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ps3 - (= number of edges of the opposing polygon Show that...

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6.851 Advanced Data Structures (Spring’10) Prof. Erik Demaine Dr. Andr´ e Schulz TA: Aleksandar Zlateski Problem 3 Due: Thursday, Feb. 25 Be sure to read the instructions on the assignments section of the class web page. Ray Shooting in Simple Polygons. 1. Let w 1 ,w 2 ,...,w k be positive weights we want to store in a weight balanced binary search tree (WBBST). We denote j k w j by W . To built the WBBST we find the unique element w r such that the sums j<r w j and j>r w j are both at most W/ 2. We make the element w r the root of the WBBST and recurse on the two remaining subsets (left and right of w r ). (a) Show that we can search for an item with weight w i in the WBBST in O (1+log( W/w i )) time. (b) To search through all home-in situations that occur during the ray shooting algorithm we use a WBBST for every concave chain of the balanced pseudo-triangulation. For an edge we choose as weight its bay-size
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Unformatted text preview: (= number of edges of the opposing polygon). Show that we can answer the searches for all home-in situations using O (log n ) time in total. 2. By adding additional interior points we can transform a balanced pseudo-triangulation into a pseudo-triangulation where every pseudo-triangle has only one concave chain. Since we add at most 6 new pseudo-triangles any line intersects at most O (log n ) of these special pseudo-triangles. Show how to triangulate the special pseudo-triangles using additional points, such that any line intersects at most O (log 2 n ) triangles. Δ (i) (ii) Figure 1: The bay-sizes of the red chain of Δ in (i) are from left to rigth: 7,3,1,6. Picture (ii) shows how to decompose a pseudo-triangle into pseudo-triangles with at most one concave chain using additional points. 1...
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