Lec12 - Convex hulls Dynamic hull (insertion) Hull...

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Unformatted text preview: Convex hulls Dynamic hull (insertion) Hull maintenance during insertion, 1 Maintain H ( S ) as points are added to S . p 1 p 2 p 3 1 3 2 Convex hulls Dynamic hull (insertion) Hull maintenance during insertion, 2 Maintain H ( S ) as points are added to S . p 4 p 5 p 6 p 7 7 6 4 5 Convex hulls Dynamic hull (insertion) Definitions Preparata static Set of object remains fixed between operations (e.g., between queries in repetitive mode). dynamic Set of objects can change between operations. Insertions and deletions allowed in general, but sometimes constrained. Implies S must be in some updatable data structure. on-line Algorithm cannot look ahead at input. off-line “Algorithm operates on all data collectively.” Static or batched dynamic. real-time Updates must be completed during fixed interarrival time (often O (log N )). Edelsbrunner static Set of objects remains fixed between operations. dynamic Set of objects can change between operations. on-line Sequence of operations (insertions, deletions, queries) not known in advance, algorithm cannot look ahead. off-line Insertions and deletions contrained to come from a prespecified and usually small set. batched Sequence of operations fully known in advance, algorithm can look ahead. Chiang, Y.-J. and Tomassia, R. (1992). “Dynamic Algorithms in Computational Geometry”, Proceedings of the IEEE , Vol. 80, No. 9, September 1992, pp. 1412-1434. Convex hulls Dynamic hull (insertion) Problem definitions ON-LINE CONVEX HULL INSTANCE: Sequence of N points in the plane p 1 , p 2 , …, p N . QUESTION: Find their convex hull in such a way that after p i is processed we have H ({ p 1 , p 2 , …, p i }). REAL-TIME CONVEX HULL INSTANCE: Sequence of N points in the plane p 1 , p 2 , …, p N . QUESTION: Find their convex hull on-line assuming constant interarrival delay. The algorithm must maintain some representation of the hull and update it as points are inserted. Can this be done and still achieve O ( N log N ) time for the entire sequence? These are dynamic problems, but note that both of these problems are constrained to insertions only . Generalizing, we have: DYNAMIC CONVEX HULL (HULL MAINTENANCE) INSTANCE: An initially empty set S and a sequence of N points in the plane p 1 , p 2 , …, p N , each of which corresponds to either an insertion or deletion from S . (Only a previously inserted point can be deleted). QUESTION: Maintain the convex hull H ( S )....
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This note was uploaded on 07/14/2011 for the course COT 5520 taught by Professor Mukherjee during the Summer '11 term at University of Central Florida.

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Lec12 - Convex hulls Dynamic hull (insertion) Hull...

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