Lec14 - Proximity Introduction Comments We consider in this...

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Comments We consider in this topic a large class of related problems that deal with proximity of points in the plane. We will: 1. Define some proximity problems and see how they are related 2. Study a classic algorithm for one of the problems 3. Introduce triangulations 4. Examine a data structure that seems to represent nearly everything we might like to know about proximity in a set of points in the plane. Overview of the topic Subtopic Source Collection of proximity problems P5.1 - P5.3 Closest pair, divide and conquer algorithm P5.4 Triangulations P6.2 Problem definitions Greedy algorithm Constrained triangulations Triangulating polygons O1 Voronoi diagrams P5.5 Definition Properties Construction Delaunay triangulations O5.3 Proximity problems and Voronoi diagrams P5.6 Proximity Introduction

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Closest pair CLOSEST PAIR INSTANCE: Set S = { p 1 , p 2 , . .., p N } of N points in the plane. QUESTION: Determine the two points of S whose mutual distance is smallest. Distance is defined as the usual Euclidean distance: distance( p i , p j ) = sqrt(( x i - x j ) 2 + ( y i - y j ) 2 ). This problem is described as “one of the fundamental questions of computational geometry” in Preparata. Brute force A brute force solution is to compute the distance for every pair of points, saving the smallest; this requires O ( dN 2 ) time. The factor d is the number of dimensions, i.e., the number of coordinates involved in each distance computation. For d = 1, we can do better than O ( N 2 ), as follows: 1. Sort the N points (which are simply numbers). O ( N log N ) 2. Scan the sorted sequence ( x 1 , x 2 , …, x N ) computing x i +1 - x i for i = 1, 2, …, N -1. Save the smallest difference. This is optimal for d = 1. For d = 2, can we do better than O ( 2N 2 ) O ( N 2 )? Proximity Proximity problems
All nearest neighbors ALL NEAREST NEIGHBORS INSTANCE: Set S = { p 1 , p 2 , . .., p N } of N points in the plane. QUESTION: Determine the “nearest neighbor” (point of minimum distance) for each point in S . Proximity Proximity problems

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Nearest neighbor relation, 1 “Nearest neighbor” is a relation on a set S as follows: point b is a nearest neighbor of point a , denoted a b , if distance( a , b ) = min distance( a , c ) c S - a ( a , b , c S ). The “nearest neighbor” relation is not symmetric, i.e., a b does not imply b a (though it could be true) . Preparata, p. 186 says: “Note also that a point is not the nearest neighbor of a unique point (i.e., “ ” is not a function).” Perhaps slightly clearer: “Note that a point is not necessarily the nearest neighbor of a unique point, i.e., “ ” is not one-to-one , nor is it onto , as a point may have more than one nearest neighbor.” Proximity Proximity problems
Nearest neighbor relation, 2 Footnote 2 on Preparata, p. 186: “Although a point can be the nearest neighbor of every other point, a point can have at most six nearest neighbors in two dimensions…”

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Lec14 - Proximity Introduction Comments We consider in this...

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