we will assume that – The input (e.g., point coordinates) are real numbers – We can perform (natural) operations on them in constant time, with perfect precision Advantage: simplicity Drawbacks: highly non-trivial issues: – Theoretical: if we allow arbitrary operations on reals, we can compress n numbers into a one number – Practical: algorithm designed for infinite precision sometimes fail on real computers
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Introduction to Algorithms April 29, 2003 L19.4 © 2003 by Piotr Indyk Closest Pair Find a closest pair among p 1 …p n Easy to do in O(n 2 ) time – For all p i p j , compute ||p i – p j || and choose the minimum We will aim for O(n log n) time
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Introduction to Algorithms April 29, 2003 L19.5 © 2003 by Piotr Indyk
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lec19 - Introduction to Algorithms 6.046J/18.401 Lecture 19...

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