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Enumerating extreme points in higher dimensions [chapter]

Th. Ottmann, S. Schuierer, S. Soundaralakshmi
1995 Lecture Notes in Computer Science  
We consider the problem of 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.  ... 
doi:10.1007/3-540-59042-0_105 fatcat:n45botec5zeavglrimbmocbn4u

Extreme Enumeration on GPU and in Clouds [chapter]

Po-Chun Kuo, Michael Schneider, Özgür Dagdelen, Jan Reichelt, Johannes Buchmann, Chen-Mou Cheng, Bo-Yin Yang
2011 Lecture Notes in Computer Science  
Specifically, our improvements to the recent 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.  ... 
doi:10.1007/978-3-642-23951-9_12 fatcat:kkvmsnsbf5dnhmrdji3la3hdcm

Parallel Improved Schnorr-Euchner Enumeration SE++ on Shared and Distributed Memory Systems, With and Without Extreme Pruning

Fábio Correia, Artur Mariano, Alberto Proença, Christian H. Bischof, Erik Agrell
2016 Journal of Wireless Mobile Networks, Ubiquitous Computing, and Dependable Applications  
It solves the SVP for lattices 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.  ... 
doi:10.22667/jowua.2016.12.31.001 dblp:journals/jowua/CorreiaMPBA16 fatcat:qq6irsu7rzffrcc6iet4umpqka

Segments in enumerating faces

Katta G. Murty, Sung-Jin Chung
1995 Mathematical programming  
The role of segments 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) .  ... 
doi:10.1007/bf01585927 fatcat:pjspc6hjergvbfbjoc6l3ddvnm

Locally optimal 2-periodic sphere packings [article]

Alexei Andreanov, Yoav Kallus
2019 arXiv   pre-print
We implement this computation 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.  ... 
arXiv:1704.08156v2 fatcat:d4eeemx3obafzn7yazfhxrydwi

A problem in enumerating extreme points, and an efficient algorithm for one class of polytopes

Katta G. Murty
2008 Optimization Letters  
Here we describe the central problem 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.  ... 
doi:10.1007/s11590-008-0103-8 fatcat:nobo7bnzobbyner2kb4e5r3pyq

Orthogonalized lattice enumeration for solving SVP

Zhongxiang Zheng, Xiaoyun Wang, Guangwu Xu, Yang Yu
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.  ... 
doi:10.1007/s11432-017-9307-0 fatcat:cgq6hkdksreo7gpveptoewe46a

The Bichromatic Rectangle Problem in High Dimensions

Jonathan Backer, J. Mark Keil
2009 Canadian Conference on Computational Geometry  
Our algorithm 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.  ... 
dblp:conf/cccg/BackerK09 fatcat:zwz6iyfaxzfxzkrl5wyfn7yyli

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.  ... 

Comparison of the upper bounds for the extreme points of the polytopes of line-stochastic tensors [article]

Fuzhen Zhang, Xiao-Dong Zhang
2021 arXiv   pre-print
As all these approaches are worthy of consideration and investigation 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  ... 
arXiv:2110.12337v2 fatcat:r6ikftgttjhsno47kzge72b6kq

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.  ... 

Weight enumeration of codes from finite spaces

Relinde P. M. J. Jurrius
2011 Designs, Codes and Cryptography  
For our calculations we use that these codes correspond to a projective system containing all the 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  ... 
doi:10.1007/s10623-011-9557-2 fatcat:hmihrf6q4bfbjnaf6ykgsb26pu

Parallel Shortest Lattice Vector Enumeration on Graphics Cards [chapter]

Jens Hermans, Michael Schneider, Johannes Buchmann, Frederik Vercauteren, Bart Preneel
2010 Lecture Notes in Computer Science  
Our implementation is almost 5 times faster 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.  ... 
doi:10.1007/978-3-642-12678-9_4 fatcat:ewdokfhpmnakfk7rr5f2lchdxa

BKZ 2.0: Better Lattice Security Estimates [chapter]

Yuanmi Chen, Phong Q. Nguyen
2011 Lecture Notes in Computer Science  
The best lattice reduction algorithm known 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  ... 
doi:10.1007/978-3-642-25385-0_1 fatcat:zdtum4wadvailaeptm2a6bat6e

Fast Lattice Point Enumeration with Minimal Overhead [chapter]

Daniele Micciancio, Michael Walter
2014 Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms  
The algorithms typically used 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.  ... 
doi:10.1137/1.9781611973730.21 dblp:conf/soda/MicciancioW15 fatcat:wf3jvwa47falzk4nuabsayq2yi
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