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131
Fast parallel matrix and gcd computations
 In Proc. of the 23rd Annual Symposium on Foundations of Computer Science (FOCS’82
, 1982
"... Parallel algorithms to compute the determinant and characteristic polynomial of matrices and the gcd of polynomials are presented. The rank of matrices and solutions of arbitrary systems of linear equations are computed by parallel Las Vegas algorithms. All algorithms work over arbitrary fields. The ..."
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Cited by 41 (1 self)
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Parallel algorithms to compute the determinant and characteristic polynomial of matrices and the gcd of polynomials are presented. The rank of matrices and solutions of arbitrary systems of linear equations are computed by parallel Las Vegas algorithms. All algorithms work over arbitrary fields. They run in parallel time O(log ~ n) (where n is the number of inputs) and use a polynomial number of processors. 1.
Relatively hyperbolic groups
 Michigan Math. J
, 1998
"... Abstract. We generalize some results of Paulin and RipsSela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following. • If G is a nonelementary relatively hyperbolic group with slender parabolic subgroups, and either G is not coHopfian or Out(G) ..."
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Cited by 33 (2 self)
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Abstract. We generalize some results of Paulin and RipsSela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following. • If G is a nonelementary relatively hyperbolic group with slender parabolic subgroups, and either G is not coHopfian or Out(G) is infinite, then G splits over a slender group. • If H is a nonparabolic subgroup of a relatively hyperbolic group, and if any isometric Haction on an Rtree is trivial, then H is Hopfian. • If G is a nonelementary relatively hyperbolic group whose peripheral subgroups are finitely generated, then G has a nonelementary relatively hyperbolic quotient that is Hopfian. • Any finitely presented group is isomorphic to a finite index subgroup of Out(H) for some group H with Kazhdan property (T). (This sharpens a result of OllivierWise). 1.
Degree spectra and computable dimension in algebraic structures
 Annals of Pure and Applied Logic 115 (2002
, 2002
"... \Lambda \Lambda ..."
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Asymmetric multiple description lattice vector quantizers
 IEEE Trans. Inf. Theory
, 2002
"... Abstract—We consider the design of asymmetric multiple description lattice quantizers that cover the entire spectrum of the distortion profile, ranging from symmetric or balanced to successively refinable. We present a solution to a labeling problem, which is an important part of the construction, a ..."
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Cited by 28 (3 self)
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Abstract—We consider the design of asymmetric multiple description lattice quantizers that cover the entire spectrum of the distortion profile, ranging from symmetric or balanced to successively refinable. We present a solution to a labeling problem, which is an important part of the construction, along with a general design procedure. The highrate asymptotic performance of the quantizer is also studied. We evaluate the ratedistortion performance of the quantizer and compare it to known informationtheoretic bounds. The highrate asymptotic analysis is compared to the performance of the quantizer. Index Terms—Cubic lattice, highrate quantization, lattice quantization, multiple descriptions, quantization, source coding, successive refinement, vector quantization. I.
Rigidity Theorems for Actions of Product Groups and Countable Borel Equivalence Relations
"... This paper is a contribution to the theory of countable Borel equivalence relations on standard Borel spaces. As usual, by a standard Borel space we mean a Polish (complete separable metric) space equipped with its #algebra of Borel sets. An equivalence relation E on a standard Borel space X is Bor ..."
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Cited by 28 (6 self)
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This paper is a contribution to the theory of countable Borel equivalence relations on standard Borel spaces. As usual, by a standard Borel space we mean a Polish (complete separable metric) space equipped with its #algebra of Borel sets. An equivalence relation E on a standard Borel space X is Borel if it is a Borel subset of X². Given two
Symmetry Breaking in Graphs
 Electronic Journal of Combinatorics
, 1996
"... A labeling of the vertices of a graph G, OE : V (G) ! f1; : : : ; rg, is said to be rdistinguishing provided no automorphism of the graph preserves all of the vertex labels. The distinguishing number of a graph G, denoted by D(G), is the minimum r such that G has an rdistinguishing labeling. T ..."
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Cited by 23 (4 self)
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A labeling of the vertices of a graph G, OE : V (G) ! f1; : : : ; rg, is said to be rdistinguishing provided no automorphism of the graph preserves all of the vertex labels. The distinguishing number of a graph G, denoted by D(G), is the minimum r such that G has an rdistinguishing labeling. The distinguishing number of the complete graph on t vertices is t. In contrast, we prove (i) given any group \Gamma, there is a graph G such that Aut(G) = \Gamma and D(G) = 2; (ii) D(G) = O(log(jAut(G)j)); (iii) if Aut(G) is abelian, then D(G) 2; (iv) if Aut(G) is dihedral, then D(G) 3; and (v) If Aut(G) = S 4 , then either D(G) = 2 or D(G) = 4. Mathematics Subject Classification 05C,20B,20F,68R 1
Transforming Curves on Surfaces
, 1999
"... We describe an optimal algorithm to decide if one closed curve on a triangulated 2manifold can be continuously transformed to another, i.e., if they are homotopic. Suppose C 1 and C 2 are two closed curves on a surface M of genus g. Further, suppose T is a triangulation of M of size n such that C ..."
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Cited by 20 (3 self)
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We describe an optimal algorithm to decide if one closed curve on a triangulated 2manifold can be continuously transformed to another, i.e., if they are homotopic. Suppose C 1 and C 2 are two closed curves on a surface M of genus g. Further, suppose T is a triangulation of M of size n such that C 1 and C 2 are represented as edgevertex sequences of lengths k 1 and k 2 in T , respectively. Then, our algorithm decides if C 1 and C 2 are homotopic in O(n+k 1 +k 2 ) time and space, provided g 6= 2 if M orientable, and g 6= 3; 4 if M is nonorientable. This as well implies an optimal algorithm to decide if a closed curve on a surface can be continuously contracted to a point. Except for three low genus cases, our algorithm completes an investigation into the computational complexity of two classical problems for surfaces posed by the mathematician Max Dehn at the beginning of this century. The novelty of our approach is in the application of methods from modern combinatorial group theory...
Optimizing Vote and Quorum Assignments for Reading and Writing Replicated Data
 IEEE Transactions on Knowledge and Data Engineering
, 1989
"... In the weighted voting protocol which is used to maintain the consistency of replicated data, the availability of the data to read and write operations not only depends on the availability of the nodes storing the data but also on the vote and quorum assignments used. We consider the problem of dete ..."
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Cited by 19 (1 self)
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In the weighted voting protocol which is used to maintain the consistency of replicated data, the availability of the data to read and write operations not only depends on the availability of the nodes storing the data but also on the vote and quorum assignments used. We consider the problem of determining the vote and quorum assignments that yield the best permormance in a distributed system where node availabilities can be different and the mix of the read and write operations is arbitrary. The optimal vote and quorum assignments depend not only on the system parameters such as node availability and operation mix, but also on the performance measure. We present an enumeration algorithm that can be used to find the vote and quorum assignments that need to be considered for achieving optimal performance. When the performance measure is data availability, an analytical method is derived to evaluate it for any vote and quorum assignment. This method and the enumeration algorithm is used ...
The Hidden Subgroup Problem and Quantum Computation Using Group Representations
 SIAM Journal on Computing
, 2003
"... The Hidden Subgroup Problem is the foundation of many quantum algorithms. An efficient solution is known for the problem over Abelian groups, employed by both Simon's algorithm and Shor's factoring and discrete log algorithms. The nonAbelian case is open; an efficient solution would give ..."
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Cited by 19 (2 self)
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The Hidden Subgroup Problem is the foundation of many quantum algorithms. An efficient solution is known for the problem over Abelian groups, employed by both Simon's algorithm and Shor's factoring and discrete log algorithms. The nonAbelian case is open; an efficient solution would give rise to an efficient quantum algorithm for Graph Isomorphism. We fully analyze a natural generalization of the Abelian case algorithm to the nonAbelian case. We show that the algorithm finds the normal core of the hidden subgroup, and that, in particular, normal subgroups can be found. We show, however, that this immediate generalization of the Abelian algorithm does not efficiently solve Graph Isomorphism. 1
Packaging mathematical structures
 THEOREM PROVING IN HIGHER ORDER LOGICS 5674
, 2009
"... This paper proposes generic design patterns to define and combine algebraic structures, using dependent records, coercions and type inference, inside the Coq system. This alternative to telescopes in particular supports multiple inheritance, maximal sharing of notations and theories, and automated ..."
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Cited by 19 (5 self)
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This paper proposes generic design patterns to define and combine algebraic structures, using dependent records, coercions and type inference, inside the Coq system. This alternative to telescopes in particular supports multiple inheritance, maximal sharing of notations and theories, and automated structure inference. Our methodology is robust enough to handle a hierarchy comprising a broad variety of algebraic structures, from types with a choice operator to algebraically closed fields. Interfaces for the structures enjoy the convenience of a classical setting, without requiring any axiom. Finally, we present two applications of our proof techniques: a key lemma for characterising the discrete logarithm, and a matrix decomposition problem.