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Enumerating extreme points in higher dimensions
[chapter]

1995
*
Lecture Notes in Computer Science
*

We consider the problem of

doi:10.1007/3-540-59042-0_105
fatcat:n45botec5zeavglrimbmocbn4u
*enumerating*all*extreme**points*of a given set P of n*points**in*d*dimensions*. ... We also present an algorithm to compute the depth of each*point*of the given set of n*points**in*d-*dimensions*. ... We*enumerate*all the*extreme**points*of a*point*set P of n*points**in*d*dimensions**in*O(nm) time where m is the number of*extreme**points*of P. ...##
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Extreme Enumeration on GPU and in Clouds
[chapter]

2011
*
Lecture Notes in Computer Science
*

Specifically, our improvements to the recent

doi:10.1007/978-3-642-23951-9_12
fatcat:kkvmsnsbf5dnhmrdji3la3hdcm
*Extreme*Pruning*in**enumeration*approach, proposed by Gama et al.*in*Eurocrypt 2010, include: (1) a more flexible bounding function*in*polynomial form; (2) code ... Our implementation allows us to find a short vector at*dimension*114 using 8 NVIDIA video cards*in*less than two days. ...*In**dimension*100, the number of finished*enumeration*trees was already too small to derive a meaningful success rate. The success rate of BKZ vanishes*in**higher**dimensions*. ...##
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Parallel Improved Schnorr-Euchner Enumeration SE++ on Shared and Distributed Memory Systems, With and Without Extreme Pruning

2016
*
Journal of Wireless Mobile Networks, Ubiquitous Computing, and Dependable Applications
*

It solves the SVP for lattices

doi:10.22667/jowua.2016.12.31.001
dblp:journals/jowua/CorreiaMPBA16
fatcat:qq6irsu7rzffrcc6iet4umpqka
*in*much*higher**dimensions**in*less time than implementations without*extreme*pruning. ...*Extreme*pruning significantly reduces the probability of finding the shortest vector of the lattice, but the execution time of*enumeration*decreases*in*a much*higher*pace. ... As shown*in*the results, this technique solves the SVP for lattices*in*much*higher**dimensions**in*less time than the implementation without*extreme*pruning. ...##
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Segments in enumerating faces

1995
*
Mathematical programming
*

The role of segments

doi:10.1007/bf01585927
fatcat:pjspc6hjergvbfbjoc6l3ddvnm
*in*the unsolved problem of*enumerating*the*extreme**points*of a convex polytope specified by a degenerate system of linear constraints,*in*time polynomial*in*the number of*extreme**points*... Using segments, we describe an algorithm that*enumerates*all the faces,*in*time polynomial*in*their number. ... Some methods for*enumerating*faces use a pivot scheme to first*enumerate*the*extreme**points*of K based on*enumerating*the feasible bases for (1) . ...##
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Locally optimal 2-periodic sphere packings
[article]

2019
*
arXiv
*
pre-print

We implement this computation

arXiv:1704.08156v2
fatcat:d4eeemx3obafzn7yazfhxrydwi
*in*d = 3, 4, and 5 and show that no 2-periodic packing surpasses the density of the optimal lattices*in*these*dimensions*. A partial*enumeration*is performed*in*d = 6. ... We generalize Voronoi's method to m > 1 and present a procedure to*enumerate*all locally optimal 2-periodic sphere packings*in*any*dimension*, provided there are finitely many. ... We are hopeful that a similar approach could be used for m = 2 to make full*enumeration**in**higher**dimensions*than d = 5 tractable, but we do not attempt to implement it*in*this work. ...##
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A problem in enumerating extreme points, and an efficient algorithm for one class of polytopes

2008
*
Optimization Letters
*

Here we describe the central problem

doi:10.1007/s11590-008-0103-8
fatcat:nobo7bnzobbyner2kb4e5r3pyq
*in*carrying out the*enumeration*efficiently after reaching a segment. ... We then discuss two procedures for*enumerating**extreme**points*, the mukkadvayam checking procedure, and the nearest*point*procedure. ... the*enumeration*of*extreme**points*of K as*in*Section 5 again. ...##
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Orthogonalized lattice enumeration for solving SVP

2018
*
Science China Information Sciences
*

*In*this paper, we consider sparse orthogonalized integer representations for shortest vectors and propose a new

*enumeration*method, called orthognalized

*enumeration*, by integrating such a representation ... Furthermore, we present a mixed BKZ method, called MBKZ, by alternately applying orthognalized

*enumeration*and other existing

*enumeration*methods. ...

*dimensions*that are much

*higher*than ever. ...

##
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The Bichromatic Rectangle Problem in High Dimensions

2009
*
Canadian Conference on Computational Geometry
*

Our algorithm

dblp:conf/cccg/BackerK09
fatcat:zwz6iyfaxzfxzkrl5wyfn7yyli
*enumerates*the set of relevant hyperrectangles (inclusion maximal axisaligned hyperrectangles that do not contain a red*point*) and counts the number of blue*points**in*each one. ... We prove asymptotically tight bounds on this quantity*in*the worst case. The techniques developed directly apply to the maximum empty rectangle problem*in*high*dimensions*. ... Still, the gap between worst case run times*in*2D suggests that we can do better than*enumeration**in*three or*higher**dimensions*. ...##
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Page 6251 of Mathematical Reviews Vol. , Issue 96j
[page]

1996
*
Mathematical Reviews
*

(D-FRBG-I; Freiburg)

*Enumerating**extreme**points**in**higher**dimensions*. (English summary ) STACS 95 (Munich, 1995), 562-570, Lecture Notes*in*Comput. Sci., 900, Springer, Berlin, 1995. ... Summary: “We consider the problem of*enumerating*all*extreme**points*of a given set P of n*points**in*d*dimensions*. ...##
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Comparison of the upper bounds for the extreme points of the polytopes of line-stochastic tensors
[article]

2021
*
arXiv
*
pre-print

As all these approaches are worthy of consideration and investigation

arXiv:2110.12337v2
fatcat:r6ikftgttjhsno47kzge72b6kq
*in*the*enumeration*problem, various bounds have been obtained. ...*In**enumerating*vertices of the polytopes of stochastic tensors, different approaches have been used: (1) Combinatorial method via Latin squares; (2) Analytic (topological) approach by using hyperplanes ... It is a fundamental and central question*in*the polytope theory to determine the number and structures of the vertices (or faces of*higher**dimensions*) for a given polytope, and this is an*extremely*difficult ...##
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Page 5116 of Mathematical Reviews Vol. , Issue 2004f
[page]

2004
*
Mathematical Reviews
*

Bounds are given for the second minimum

*higher*weight and a Gieason-type theorem is derived for the second*higher*weight*enumerator*. ... These codes have weight*enumerators*for which no*extremal*self-dual codes were previously known to exist.” 2004f:94094 94B05 Dougherty, Steven T. (1-SCRN; Scranton, PA); Gulliver, T. ...##
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Weight enumeration of codes from finite spaces

2011
*
Designs, Codes and Cryptography
*

For our calculations we use that these codes correspond to a projective system containing all the

doi:10.1007/s10623-011-9557-2
fatcat:hmihrf6q4bfbjnaf6ykgsb26pu
*points**in*a finite projective or affine space. ... As a result from the geometric method we use for the weight*enumeration*, we also completely determine the set of supports of subcodes and words*in*an extension code. ... Acknowledgments The author would like to thank Vladimir Tonchev for coming up with the question about the weight*enumerator*of the extension codes of the Simplex code, and for his encouraging conversations ...##
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Parallel Shortest Lattice Vector Enumeration on Graphics Cards
[chapter]

2010
*
Lecture Notes in Computer Science
*

Our implementation is almost 5 times faster

doi:10.1007/978-3-642-12678-9_4
fatcat:ewdokfhpmnakfk7rr5f2lchdxa
*in*high lattice*dimensions*. Exhaustive search is one of the main building blocks for lattice basis reduction*in*cryptanalysis. ... Our work results*in*an advance*in*practical lattice reduction. ... We thankÖzgür Dagdelen for creating some of the initial ideas of parallelizing lattice*enumeration*and Benjamin Milde, Chen-Mou Cheng, and Bo-Yin Yang for the nice discussions and helpful ideas. ...##
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BKZ 2.0: Better Lattice Security Estimates
[chapter]

2011
*
Lecture Notes in Computer Science
*

The best lattice reduction algorithm known

doi:10.1007/978-3-642-25385-0_1
fatcat:zdtum4wadvailaeptm2a6bat6e
*in*practice for high*dimension*is Schnorr-Euchner's BKZ: all security estimates of lattice cryptosystems are based on NTL's old implementation of BKZ. ... We propose an efficient simulation algorithm to model the behaviour of BKZ*in*high*dimension*with high blocksize ≥ 50, which can predict approximately both the output quality and the running time, thereby ... Second, recent progress [10]*in**enumeration*shows that*enumeration*can now be performed*in*much*higher**dimension*(e.g. β ≈ 110) than previously imagined, but no approximate value of c(β, n) is known ...##
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Fast Lattice Point Enumeration with Minimal Overhead
[chapter]

2014
*
Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms
*

The algorithms typically used

doi:10.1137/1.9781611973730.21
dblp:conf/soda/MicciancioW15
fatcat:wf3jvwa47falzk4nuabsayq2yi
*in*practice have worst-case asymptotic running time 2 O(n 2 ) , but perform*extremely*well*in*practice, at least for all values of the lattice*dimension*for which experimentation ... However, there is an uncomfortable gap between our theoretical understanding and practical performance of lattice*point**enumeration*algorithms. ...*dimension*n gets*higher*. ...
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