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Exercise 2. Here is a classic geometric problem, in the same application domain as the Center Selection problem: Given n points with Cartesian coordinates (₁, 3) in the plane, and positive weights w, find one(!) point (x, y) that minimizes the weighted sum of the Euclidean distances to the given points. For formal clarity: We want to minimize n Σwi i-1 n •√(x- i-1 (x − x)² + (y - y₁)². This problem is at least not very easy to solve exactly. In the following we therefore propose a rough but rather quick and simple approximation algorithm: Instead of the Euclidean distance, take the Manhattan distance and minimize 1 w₁(x − x₂|+|y-yil). 2.1. Explain how the point that minimizes the weigthed sum of Manhattan distances can be found in polynonial time. That is: Sketch an algorithm, argue why it is correct, and explain your time bound. Try to keep the time bound as low as possible. 2.2. Show that this algorithm has an approximation ratio √2. More specifi- cally: The point found by the algorithm has a weighted sum of Euclidean(!) distances that is at most √2 times the optimal sum. Advice: In 2.1, split the problem in two "independent" one-dimensional problems, that is, work separately in 2- and y-direction. To get a correct idea where to find the optimal coordinate z (and y), it might be good to study examples with very small n first. In 2.2, first write down in general terms what the approximation ratio of the proposed algorithm is, then do the necessary geometry. Finally check whether you have really achieved the goal of proving the claimed approxiamtion ratio. (It is easy to forget this goal and stop half-way.)

Exercise 2. Here is a classic geometric problem, in the same application domain as the Center Selection problem: Given n points with Cartesian coordinates (₁, 3) in the plane, and positive weights w, find one(!) point (x, y) that minimizes the weighted sum of the Euclidean distances to the given points. For formal clarity: We want to minimize n Σwi i-1 n •√(x- i-1 (x − x)² + (y - y₁)². This problem is at least not very easy to solve exactly. In the following we therefore propose a rough but rather quick and simple approximation algorithm: Instead of the Euclidean distance, take the Manhattan distance and minimize 1 w₁(x − x₂|+|y-yil). 2.1. Explain how the point that minimizes the weigthed sum of Manhattan distances can be found in polynonial time. That is: Sketch an algorithm, argue why it is correct, and explain your time bound. Try to keep the time bound as low as possible. 2.2. Show that this algorithm has an approximation ratio √2. More specifi- cally: The point found by the algorithm has a weighted sum of Euclidean(!) distances that is at most √2 times the optimal sum. Advice: In 2.1, split the problem in two "independent" one-dimensional problems, that is, work separately in 2- and y-direction. To get a correct idea where to find the optimal coordinate z (and y), it might be good to study examples with very small n first. In 2.2, first write down in general terms what the approximation ratio of the proposed algorithm is, then do the necessary geometry. Finally check whether you have really achieved the goal of proving the claimed approxiamtion ratio. (It is easy to forget this goal and stop half-way.)

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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Exercise 2. Here is a classic geometric problem, in the same application domain as the Center Selection problem: Given n points with Cartesian coordinates (₁, 3) in the plane, and positive weights w, find one(!) point (x, y) that minimizes the weighted sum of the Euclidean distances to the given points. For formal clarity: We want to minimize n Σwi i-1 n •√(x- i-1 (x − x)² + (y - y₁)². This problem is at least not very easy to solve exactly. In the following we therefore propose a rough but rather quick and simple approximation algorithm: Instead of the Euclidean distance, take the Manhattan distance and minimize 1 w₁(x − x₂|+|y-yil). 2.1. Explain how the point that minimizes the weigthed sum of Manhattan distances can be found in polynonial time. That is: Sketch an algorithm, argue why it is correct, and explain your time bound. Try to keep the time bound as low as possible. 2.2. Show that this algorithm has an approximation ratio √2. More specifi- cally: The point found by the algorithm has a weighted sum of Euclidean(!) distances that is at most √2 times the optimal sum. Advice: In 2.1, split the problem in two "independent" one-dimensional problems, that is, work separately in 2- and y-direction. To get a correct idea where to find the optimal coordinate z (and y), it might be good to study examples with very small n first. In 2.2, first write down in general terms what the approximation ratio of the proposed algorithm is, then do the necessary geometry. Finally check whether you have really achieved the goal of proving the claimed approxiamtion ratio. (It is easy to forget this goal and stop half-way.)
Transcribed Image Text:Exercise 2. Here is a classic geometric problem, in the same application domain as the Center Selection problem: Given n points with Cartesian coordinates (₁, 3) in the plane, and positive weights w, find one(!) point (x, y) that minimizes the weighted sum of the Euclidean distances to the given points. For formal clarity: We want to minimize n Σwi i-1 n •√(x- i-1 (x − x)² + (y - y₁)². This problem is at least not very easy to solve exactly. In the following we therefore propose a rough but rather quick and simple approximation algorithm: Instead of the Euclidean distance, take the Manhattan distance and minimize 1 w₁(x − x₂|+|y-yil). 2.1. Explain how the point that minimizes the weigthed sum of Manhattan distances can be found in polynonial time. That is: Sketch an algorithm, argue why it is correct, and explain your time bound. Try to keep the time bound as low as possible. 2.2. Show that this algorithm has an approximation ratio √2. More specifi- cally: The point found by the algorithm has a weighted sum of Euclidean(!) distances that is at most √2 times the optimal sum. Advice: In 2.1, split the problem in two "independent" one-dimensional problems, that is, work separately in 2- and y-direction. To get a correct idea where to find the optimal coordinate z (and y), it might be good to study examples with very small n first. In 2.2, first write down in general terms what the approximation ratio of the proposed algorithm is, then do the necessary geometry. Finally check whether you have really achieved the goal of proving the claimed approxiamtion ratio. (It is easy to forget this goal and stop half-way.)
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