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122
What Do We Know About The Product Replacement Algorithm?
 in: Groups ann Computation III
, 2000
"... . The product replacement algorithm is a commonly used heuristic to generate random group elements in a finite group G, by running a random walk on generating ktuples of G. While experiments showed outstanding performance, until recently there was little theoretical explanation. We give an exten ..."
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Cited by 47 (8 self)
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. The product replacement algorithm is a commonly used heuristic to generate random group elements in a finite group G, by running a random walk on generating ktuples of G. While experiments showed outstanding performance, until recently there was little theoretical explanation. We give an extensive review of both positive and negative theoretical results in the analysis of the algorithm. Introduction In the past few decades the study of groups by means of computations has become a wonderful success story. The whole new field, Computational Group Theory, was developed out of needs to discover and prove new results on finite groups. More recently, the probabilistic method became an important tool for creating faster and better algorithms. A number of applications were developed which assume a fast access to (nearly) uniform group elements. This led to a development of the so called "product replacement algorithm", which is a commonly used heuristic to generate random group elemen...
Quantum algorithms for solvable groups
 In Proceedings of the 33rd ACM Symposium on Theory of Computing
, 2001
"... ABSTRACT In this paper we give a polynomialtime quantum algorithm for computing orders of solvable groups. Several other problems, such as testing membership in solvable groups, testing equality of subgroups in a given solvable group, and testing normality of a subgroup in a given solvable group, r ..."
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Cited by 45 (1 self)
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ABSTRACT In this paper we give a polynomialtime quantum algorithm for computing orders of solvable groups. Several other problems, such as testing membership in solvable groups, testing equality of subgroups in a given solvable group, and testing normality of a subgroup in a given solvable group, reduce to computing orders of solvable groups and therefore admit polynomialtime quantum algorithms as well. Our algorithm works in the setting of blackbox groups, wherein none of these problems have polynomialtime classical algorithms. As an important byproduct, our algorithm is able to produce a pure quantum state that is uniform over the elements in any chosen subgroup of a solvable group, which yields a natural way to apply existing quantum algorithms to factor groups of solvable groups. 1.
On efficient sparse integer matrix Smith normal form computations
, 2001
"... We present a new algorithm to compute the Integer Smith normal form of large sparse matrices. We reduce the computation of the Smith form to independent, and therefore parallel, computations modulo powers of wordsize primes. Consequently, the algorithm does not suffer from coefficient growth. W ..."
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Cited by 42 (20 self)
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We present a new algorithm to compute the Integer Smith normal form of large sparse matrices. We reduce the computation of the Smith form to independent, and therefore parallel, computations modulo powers of wordsize primes. Consequently, the algorithm does not suffer from coefficient growth. We have implemented several variants of this algorithm (Elimination and/or BlackBox techniques) since practical performance depends strongly on the memory available. Our method has proven useful in algebraic topology for the computation of the homology of some large simplicial complexes.
On Lattice Reduction for Polynomial Matrices
 Journal of Symbolic Computation
, 2000
"... A simple algorithm for transformation to weak Popov form  essentially lattice reduction for polynomial matrices  is described and analyzed. The algorithm is adapted and applied to various tasks involving polynomial matrices: rank profile and determinant computation; unimodular triangular factori ..."
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Cited by 39 (2 self)
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A simple algorithm for transformation to weak Popov form  essentially lattice reduction for polynomial matrices  is described and analyzed. The algorithm is adapted and applied to various tasks involving polynomial matrices: rank profile and determinant computation; unimodular triangular factorization; transformation to Hermite and Popov canonical form; rational and diophantine linear system solving; short vector computation.
Computing Simplicial Homology Based on Efficient Smith Normal Form Algorithms
, 2002
"... We recall that the calculation of homology with integer coecients of a simplicial complex reduces to the calculation of the Smith Normal Form of the boundary matrices which in general are sparse. We provide a review of several algorithms for the calculation of Smith Normal Form of sparse matrices an ..."
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Cited by 36 (2 self)
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We recall that the calculation of homology with integer coecients of a simplicial complex reduces to the calculation of the Smith Normal Form of the boundary matrices which in general are sparse. We provide a review of several algorithms for the calculation of Smith Normal Form of sparse matrices and compare their running times for actual boundary matrices. Then we describe alternative approaches to the calculation of simplicial homology. The last section then describes motivating examples and actual experiments with the GAP package that was implemented by the authors. These examples also include as an example of other homology theories some calculations of Lie algebra homology.
Parallel algorithms for matrix normal forms
 LINEAR ALGEBRA AND ITS APPLICATIONS
, 1990
"... Here we offer a new randomized parallel algorithm that determines the Smith normal form of a matrix with entries being univariate polynomials with coefficients in an arbitrary field. The algorithm has two important advantages over our previous one: the multipliers relating the Smith form to the inpu ..."
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Cited by 35 (3 self)
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Here we offer a new randomized parallel algorithm that determines the Smith normal form of a matrix with entries being univariate polynomials with coefficients in an arbitrary field. The algorithm has two important advantages over our previous one: the multipliers relating the Smith form to the input matrix are computed, and the algorithm is probabilistic of Las Veg as type, i.e., always finds the correct answer. The Smith form algorithm is also a good sequential algorithm. Our algorithm reduces the problem of Smith form computation to two Hermite form computations. Thus the Smith form problem has complexity asymptotically that of the Hermite form problem. We also construct fast parallel algorithms for Jordan normal form and testing similarity of matrices. Both the similarity and nonsimilarity problems are in the complexity class RNC for the usual coefficient fields, i.e., they can be probabilistically decided in polylogarithmic time using polynomially many processors.
An Interpolating Sequent Calculus for QuantifierFree Presburger Arithmetic
 In Proc. of IJCAR
, 2010
"... Abstract. Craig interpolation has become a versatile tool in formal verification, for instance to generate intermediate assertions for safety analysis of programs. Interpolants are typically determined by annotating the steps of an unsatisfiability proof with partial interpolants. In this paper, we ..."
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Abstract. Craig interpolation has become a versatile tool in formal verification, for instance to generate intermediate assertions for safety analysis of programs. Interpolants are typically determined by annotating the steps of an unsatisfiability proof with partial interpolants. In this paper, we consider Craig interpolation for full quantifierfree Presburger arithmetic (QFPA), for which currently no efficient interpolation procedures are known. Closing this gap, we introduce an interpolating sequent calculus for QFPA and prove it to be sound and complete. We have extended the Princess theorem prover to generate interpolating proofs, and applied it to a large number of publicly available linear integer arithmetic benchmarks. The results indicate the robustness and efficiency of our proofbased interpolation procedure. 1
On Computing the Homology Type of a Triangulation
, 1994
"... :We analyze an algorithm for computing the homology type of a triangulation. By triangulation we mean a finite simplicial complex; its homology type is given by its homology groups (with integer coefficients). The algorithm could be used in computeraided design to tell whether two finiteelement me ..."
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Cited by 20 (0 self)
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:We analyze an algorithm for computing the homology type of a triangulation. By triangulation we mean a finite simplicial complex; its homology type is given by its homology groups (with integer coefficients). The algorithm could be used in computeraided design to tell whether two finiteelement meshes or B'ezierspline surfaces are of the same "topological type," and whether they can be embedded in R³. Homology computation is a purely combinatorial problem of considerable intrinsic interest. While the worstcase bounds we obtain for this algorithm are poor, we argue that many triangulations (in general) and virtually all triangulations in design are very "sparse," in a sense we make precise. We formalize this sparseness measure, and perform a probabilistic analysis of the sparse case to show that the expected running time of the algorithm is roughly quadratic in the geometric complexity (number of simplices) and linear in the dimension.
Probabilistic Computation of the Smith Normal Form of a Sparse Integer Matrix
, 1995
"... We present a new probabilistic algorithms to compute the Smith normal form of a sparse integer matrix A 2 Z m\Thetan . The algorithm treats A as a "blackbox"; A is only used to compute matrixvector products and we don't access individual entries in A directly. The algorithm requir ..."
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Cited by 19 (4 self)
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We present a new probabilistic algorithms to compute the Smith normal form of a sparse integer matrix A 2 Z m\Thetan . The algorithm treats A as a "blackbox"; A is only used to compute matrixvector products and we don't access individual entries in A directly. The algorithm requires about O(m 2 log kAk) such blackbox evaluations reduced modulo wordsized primes p on vectors in Z n\Theta1 p , plus O(m 2 n log kAk) additional bit operations. For sparse matrices this represents a substantial improvement over previously known algorithms. For example, on an n \Theta n integer matrix A with O(n log n) nonzero entries, only about O(n 3 log 2 kAk) bit operations are required to find the Smith form. The new algorithm suffers from no "fillin" or intermediate value explosion, and uses very little additional space. It also admits a simple coarse grain parallelization. The algorithm is probabilistic of the Monte Carlo type  on any input it returns the correct answer with a contr...