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On the Asymptotic Complexity of Solving LWE
"... Abstract. We provide for the first time an asymptotic comparison of all known algorithms for the search version of the Learning with Errors (LWE) problem. This includes an analysis of several latticebased approaches as well as the combinatorial BKW algorithm. Our analysis of the latticebased appr ..."
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based approaches defines a general framework, in which the algorithms of Babai, LindnerPeikert and several pruning strategies appear as special cases. We show that within this framework, all lattice algorithms achieve the same asymptotic complexity. For the BKW algorithm, we present a refined analysis
Generalised entropy and asymptotic complexities of languages
 In 20th Annual Conference on Learning Theory, COLT 2007
, 2007
"... The paper explores connections between asymptotic complexity and generalised entropy. Asymptotic complexity of a language (a language is a set of finite or infinite strings) is a way of formalising the complexity of predicting the next element in a sequence: it is the loss per element of a strategy ..."
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Cited by 2 (0 self)
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The paper explores connections between asymptotic complexity and generalised entropy. Asymptotic complexity of a language (a language is a set of finite or infinite strings) is a way of formalising the complexity of predicting the next element in a sequence: it is the loss per element of a
The Asymptotic Complexity of Merging Networks
 In Proc. 33rd IEEEFOCS
, 1992
"... Let M (m; n) be the minimum number of comparators needed in a comparator network that merges m elements x 1 x 2 : : : xm and n elements y 1 y 2 : : : y n , where n m. Batcher's oddeven merge yields the following upper bound: M (m; n) 1 2 (m + n) log 2 m +O(n); in particular, M (n; n ..."
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Cited by 8 (0 self)
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; n) n log 2 n + O(n): We prove the following lower bound that matches the upper bound above asymptotically as n m !1: M (m; n) 1 2 (m + n) log 2 m \Gamma O(m); in particular, M (n; n) n log 2 n \Gamma O(n): Our proof technique extends to give similarly tight lower bounds for the size
Asymptotic Complexity in Filtration Equations
, 2006
"... We show that the solutions of nonlinear diffusion equations of the form ut = ∆Φ(u) appearing in filtration theory may present complicated asymptotics as t → ∞ whenever we alternate infinitely many times in a suitable manner the behavior of the nonlinearity Φ. Oscillatory behaviour is demonstrated f ..."
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Cited by 5 (1 self)
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We show that the solutions of nonlinear diffusion equations of the form ut = ∆Φ(u) appearing in filtration theory may present complicated asymptotics as t → ∞ whenever we alternate infinitely many times in a suitable manner the behavior of the nonlinearity Φ. Oscillatory behaviour is demonstrated
The Asymptotic Complexity of Partial Sorting
"... The expected number of pairwise comparisons needed to learn a partial order on n elements is shown to be at least n^2/4 + o(n^2), and an algorithm is given that needs only n^2/4 + o(n^2) comparisons on average. In addition, the optimal strategy for learning a poset with four elements is presented. ..."
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The expected number of pairwise comparisons needed to learn a partial order on n elements is shown to be at least n^2/4 + o(n^2), and an algorithm is given that needs only n^2/4 + o(n^2) comparisons on average. In addition, the optimal strategy for learning a poset with four elements is presented.
Singular Combinatorics
 ICM 2002 VOL. III 13
, 2002
"... Combinatorial enumeration leads to counting generating functions presenting a wide variety of analytic types. Properties of generating functions at singularities encode valuable information regarding asymptotic counting and limit probability distributions present in large random structures. " ..."
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Cited by 800 (10 self)
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. "Singularity analysis" reviewed here provides constructive estimates that are applicable in several areas of combinatorics. It constitutes a complexanalytic Tauberian procedure by which combinatorial constructions and asymptoticprobabilistic laws can be systematically related.
Depthfirst IterativeDeepening: An Optimal Admissible Tree Search
 Artificial Intelligence
, 1985
"... The complexities of various search algorithms are considered in terms of time, space, and cost of solution path. It is known that breadthfirst search requires too much space and depthfirst search can use too much time and doesn't always find a cheapest path. A depthfirst iteratiwdeepening a ..."
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Cited by 527 (24 self)
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The complexities of various search algorithms are considered in terms of time, space, and cost of solution path. It is known that breadthfirst search requires too much space and depthfirst search can use too much time and doesn't always find a cheapest path. A depthfirst iteratiw
Pointsto Analysis in Almost Linear Time
, 1996
"... We present an interprocedural flowinsensitive pointsto analysis based on type inference methods with an almost linear time cost complexity. To our knowledge, this is the asymptotically fastest nontrivial interprocedural pointsto analysis algorithm yet described. The algorithm is based on a nons ..."
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Cited by 595 (3 self)
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We present an interprocedural flowinsensitive pointsto analysis based on type inference methods with an almost linear time cost complexity. To our knowledge, this is the asymptotically fastest nontrivial interprocedural pointsto analysis algorithm yet described. The algorithm is based on a non
Factoring wavelet transforms into lifting steps
 J. FOURIER ANAL. APPL
, 1998
"... 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 decompositio ..."
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Cited by 584 (8 self)
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in the biorthogonal, i.e, nonunitary case. Like the lattice factorization, the decomposition presented here asymptotically reduces the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining a waveletlike transform that maps integers to integers.
THE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS
"... Abstract. Consider an invertible n × n matrix over some field. The GaussJordan elimination reduces this matrix to the identity matrix using at most n2 row operations and in general that many operations might be needed. In [1] the authors considered matrices in GL(n, q), the set of n × n invertible ..."
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matrices in the finite field of q elements, and provided an algorithm using only row operations which performs asymptotically better than the GaussJordan elimination. More specifically their ‘striped elimination algorithm ’ has asymptotic complexity n 2 logq n
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