The.Algorithm.Design.Manual.Springer-Verlag.1998
The.Algorithm.Design.Manual.Springer-Verlag.1998 The.Algorithm.Design.Manual.Springer-Verlag.1998
Voronoi Diagrams Aurenhammer [Aur91] and Fortune [For92] provide excellent surveys on Voronoi diagrams and associated variants such as power diagrams. The first algorithm for constructing Voronoi diagrams was based on divide-and-conquer and is due to Shamos and Hoey [SH75]. Good expositions of Fortune's sweepline algorithm for constructing Voronoi diagrams in [For87] include [O'R94]. Good expositions on the relationship between Delaunay triangulations and (d+1)-dimensional convex hulls [ES86] include [O'R94]. In a kth-order Voronoi diagram, we partition the plane such that each point in a region is closest to the same set of k sites. Using the algorithm of [ES86], the complete set of kth-order Voronoi diagrams can be constructed in time. By doing point location on this structure, the k nearest neighbors to a query point can be found in . Expositions on kth-order Voronoi diagrams include [O'R94, PS85]. The smallest enclosing circle problem can be solved in time using (n-1)st order Voronoi diagrams [PS85]. In fact, there exist linear-time algorithms based on low-dimensional linear programming [Meg83]. A linear algorithm for computing the Voronoi diagram of a convex polygon is given by [AGSS89]. Related Problems: Nearest neighbor search (see page ), point location (see page ), triangulation (see page ). Next: Nearest Neighbor Search Up: Computational Geometry Previous: Triangulation Algorithms Mon Jun 2 23:33:50 EDT 1997 file:///E|/BOOK/BOOK4/NODE187.HTM (4 of 4) [19/1/2003 1:31:41]
Nearest Neighbor Search Next: Range Search Up: Computational Geometry Previous: Voronoi Diagrams Nearest Neighbor Search Input description: A set S of n points in d dimensions; a query point q. Problem description: Which point in S is closest to q? Discussion: The need to quickly find the nearest neighbor to a query point arises in a variety of geometric applications. The classic example in two dimensions is designing a system to dispatch emergency vehicles to the scene of a fire. Once the dispatcher learns the location of the fire, she uses a map to find the firehouse closest to this point so as to minimize transportation delays. This situation occurs in any application mapping customers to service providers. Nearest-neighbor search is also important in classification. Suppose we are given a collection of data about people (say age, height, weight, years of education, sex, and income level) each of whom has been labeled as Democrat or Republican. We seek a classifier to decide which way a different person is likely to vote. Each of the people in our data set is represented by a party-labeled point in d-dimensional space. A simple classifier can be built by assigning to the new point the party affiliation of its nearest neighbor. Such nearest-neighbor classifiers are widely used, often in high-dimensional spaces. The vector- file:///E|/BOOK/BOOK4/NODE188.HTM (1 of 4) [19/1/2003 1:31:43]
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Nearest Neighbor Search<br />
Next: Range Search Up: Computational Geometry Previous: Voronoi Diagrams<br />
Nearest Neighbor Search<br />
Input description: A set S of n points in d dimensions; a query point q.<br />
Problem description: Which point in S is closest to q?<br />
Discussion: <strong>The</strong> need to quickly find the nearest neighbor to a query point arises in a variety of<br />
geometric applications. <strong>The</strong> classic example in two dimensions is designing a system to dispatch<br />
emergency vehicles to the scene of a fire. Once the dispatcher learns the location of the fire, she uses a<br />
map to find the firehouse closest to this point so as to minimize transportation delays. This situation<br />
occurs in any application mapping customers to service providers.<br />
Nearest-neighbor search is also important in classification. Suppose we are given a collection of data<br />
about people (say age, height, weight, years of education, sex, and income level) each of whom has been<br />
labeled as Democrat or Republican. We seek a classifier to decide which way a different person is likely<br />
to vote. Each of the people in our data set is represented by a party-labeled point in d-dimensional space.<br />
A simple classifier can be built by assigning to the new point the party affiliation of its nearest neighbor.<br />
Such nearest-neighbor classifiers are widely used, often in high-dimensional spaces. <strong>The</strong> vector-<br />
file:///E|/BOOK/BOOK4/NODE188.HTM (1 of 4) [19/1/2003 1:31:43]
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