Pub Date : 2020-06-30DOI: 10.1201/9781420035315.ch15
S. Golomb, D. Klarner
recurrence relation a n = 5 a n − 1 − 7 a n − 2 + 4 a n − 3 for n ≥ 5 with initial values a 1 = 1 , a 2 = 2 , a 3 = 6 , and a 4 = 19
recurrence关系a n = 5 n−1−7 a n−2 + 4 (n−3 for n≥5和价值观名字的首字母a = 1, a = 2, 3 = 6,和a - 4 = 19
{"title":"Polyominoes","authors":"S. Golomb, D. Klarner","doi":"10.1201/9781420035315.ch15","DOIUrl":"https://doi.org/10.1201/9781420035315.ch15","url":null,"abstract":"recurrence relation a n = 5 a n − 1 − 7 a n − 2 + 4 a n − 3 for n ≥ 5 with initial values a 1 = 1 , a 2 = 2 , a 3 = 6 , and a 4 = 19","PeriodicalId":156768,"journal":{"name":"Handbook of Discrete and Computational Geometry, 2nd Ed.","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127427911","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-10-03DOI: 10.1201/9781420035315.pt6
M. Dyer, N. Megiddo, E. Welzl
{"title":"Linear programming","authors":"M. Dyer, N. Megiddo, E. Welzl","doi":"10.1201/9781420035315.pt6","DOIUrl":"https://doi.org/10.1201/9781420035315.pt6","url":null,"abstract":"","PeriodicalId":156768,"journal":{"name":"Handbook of Discrete and Computational Geometry, 2nd Ed.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124246399","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2004-04-13DOI: 10.1201/9781420035315.ch44
B. Chazelle
Discrepancy theory investigates how uniform nonrandom structures can be. For example, given n points in the plane, how should we color them red and blue so as to minimize the difference between the number of red points and the number of blue ones within any disk? Or, how should we place n points in the unit square so that the number of points that lie within any given triangle in the square is as close as possible to n times the area of the triangle? Questions of this nature have direct relevance to computational geometry for two reasons. One of them is their close association with the problem of derandomizing probabilistic algorithms. Such algorithms are often based on random sampling and discrepancy theory provides tools for carrying out the sampling deterministically. This has led to the intriguing fact that virtually all of the important problems in low-dimensional computational geometry can be solved as efficiently deterministically as probabilistically. The second application of discrepancy theory to computational geometry is in the area of lower bounds for multidimensional searching. The complexity of these problems is often tied to spectral properties of geometric set systems, which themselves lie at the heart of geometric discrepancy theory.
{"title":"The discrepancy method in computational geometry","authors":"B. Chazelle","doi":"10.1201/9781420035315.ch44","DOIUrl":"https://doi.org/10.1201/9781420035315.ch44","url":null,"abstract":"Discrepancy theory investigates how uniform nonrandom structures can be. For example, given n points in the plane, how should we color them red and blue so as to minimize the difference between the number of red points and the number of blue ones within any disk? Or, how should we place n points in the unit square so that the number of points that lie within any given triangle in the square is as close as possible to n times the area of the triangle? Questions of this nature have direct relevance to computational geometry for two reasons. One of them is their close association with the problem of derandomizing probabilistic algorithms. Such algorithms are often based on random sampling and discrepancy theory provides tools for carrying out the sampling deterministically. This has led to the intriguing fact that virtually all of the important problems in low-dimensional computational geometry can be solved as efficiently deterministically as probabilistically. The second application of discrepancy theory to computational geometry is in the area of lower bounds for multidimensional searching. The complexity of these problems is often tied to spectral properties of geometric set systems, which themselves lie at the heart of geometric discrepancy theory.","PeriodicalId":156768,"journal":{"name":"Handbook of Discrete and Computational Geometry, 2nd Ed.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131304751","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2004-04-13DOI: 10.1201/9781420035315.ch65
Lutz Kettner, S. Näher
Over the past decades, two major software libraries that support a wide range of geometric computing have been developed: Leda, the Library of Efficient Data Types and Algorithms, and Cgal, the Computational Geometry Algorithms Library. We start with an introduction of common aspects of both libraries and major differences. We continue with sections that describe each library in detail. Both libraries are written in C++. Leda is based on the object-oriented paradigm and Cgal is based on the generic programming paradigm. They provide a collection of flexible, efficient, and correct software components for computational geometry. Users should be able to easily include existing functionality into their programs. Additionally, both libraries have been designed to serve as platforms for the implementation of new algorithms. Correctness is of crucial importance for a library, even more so in the case of geometric algorithms where correctness is harder to achieve than in other areas of software construction. Two well-known reasons are the exact arithmetic assumption and the nondegeneracy assumption that are often used in geometric algorithms. However, both assumptions usually do not hold: floating point arithmetic is not exact and inputs are frequently degenerate. See Chapter 45 for details.
{"title":"Two computational geometry libraries: LEDA and CGAL","authors":"Lutz Kettner, S. Näher","doi":"10.1201/9781420035315.ch65","DOIUrl":"https://doi.org/10.1201/9781420035315.ch65","url":null,"abstract":"Over the past decades, two major software libraries that support a wide range of geometric computing have been developed: Leda, the Library of Efficient Data Types and Algorithms, and Cgal, the Computational Geometry Algorithms Library. We start with an introduction of common aspects of both libraries and major differences. We continue with sections that describe each library in detail. Both libraries are written in C++. Leda is based on the object-oriented paradigm and Cgal is based on the generic programming paradigm. They provide a collection of flexible, efficient, and correct software components for computational geometry. Users should be able to easily include existing functionality into their programs. Additionally, both libraries have been designed to serve as platforms for the implementation of new algorithms. Correctness is of crucial importance for a library, even more so in the case of geometric algorithms where correctness is harder to achieve than in other areas of software construction. Two well-known reasons are the exact arithmetic assumption and the nondegeneracy assumption that are often used in geometric algorithms. However, both assumptions usually do not hold: floating point arithmetic is not exact and inputs are frequently degenerate. See Chapter 45 for details.","PeriodicalId":156768,"journal":{"name":"Handbook of Discrete and Computational Geometry, 2nd Ed.","volume":"160 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123026435","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2004-04-13DOI: 10.1201/9781420035315.ch9
R. Connelly, E. Demaine
There is a long and involved history of linkages starting at least in the nineteenth century with the advent of complicated and intricate mechanical engineering. Some of these practical problems led to interesting, nontrivial geometric problems, and in modern mathematics there has been significant progress on even very basic questions. Over the years, several different points of view have been taken by various groups of people. We will attempt to survey these different perspectives and results obtained in the exploration of linkages.
{"title":"Geometry and Topology of Polygonal Linkages","authors":"R. Connelly, E. Demaine","doi":"10.1201/9781420035315.ch9","DOIUrl":"https://doi.org/10.1201/9781420035315.ch9","url":null,"abstract":"There is a long and involved history of linkages starting at least in the nineteenth century with the advent of complicated and intricate mechanical engineering. Some of these practical problems led to interesting, nontrivial geometric problems, and in modern mathematics there has been significant progress on even very basic questions. Over the years, several different points of view have been taken by various groups of people. We will attempt to survey these different perspectives and results obtained in the exploration of linkages.","PeriodicalId":156768,"journal":{"name":"Handbook of Discrete and Computational Geometry, 2nd Ed.","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126547748","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2004-04-13DOI: 10.1201/9781420035315.ch8
P. Indyk, J. Matoušek
{"title":"Low-Distortion Embeddings of Finite Metric Spaces","authors":"P. Indyk, J. Matoušek","doi":"10.1201/9781420035315.ch8","DOIUrl":"https://doi.org/10.1201/9781420035315.ch8","url":null,"abstract":"","PeriodicalId":156768,"journal":{"name":"Handbook of Discrete and Computational Geometry, 2nd Ed.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128904078","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Geometric Graph Theory","authors":"J. Pach","doi":"10.1201/9781420035315.ch10","DOIUrl":"https://doi.org/10.1201/9781420035315.ch10","url":null,"abstract":"Note: Professor Pach's number: [172]; 2nd edition Reference DCG-CHAPTER-2008-027 Record created on 2008-11-18, modified on 2017-05-12","PeriodicalId":156768,"journal":{"name":"Handbook of Discrete and Computational Geometry, 2nd Ed.","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128605737","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2004-04-13DOI: 10.1201/9781420035315.ch63
H. Edelsbrunner
{"title":"Biological applications of computational topology","authors":"H. Edelsbrunner","doi":"10.1201/9781420035315.ch63","DOIUrl":"https://doi.org/10.1201/9781420035315.ch63","url":null,"abstract":"","PeriodicalId":156768,"journal":{"name":"Handbook of Discrete and Computational Geometry, 2nd Ed.","volume":"72 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120873716","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2004-04-13DOI: 10.1201/9781420035315.ch54
J. Rossignac
{"title":"Surface simplification and 3D geometry compression","authors":"J. Rossignac","doi":"10.1201/9781420035315.ch54","DOIUrl":"https://doi.org/10.1201/9781420035315.ch54","url":null,"abstract":"","PeriodicalId":156768,"journal":{"name":"Handbook of Discrete and Computational Geometry, 2nd Ed.","volume":"76 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126217428","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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