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Index Terms
- Topology for Computing (Cambridge Monographs on Applied and Computational Mathematics)
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Reviews
Mukkai S. Krishnamoorthy
Topology is considered an esoteric subject for computer scientists. This book is an attempt to demystify it and make it more accessible to them. The book is an extended version of the author's PhD thesis. It has three major sections: "Mathematics," "Algorithms," and "Applications." The first several chapters on topology (the mathematical portion) are very well written; the material is presented succinctly and elegantly. By providing a number of different perspectives for each definition in short succession, the author gives the reader a thorough understanding of a number of nontrivial topics, such as the ideas of a filtration and a simplex. Additionally, the early discussion of the standard tools of algebraic topology as functors provides a much deeper understanding of the ideas. The discussion of algebraic topology is well done, as the author notes precisely why homotopy is impractical as a computational tool. The fundamental ideas about group theory are presented adequately. The necessary theorems are presented quickly, but with little additional insight. The chapter on homology is very well done; there are very few proofs, but if one thinks about the theorems, sufficient insight is provided to perform computations. The second part deals with algorithms. Four types-persistence algorithms, topological simplifications (rendering algorithms), Morse-Smale complex algorithms, and linking algorithms-are considered. Pseudocode is presented for each of the algorithms. Many of the ideas in this section are new, and are the result of the author's doctoral research. Formulation of persistence algorithms is done with great care and precision. Visualization helps in understanding some of the difficult concepts. Results from graph theory are also presented. Reordering algorithms are introduced for topological simplification. The two reordering algorithms, Betti number algorithms and migration algorithms, are presented, and a number of examples are provided to illustrate them. Proofs of theorems are also presented in detail. Morse-Smale complex algorithms and linking number algorithms are described in the next two chapters. Again, proofs of the theorems are presented. The last section describes the software that was developed by the author for implementing the presented algorithms. Many examples are solved using this software. Further, a scientific study of the performance of the algorithms is presented, with various statistical measures analyzed. Case studies include computational molecular biology, the visualization of crystalline solids, and a number of interesting surfaces. Online Computing Reviews Service
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