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Computational Geometry: An Introduction Through Randomized Algorithms

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This introduction to computational geometry is designed for beginners. It emphasizes simple randomized methods, developing basic principles with the help of planar applications, beginning with deterministic algorithms and shifting to randomized algorithms as the problems become more complex. It also explores higher dimensional advanced applications and provides exercises.

This book is based on lectures given to graduate students at the University of Chicago. It is intended to provide a rapid and concise introduction to computational geometry. No prior familiarity with computational geometry is assumed. A modest undergraduate background in computer science or a related field should suffice.

My goal is to describe some basic problems in computational geometry and the simplest known algorithms for them. It so happens that several of these algorithms are randomized. That is why we have chosen randomized methods to provide an introduction to computational geometry. There is another feature of randomized methods that makes them ideal for this task: They are all based on a few basic principles, which can be applied systematically to a large number of apparently dissimilar problems. Thus, it becomes possible to provide through randomized algorithms a unified, broad perspective of computational geometry. I have tried to give an account that brings out this simplicity and unity of randomized algorithms and also their depth.

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