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Smallest enclosing circles

Consider the following problem: given a set V with N points in the same plan that have ? integer coordinates?

and an integer k (0

covering the N points from V and have the smallest radius r . An equivalent formulation is?

following: given a point x and a finite set of points S, define?

dist (x, S) = min {dist (x, y) |y in S} and radius (S) = max {dist (x, S) |x in V}. The scope is to?

? find a subset of cardinal k which minimizes the radius (S).?

The following are required:?

1. To describe a particular case, netrivial, when N = 11 and k = 3.?

2. To write a program that determines the optimal solution always. It will only describe the main idea?

and time complexity of the program. It will also describe three instances?

whose performance requires between 5 and 10 minutes. For each instance we must specifie the exact time of execution

. The program will be attached as an annex.?

3. Consider a greedy algorithm that at each step chooses a point x(that wasn't already chosen)?

and is at maximum distance from the chosen centers, in other words it maximizes dist (x, S). The first center is?

chosen arbitrarily. Prove that the greedy algorithm is an approximation algorithm with ratio?

bounded by 2.?

4. To write a program that implements the greedy algorithm. To run this algorithm?

together with the algorithm implemented at point 2 over at least 10 instances and check the property from 3.?

For each instance state the optimal solution, approximate solution, the ratio of?

two solutions and lead times. The program will be attached as an annex.

Programming language to be used: C or C++ or C# or Java

## Deliverables

1) All deliverables will be considered "work made for hire" under U.S. Copyright law. Employer will receive exclusive and complete copyrights to all work purchased. (No GPL, GNU, 3rd party components, etc. unless all copyright ramifications are explained AND AGREED TO by the employer on the site per the worker's Worker Legal Agreement).

2) Complete and fully-functional working program(s) in executable form as well as complete source code of all work done.

3) Deliverables must be in ready-to-run condition, as follows (depending on the nature of the deliverables):

a) For web sites or other server-side deliverables intended to only ever exist in one place in the Employer's environment--Deliverables must be installed by the Worker in ready-to-run condition in the Employer's environment.

b) For all others including desktop software or software the employer intends to distribute: A software installation package that will install the software in ready-to-run condition on the platform(s) specified in this project.

## Platform

C/C++/C#/Java

Habilidades: Vale Tudo, Programação C, Programação C#, Engenharia, Java, Gestão de projetos, Arquitetura de software, Teste de Software

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