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184,049
The lattice dimension of a tree
, 2004
"... The lattice dimension of a graph G is the minimal dimension of a cubic lattice in which G can be isometrically embedded. In this note we prove that the lattice dimension of a tree with n leaves is ⌈n/2⌉. In the paper, T is a tree with q edges and n leaves. By the Djoković theorem [3] (see also [1, 2 ..."
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Cited by 3 (0 self)
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The lattice dimension of a graph G is the minimal dimension of a cubic lattice in which G can be isometrically embedded. In this note we prove that the lattice dimension of a tree with n leaves is ⌈n/2⌉. In the paper, T is a tree with q edges and n leaves. By the Djoković theorem [3] (see also [1
The Lattice Dimension of a Graph
, 2004
"... Abstract. We describe a polynomial time algorithm for, given an undirected graph G, finding the minimum dimension d such that G may be isometrically embedded into the ddimensional integer lattice Z d. 1 ..."
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Abstract. We describe a polynomial time algorithm for, given an undirected graph G, finding the minimum dimension d such that G may be isometrically embedded into the ddimensional integer lattice Z d. 1
The Lattice Dimension of Benzenoid Systems
, 2006
"... A labeling of vertices of a benzenoid system B is proposed that reflects the graph distance in B and is significantly shorter that the labeling obtained from a hypercube embedding of B. The new labeling corresponds to an embedding of B into the integer lattice Z^d and is shown to be optimal for all ..."
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Cited by 1 (1 self)
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A labeling of vertices of a benzenoid system B is proposed that reflects the graph distance in B and is significantly shorter that the labeling obtained from a hypercube embedding of B. The new labeling corresponds to an embedding of B into the integer lattice Z^d and is shown to be optimal for all
Fully homomorphic encryption using ideal lattices
 In Proc. STOC
, 2009
"... We propose a fully homomorphic encryption scheme – i.e., a scheme that allows one to evaluate circuits over encrypted data without being able to decrypt. Our solution comes in three steps. First, we provide a general result – that, to construct an encryption scheme that permits evaluation of arbitra ..."
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Cited by 642 (17 self)
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that is represented as a lattice), as needed to evaluate general circuits. Unfortunately, our initial scheme is not quite bootstrappable – i.e., the depth that the scheme can correctly evaluate can be logarithmic in the lattice dimension, just like the depth of the decryption circuit, but the latter is greater than
Implementing data cubes efficiently
 In SIGMOD
, 1996
"... Decision support applications involve complex queries on very large databases. Since response times should be small, query optimization is critical. Users typically view the data as multidimensional data cubes. Each cell of the data cube is a view consisting of an aggregation of interest, like total ..."
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Cited by 545 (1 self)
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to materializing the data cube. In this paper, we investigate the issue of which cells (views) to materialize when it is too expensive to materialize all views. A lattice framework is used to express dependencies among views. We present greedy algorithms that work off this lattice and determine a good set of views
Factoring wavelet transforms into lifting steps
 J. Fourier Anal. Appl
, 1998
"... ABSTRACT. This paper is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This dec ..."
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Cited by 573 (8 self)
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present here a selfcontained derivation, building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used
RSA Cryptanalysis with Increased Bounds on the Secret Exponent using Less Lattice Dimension
"... Abstract. We consider RSA with N = pq, q < p < 2q, public encryption exponent e and private decryption exponent d. Boneh and Durfee (Eurocrypt 1999, IEEEIT 2000) used Coppersmith’s method (Journal of Cryptology, 1997) to factorize N using e when d < N 0.292, the theoretical bound. Related ..."
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Cited by 5 (1 self)
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by large lattice dimensions. In this paper we present theoretical results as well as experimental evidences to extend the bound of d for which RSA is weak. This requires the knowledge of a few most significant bits of p (alternatively these bits need to be searched exhaustively). We provide experimental
A New Kind of Science
, 2002
"... “Somebody says, ‘You know, you people always say that space is continuous. How do you know when you get to a small enough dimension that there really are enough points in between, that it isn’t just a lot of dots separated by little distances? ’ Or they say, ‘You know those quantum mechanical amplit ..."
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Cited by 850 (0 self)
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“Somebody says, ‘You know, you people always say that space is continuous. How do you know when you get to a small enough dimension that there really are enough points in between, that it isn’t just a lot of dots separated by little distances? ’ Or they say, ‘You know those quantum mechanical
Virtual Time and Global States of Distributed Systems
 PARALLEL AND DISTRIBUTED ALGORITHMS
, 1988
"... A distributed system can be characterized by the fact that the global state is distributed and that a common time base does not exist. However, the notion of time is an important concept in every day life of our decentralized "real world" and helps to solve problems like getting a consiste ..."
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Cited by 741 (6 self)
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orderedand form a lattice. By using timestamps and a simple clock update mechanism the structure of causality is represented in an isomorphic way. The new model of time has a close analogy to Minkowski's relativistic spacetime and leads among others to an interesting characterization of the global
Results 1  10
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184,049