MIT6_851S10_assn03_sol - time spent searching the WBBST for...

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6.851 Advanced Data Structures (Spring’10) Prof. Erik Demaine Dr. Andr´ e Schulz TA: Aleksandar Zlateski Problem 3 Sample Solutions Ray Shooting in Simple Polygons With every step we reduce the WBBST total weight of the current subtree by at least a factor of 2. We Fnish once we reach a subtree with weight ω i . Hence, we solve the recurrence T ( ω ) = 1 + T ( ω/ 2). The base case is T ( ω i ) = 1, so we get T (Ω) = O (1 + log i )). Suppose each concave chain in the balanced pseudo-triangulation is stored in a WBBST, where the weight of an edge i equals the number edges in the opposing polygon ω i . We consider two adjacent pseudo-triangles, t a and t b , crossed by the ray in this algorithm. Let i be the edge the ray crosses to move from t a into t b . In t b the ray homes-in on the next edge it crosses, i + 1, in a concave chain, which has at most ω i edges, and so the total
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Unformatted text preview: time spent searching the WBBST for the home-in chain in tb is O (log( ω i /ω i +1 )). The sum telescopes, and its result is the difference in the logs of two pseudo-triangle sizes, which is no larger than O (log n ). The ray-shooting algorithm traverses no more than O (log n ) triangles in total, giving the total runtime of O (log n ). 1 MIT OpenCourseWare 6.851 Advanced Data Structures Spring 2010 For information about citing these materials or our Terms of Use, visit: ....
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This note was uploaded on 03/31/2011 for the course EECS 6.851 taught by Professor Erikdemaine during the Spring '10 term at MIT.

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MIT6_851S10_assn03_sol - time spent searching the WBBST for...

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