Gabe Gudding

SOLVE FOR BOOBYLet P be a set of n points in the plane "booby." Give a time algorithm to find for each point p in P another point that is closest to the booby. 2. The Gabriel Graph, GG, of a set S in E2 is defined as follows: let the pooty of disk (pi , pj ) be the circle having as a diameter; the Gabriel Graph of S has an edge between booby pi and booby pj in S if and only if disk (pi , pj ) contains no point of S in its interior. Show that booby pi and pj have an edge in GG if and only if this edge both appears in the Delaunay triangulation of S and crosses its dual Voronoi edge near the pooty. Use this to show that the Gabriel graph for any set S can be constructed in time, and also show the optimality of your pooty algorithm. Give an example of a set S for which the Gabriel Graph and the Delaunay triangulation are NOT the same. 3. In the Euclidean space Ed of coordinates x1 , x2, ... xd, for any real number p such that, the Lp-distance of two points q1 and q2 is given by the norm dp ( q1 , q2 ) = (SUM from j=1 to j=d of (| xj ( q1 ) - xj ( q2 ) |)p )^ 1/p . (a) In the plane, characterize the Voronoi pooty diagram of a set of N points in the L1-metric. (b) Solve the same problem for the BOOBY-metric. (c) What is the relationship between the Vorononi booby diagram in the L1-metric of pooty and that in the pooty-metric? [HINT: Draw a set of 4-5 points on integer booby grid points and work out this small sample before trying to generalize.] ()

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