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# HW1 - sites We deﬁne the Voronoi diagram of P as the...

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Fall Semester 2008 CS300 Algorithms Homework #1 (due 10/06) 1. Take the following list of functions and arrange them in ascending order of growth rate. That is, if function g ( n ) immediately follows function f ( n ) in your list, then it should be the case that f ( n ) is O ( g ( n )). n 2 . 5 , 2 n, ( n + 10) , 10 n , 100 n , n 2 log n 2. Assume you have functions f and g such that f ( n ) is O ( g ( n )). For each of the following statements, decide whether you think it is true or false and give a proof or counterexample . (a) 2 f ( n ) is O (2 g ( n ) ). (b) f ( n ) 2 is O ( g ( n ) 2 ). 3. Denote the Euclidean distance between two points p and q by dist( p, q ). In the plane we have dist( p, q ) = q ( p x - q x ) 2 + ( p y - q y ) 2 Let P := { p 1 , p 2 , ··· , p n } be a set of n distinct points in the plane; these points are the
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Unformatted text preview: sites. We deﬁne the Voronoi diagram of P as the subdivision of the plane into n cells, one for each site in P , with the property that a point q lies in the cell corresponding to a site p i if and only if dist( q, p i ) < dist( q, p j ) for each p j ∈ P with j ± = i . (See Fig. 1). The lower bound on time complexity for a sorting problem is Ω( n log n ), where n is the Figure 1: Voronoi diagram for a set of points P input size. Show that the lower bound of time complexity of computing Voronoi diagrams is Ω( n log n ) from the lower bound for the sorting problem. 1...
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