Generalized FFTs are eecient algorithms for computing a Fourier transform of a function deened on nite group, or a bandlimited function de-ned on a compact group. The development of such algorithms has been accompanied and motivated by a growing number of both potential and realized applications. This paper will attempt to survey some of these applications. Appendices include some more detailed examples. 1. A brief history The now classical" Fast Fourier Transform (FFT) has a long and interesting history. Originally discovered by Gauss, and later made famous after being rediscovered by Cooley and Tukey 21], it may be viewed as an algorithm which eeciently computes the discrete Fourier transform or DFT. In between Gauss and Cooley-Tukey others developed special cases of the algorithm, usually motivated by the need to make eecient data analysis of one sort or another. To cite but a few examples, Gauss was interested in eeciently interpolating the orbits of asteroids 43]; Danielson and Lanczos were concerned with x-ray diiraction 23]; Yates 103] and Good 47] needed the algorithm for statistics; Cooley and Tukey were interested in eecient time series analysis and digital signal processing 21]. For thorough historical overviews see 19, 20, 50]. Recently, there has developed a growing literature related to the construction of algorithms which generalize the FFT from the point of view of the theory of group representations (see e.g., 5, 17, 18, 29, 82]). These sorts of generalizations are natural" as mathematical constructs, but in point of fact, they too have been motivated by applications. For example, the seemingly earliest construction of nonabelian" FFTs (due to Willsky) was motivated by the search for new eecient lters 102]. Later constructions have been motivated by applications such as eecient data analysis (cf. 26]) and circuit design (cf. 6]), just to name a few examples. The purpose of this paper is to survey some of the applications of generalized FFTs and thereby (hopefully!) motivate further work in this direction. 1 2 DANIEL N. ROCKMORE One early version of the FFT is due to the statistician Yates. He was interested in the eecient analysis of data from factorial designs. Section 2 reviews this algorithm and then explains in some detail its generalization in the form of eecient computation of spectral analysis for data on a nite group or its quotient. This is illustrated by a brief discussion of one of the more successful applications to date …
{"title":"Some applications of generalized FFT's","authors":"D. Rockmore","doi":"10.1090/dimacs/028/19","DOIUrl":"https://doi.org/10.1090/dimacs/028/19","url":null,"abstract":"Generalized FFTs are eecient algorithms for computing a Fourier transform of a function deened on nite group, or a bandlimited function de-ned on a compact group. The development of such algorithms has been accompanied and motivated by a growing number of both potential and realized applications. This paper will attempt to survey some of these applications. Appendices include some more detailed examples. 1. A brief history The now classical\" Fast Fourier Transform (FFT) has a long and interesting history. Originally discovered by Gauss, and later made famous after being rediscovered by Cooley and Tukey 21], it may be viewed as an algorithm which eeciently computes the discrete Fourier transform or DFT. In between Gauss and Cooley-Tukey others developed special cases of the algorithm, usually motivated by the need to make eecient data analysis of one sort or another. To cite but a few examples, Gauss was interested in eeciently interpolating the orbits of asteroids 43]; Danielson and Lanczos were concerned with x-ray diiraction 23]; Yates 103] and Good 47] needed the algorithm for statistics; Cooley and Tukey were interested in eecient time series analysis and digital signal processing 21]. For thorough historical overviews see 19, 20, 50]. Recently, there has developed a growing literature related to the construction of algorithms which generalize the FFT from the point of view of the theory of group representations (see e.g., 5, 17, 18, 29, 82]). These sorts of generalizations are natural\" as mathematical constructs, but in point of fact, they too have been motivated by applications. For example, the seemingly earliest construction of nonabelian\" FFTs (due to Willsky) was motivated by the search for new eecient lters 102]. Later constructions have been motivated by applications such as eecient data analysis (cf. 26]) and circuit design (cf. 6]), just to name a few examples. The purpose of this paper is to survey some of the applications of generalized FFTs and thereby (hopefully!) motivate further work in this direction. 1 2 DANIEL N. ROCKMORE One early version of the FFT is due to the statistician Yates. He was interested in the eecient analysis of data from factorial designs. Section 2 reviews this algorithm and then explains in some detail its generalization in the form of eecient computation of spectral analysis for data on a nite group or its quotient. This is illustrated by a brief discussion of one of the more successful applications to date …","PeriodicalId":342609,"journal":{"name":"Groups And Computation","volume":"323 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1997-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116001505","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}
In this paper we survey some recent work directed towards generalizing the fast Fourier transform (FFT). We work primarily from the point of view of group representation theory. In this setting the classical FFT can be viewed as a family of efficient algorithms for computing the Fourier transform of either a function defined on a finite abelian group, or a bandlimited function on a compact abelian group. We discuss generalizations of the FFT to arbitrary finite groups and compact Lie groups.
{"title":"Generalized FFT's- A survey of some recent results","authors":"D. Maslen, D. Rockmore","doi":"10.1090/dimacs/028/13","DOIUrl":"https://doi.org/10.1090/dimacs/028/13","url":null,"abstract":"In this paper we survey some recent work directed towards generalizing the fast Fourier transform (FFT). We work primarily from the point of view of group representation theory. In this setting the classical FFT can be viewed as a family of efficient algorithms for computing the Fourier transform of either a function defined on a finite abelian group, or a bandlimited function on a compact abelian group. We discuss generalizations of the FFT to arbitrary finite groups and compact Lie groups.","PeriodicalId":342609,"journal":{"name":"Groups And Computation","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1996-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130768426","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":"Permutation Groups and Polynomial-Time Computation","authors":"E. Luks","doi":"10.1090/dimacs/011/11","DOIUrl":"https://doi.org/10.1090/dimacs/011/11","url":null,"abstract":"","PeriodicalId":342609,"journal":{"name":"Groups And Computation","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1996-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133041022","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}
This paper considers a permutation group G = 〈S〉 of degree n with t orbits such that the action on each orbit is primitive. It presents a O(tn2 logc(n)) time Monte Carlo group membership algorithm for some constant c. The algorithm is notable for its use of a new theorem showing how to find O(t log n) generators in O (̃|S|n) time under a more general form of the above hypotheses. The algorithm relies on new combinatorial methods for computing with groups [CF92] and previous work of Babai, Luks and Seress [BLS88]. In addition, it makes extensive use of a structure theorem for primitive groups by Cameron [Cam81], which can be derived from results of Kantor [Kan79] and the classification of finite simple groups.
{"title":"Group Membership for Groups with Primitive Orbits Namita Sarawagi, Gene Cooperman, and 253","authors":"L. Finkelstein","doi":"10.1090/dimacs/011/17","DOIUrl":"https://doi.org/10.1090/dimacs/011/17","url":null,"abstract":"This paper considers a permutation group G = 〈S〉 of degree n with t orbits such that the action on each orbit is primitive. It presents a O(tn2 logc(n)) time Monte Carlo group membership algorithm for some constant c. The algorithm is notable for its use of a new theorem showing how to find O(t log n) generators in O (̃|S|n) time under a more general form of the above hypotheses. The algorithm relies on new combinatorial methods for computing with groups [CF92] and previous work of Babai, Luks and Seress [BLS88]. In addition, it makes extensive use of a structure theorem for primitive groups by Cameron [Cam81], which can be derived from results of Kantor [Kan79] and the classification of finite simple groups.","PeriodicalId":342609,"journal":{"name":"Groups And Computation","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123827767","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}
Let K be a number eld. We present several algorithms for working with polycyclic-by-nite subgroups of GL(n; K). Let G be a subgroup of GL(n; K) given by a nite generatingset of matrices. We describe an algorithm for deciding whether or not G is polycyclic-by-nite. For polycyclic-by-nite G, we describe an algorithm for deciding whether or not a given matrix is an element of G. We also describe an algorithm for deciding whether or not G is solvable-by-nite, providing an alternative to the algorithm proposed by Beals. Preliminary experiments indicate that the algorithms described in this paper are suitable for computer implementation. Further experimentation is needed to determine the range of input for which they are practical. 1. Introduction 1.1. Notation and deenitions. Throughout this article, let Z denote the ring of integers, Q the eld of rationals, and C the eld of complex numbers. Let R denote either Z or a number eld. If p is a prime, then the eld of p-adic numbers is denoted by Q p , its algebraic closure by Q p , and the ring of p-adic integers by Z p. The eld with p elements is denoted by F p .
{"title":"Algorithms for polycyclic-by-finite matrix groups","authors":"Gretchen Ostheimer","doi":"10.1090/dimacs/028/17","DOIUrl":"https://doi.org/10.1090/dimacs/028/17","url":null,"abstract":"Let K be a number eld. We present several algorithms for working with polycyclic-by-nite subgroups of GL(n; K). Let G be a subgroup of GL(n; K) given by a nite generatingset of matrices. We describe an algorithm for deciding whether or not G is polycyclic-by-nite. For polycyclic-by-nite G, we describe an algorithm for deciding whether or not a given matrix is an element of G. We also describe an algorithm for deciding whether or not G is solvable-by-nite, providing an alternative to the algorithm proposed by Beals. Preliminary experiments indicate that the algorithms described in this paper are suitable for computer implementation. Further experimentation is needed to determine the range of input for which they are practical. 1. Introduction 1.1. Notation and deenitions. Throughout this article, let Z denote the ring of integers, Q the eld of rationals, and C the eld of complex numbers. Let R denote either Z or a number eld. If p is a prime, then the eld of p-adic numbers is denoted by Q p , its algebraic closure by Q p , and the ring of p-adic integers by Z p. The eld with p elements is denoted by F p .","PeriodicalId":342609,"journal":{"name":"Groups And Computation","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117161025","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}
We study the time complexity of McKay’s algorithm to compute canonical forms and automorphism groups of graphs. The algo rithm is based on a type of backtrack search, and it performs pruning by disc overed automorphisms and by hashing partial information of vertex labelin gs. In practice, the algorithm is implemented in the nautypackage. We obtain colorings of Fürer’s graphs that allow the algorithm to compute their canonical f orms in polynomial time. We then prove an exponential lower bound of the algorit hm for connected 3-regular graphs of color-class size 4 using Fürer’s construction. We conducted experiments withnautyfor these graphs. Our experimental results also indicate the same exponential lower bound.
{"title":"The complexity of McKay's canonical labeling algorithm","authors":"Takunari Miyazaki","doi":"10.1090/dimacs/028/14","DOIUrl":"https://doi.org/10.1090/dimacs/028/14","url":null,"abstract":"We study the time complexity of McKay’s algorithm to compute canonical forms and automorphism groups of graphs. The algo rithm is based on a type of backtrack search, and it performs pruning by disc overed automorphisms and by hashing partial information of vertex labelin gs. In practice, the algorithm is implemented in the nautypackage. We obtain colorings of Fürer’s graphs that allow the algorithm to compute their canonical f orms in polynomial time. We then prove an exponential lower bound of the algorit hm for connected 3-regular graphs of color-class size 4 using Fürer’s construction. We conducted experiments withnautyfor these graphs. Our experimental results also indicate the same exponential lower bound.","PeriodicalId":342609,"journal":{"name":"Groups And Computation","volume":"105 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124810103","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":"Parallel Computation of Sylow Subgroups in Solvable Groups","authors":"P. D. Mark","doi":"10.1090/dimacs/011/12","DOIUrl":"https://doi.org/10.1090/dimacs/011/12","url":null,"abstract":"","PeriodicalId":342609,"journal":{"name":"Groups And Computation","volume":"106 11","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114098442","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":"Complexity Issues in Infinite Group Theory","authors":"C. Sims","doi":"10.1090/dimacs/011/19","DOIUrl":"https://doi.org/10.1090/dimacs/011/19","url":null,"abstract":"","PeriodicalId":342609,"journal":{"name":"Groups And Computation","volume":"29 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133204647","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":"A non-constructive recognition algorithm for the special linear and other classical groups","authors":"F. Celler, C. Leedham-Green","doi":"10.1090/dimacs/028/05","DOIUrl":"https://doi.org/10.1090/dimacs/028/05","url":null,"abstract":"","PeriodicalId":342609,"journal":{"name":"Groups And Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123078622","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}
We describe an experimental approach to studying infinite groups using a software package called Magnus being developed by the New York Group Theory Cooperative. This approach emphasises infinite groups and partial and experimental computation. These computations are frequently inconclusive and may only occasionally succeed. Such experimentation can guide theoretical development and lead to new and interesting questions.
{"title":"Experimenting and computing with infinite groups","authors":"G. Baumslag, Charles F. Miller","doi":"10.1090/dimacs/028/02","DOIUrl":"https://doi.org/10.1090/dimacs/028/02","url":null,"abstract":"We describe an experimental approach to studying infinite groups using a software package called Magnus being developed by the New York Group Theory Cooperative. This approach emphasises infinite groups and partial and experimental computation. These computations are frequently inconclusive and may only occasionally succeed. Such experimentation can guide theoretical development and lead to new and interesting questions.","PeriodicalId":342609,"journal":{"name":"Groups And Computation","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122733883","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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