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75
When trees collide: An approximation algorithm for the generalized Steiner problem on networks
, 1994
"... We give the first approximation algorithm for the generalized network Steiner problem, a problem in network design. An instance consists of a network with linkcosts and, for each pair fi; jg of nodes, an edgeconnectivity requirement r ij . The goal is to find a minimumcost network using the a ..."
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Cited by 211 (31 self)
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We give the first approximation algorithm for the generalized network Steiner problem, a problem in network design. An instance consists of a network with linkcosts and, for each pair fi; jg of nodes, an edgeconnectivity requirement r ij . The goal is to find a minimumcost network using the available links and satisfying the requirements. Our algorithm outputs a solution whose cost is within 2dlog 2 (r + 1)e of optimal, where r is the highest requirement value. In the course of proving the performance guarantee, we prove a combinatorial minmax approximate equality relating minimumcost networks to maximum packings of certain kinds of cuts. As a consequence of the proof of this theorem, we obtain an approximation algorithm for optimally packing these cuts; we show that this algorithm has application to estimating the reliability of a probabilistic network.
A Sieve Algorithm for the Shortest Lattice Vector Problem
, 2001
"... We present a randomized 2 O(n) time algorithm to compute a shortest nonzero vector in an ndimensional rational lattice. The best known time upper bound for this problem was 2 O(n log n) ..."
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Cited by 148 (3 self)
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We present a randomized 2 O(n) time algorithm to compute a shortest nonzero vector in an ndimensional rational lattice. The best known time upper bound for this problem was 2 O(n log n)
Cryptanalysis of RSA with Private Key d Less Than N^0.292
 IEEE Transactions on Information Theory
, 2000
"... We show that if the private exponent d used in the RSA publickey cryptosystem is less than N^0.292 then the system is insecure. This is the first improvement over an old result of Wiener showing that when d is less than N^0.25 the RSA system is insecure. We hope our approach can be used to eventual ..."
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Cited by 114 (5 self)
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We show that if the private exponent d used in the RSA publickey cryptosystem is less than N^0.292 then the system is insecure. This is the first improvement over an old result of Wiener showing that when d is less than N^0.25 the RSA system is insecure. We hope our approach can be used to eventually improve the bound to d less than N^0.5.
An Algorithmic Theory of Lattice Points in Polyhedra
, 1999
"... We discuss topics related to lattice points in rational polyhedra, including efficient enumeration of lattice points, “short” generating functions for lattice points in rational polyhedra, relations to classical and higherdimensional Dedekind sums, complexity of the Presburger arithmetic, efficien ..."
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Cited by 91 (6 self)
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We discuss topics related to lattice points in rational polyhedra, including efficient enumeration of lattice points, “short” generating functions for lattice points in rational polyhedra, relations to classical and higherdimensional Dedekind sums, complexity of the Presburger arithmetic, efficient computations with rational functions, and others. Although the main slant is algorithmic, structural results are discussed, such as relations to the general theory of valuations on polyhedra and connections with the theory of toric varieties. The paper surveys known results and presents some new results and connections.
The Load, Capacity and Availability of Quorum Systems
, 1998
"... A quorum system is a collection of sets (quorums) every two of which intersect. Quorum systems have been used for many applications in the area of distributed systems, including mutual exclusion, data replication and dissemination of information Given a strategy to pick quorums, the load L(S) is th ..."
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Cited by 89 (12 self)
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A quorum system is a collection of sets (quorums) every two of which intersect. Quorum systems have been used for many applications in the area of distributed systems, including mutual exclusion, data replication and dissemination of information Given a strategy to pick quorums, the load L(S) is the minimal access probability of the busiest element, minimizing over the strategies. The capacity Cap(S) is the highest quorum accesses rate that S can handle, so Cap(S) = 1=L(S).
The Two Faces of Lattices in Cryptology
, 2001
"... Lattices are regular arrangements of points in ndimensional space, whose study appeared in the 19th century in both number theory and crystallography. Since the appearance of the celebrated LenstraLenstra Lov'asz lattice basis reduction algorithm twenty years ago, lattices have had surprising ..."
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Cited by 67 (16 self)
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Lattices are regular arrangements of points in ndimensional space, whose study appeared in the 19th century in both number theory and crystallography. Since the appearance of the celebrated LenstraLenstra Lov'asz lattice basis reduction algorithm twenty years ago, lattices have had surprising applications in cryptology. Until recently, the applications of lattices to cryptology were only negative, as lattices were used to break various cryptographic schemes. Paradoxically, several positive cryptographic applications of lattices have emerged in the past five years: there now exist publickey cryptosystems based on the hardness of lattice problems, and lattices play a crucial role in a few security proofs.
An Improved WorstCase to AverageCase Connection for Lattice Problems (extended abstract)
 In FOCS
, 1997
"... We improve a connection of the worstcase complexity and the averagecase complexity of some wellknown lattice problems. This fascinating connection was first discovered by Ajtai [1] in 1996. We improve the exponent of this connection from 8 to 3:5 + ffl. Department of Computer Science, State Unive ..."
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Cited by 54 (10 self)
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We improve a connection of the worstcase complexity and the averagecase complexity of some wellknown lattice problems. This fascinating connection was first discovered by Ajtai [1] in 1996. We improve the exponent of this connection from 8 to 3:5 + ffl. Department of Computer Science, State University of New York at Buffalo, Buffalo, NY 14260. Research supported in part by NSF grants CCR9319393 and CCR9634665, and an Alfred P. Sloan Fellowship. Email: cai@cs.buffalo.edu y Department of Computer Science, State University of New York at Buffalo, Buffalo, NY 14260. Research supported in part by NSF grants CCR9319393 and CCR9634665. Email: apn@cs.buffalo.edu 1 Introduction A lattice L is a discrete additive subgroup of R n . There are many fascinating problems concerning lattices, both from a structural and from an algorithmic point of view [12, 20, 11, 13]. The study of lattice problems can be traced back to Gauss, Dirichlet and Hermite, among others [8, 6, 14]. The subje...
LatticeBased Memory Allocation
, 2003
"... We investigate the problem of memory reuse, for reducing the necessary memory size, in the context of compilation of dedicated processors. Memory reuse is a wellknown concept when allocating registers (i.e., scalar variables). Its (recent) extension to arrays was studied mainly by Lefebvre and Feau ..."
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Cited by 45 (4 self)
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We investigate the problem of memory reuse, for reducing the necessary memory size, in the context of compilation of dedicated processors. Memory reuse is a wellknown concept when allocating registers (i.e., scalar variables). Its (recent) extension to arrays was studied mainly by Lefebvre and Feautrier (for loop parallelization) and by Quillereand Rajopadhye (for circuit synthesis based on recurrence equations) . Both consider a#ne mappings of indices to data, with modulo expressions in the first and (mainly) projections in the second. We develop a mathematical framework based on (integral) critical lattices that subsumes all previous approaches and gives new insights into the problem. Our technique consists first in building an abstract representation of conflicting indices (equivalent in a multidimensional space to the interference graph for register allocation), then in defining an integral lattice, admissible for the set of differences of conflicting indices, used to build a valid modular allocation. We also show the link with critical lattices, successive minima, and basis reduction, and we analyze various strategies for latticebased memory allocation.