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The Clarkson-Shor technique revisited and extended

Micha Sharir
2001 Proceedings of the seventeenth annual symposium on Computational geometry - SCG '01  
that copies bear this notice and the full citation on the first page.  ...  To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.  ...  Triangles and crossing segments in a 3dimensional point set  ... 
doi:10.1145/378583.378680 dblp:conf/compgeom/Sharir01 fatcat:h4zqv3n6xng4lb7ilhdiqjdnny

The Clarkson–Shor Technique Revisited and Extended

MICHA SHARIR
2003 Combinatorics, probability & computing  
We provide an alternative, simpler and more general derivation of the Clarkson-Shor probabilistic technique [7] and use it to obtain several extensions and new combinatorial bounds. †  ...  Finally, thanks are extended to an anonymous referee for careful reading of the paper and helpful comments.  ...  Acknowledgements The author wishes to thank Sariel Har-Peled, Boris Aronov, Jirka Matousek, János Pach, Shakhar Smorodinsky and Emo Welzl for useful discussions concerning the problems studied in this  ... 
doi:10.1017/s0963548302005527 fatcat:3hkc5opwlrcazijwwdjqyge7ba

Page 2311 of Mathematical Reviews Vol. , Issue 2004c [page]

2004 Mathematical Reviews  
Mutafchiev (BG-AOS-IMI,; Sofia) 2004c:60023 60C05 05A05 Sharir, Micha (IL-TLAV-SC; Tel Aviv) The Clarkson-Shor technique revisited and extended. (English summary) Combin. Probab.  ...  The present paper provides an al- ternative extension of Clarkson-Shor technique [K. L. Clarkson and P. W. Shor, Discrete Comput.  ... 

Two Proofs for Shallow Packings

Kunal Dutta, Esther Ezra, Arijit Ghosh
2016 Discrete & Computational Geometry  
We present two proofs, the first is an extension of Haussler's approach, and the second extends the proof of Chazelle, originally presented as a simplification for Haussler's proof.  ...  The distinction between primal and dual set systems in geometry is essential, and set systems of both kinds appear in numerous geometric applications, see, once again [Hp11] and the references therein.  ...  The second author wishes to thank Boris Aronov, Sariel Har-Peled, Aryeh Kontorovich, and Wolfgang Mulzer for useful discussions and suggestions.  ... 
doi:10.1007/s00454-016-9824-0 fatcat:mxmfjebuefdmxlybmgd3pjsdnu

Shallow Packings in Geometry [article]

Esther Ezra
2014 arXiv   pre-print
We refine this bound for the scenario where, for any subset, X' ⊆ X of size m < n and for any parameter 1 < k < m, the number of vectors of length at most k in the restriction of to X' is only O(m^d_1  ...  k^d-d_1), for a fixed integer d > 0 and a real parameter 1 < d_1 < d (this generalizes the standard notion of bounded primal shatter dimension when d_1 = d).  ...  The authors wishes to thank Boris Aronov, Sariel Har-Peled, Aryeh Kontorovich, and Wolfgang Mulzer for useful discussions and suggestions.  ... 
arXiv:1412.5215v1 fatcat:oc24vbipmvafdm4c6eykppo4sa

Epsilon-Nets for Halfspaces Revisited [article]

Sariel Har-Peled and Haim Kaplan and Micha Sharir and Shakhar Smorodinsky
2014 arXiv   pre-print
We also consider several related variants of this result, including the case of points and pseudo-disks in the plane.  ...  This follows from Euler's formula for the 0-level (as in the first proof), combined with the random sampling technique of Clarkson and Shor [CS89] (or, alternatively, the simpler scheme of Tagansky  ...  As shown by Clarkson and Shor [CS89] , the overall number of vertices of A(P * ) at level at most 3k is O(nk 2 ).  ... 
arXiv:1410.3154v1 fatcat:5xbfyack7vdd5niibc2vf44gya

Shortest Path in a Polygon using Sublinear Space [article]

Sariel Har-Peled
2015 arXiv   pre-print
Specifically, given a simple polygon with n vertices in a read only memory, and additional working memory of size , the new algorithm computes the shortest path (in ) in O( n^2 / ) expected time.  ...  ]#1(#2) We resolve an open problem due to Tetsuo Asano, showing how to compute the shortest path in a polygon, given in a read only memory, using sublinear space and subquadratic time.  ...  The author thanks him for the talk, and the subsequent discussions. The author also thanks Pankaj Agarwal, Chandra Chekuri, Jeff Erickson and Bernd Gärtner for useful discussions.  ... 
arXiv:1412.0779v2 fatcat:3mlsshmuybbzhiypk2unsjbnou

On Constant Factors in Comparison-Based Geometric Algorithms and Data Structures

Timothy M. Chan, Patrick Lee
2015 Discrete & Computational Geometry  
However, for nonorthogonal problems, the definition of a "comparison" needs to be extended to include testing of certain predicates (e.g., determining whether three points are in clockwise order), and  ...  that require nontrivial interesting techniques to find the best constant factors, not just straightforward modifications of existing algorithms.  ...  The reduction from 3D maxima to 2D point location uses Clarkson-Shor-style sampling [17] .  ... 
doi:10.1007/s00454-015-9677-y fatcat:kfgu6lq3y5citgx7d5xgj6mbze

On Constant Factors in Comparison-Based Geometric Algorithms and Data Structures

Timothy M. Chan, Patrick Lee
2014 Annual Symposium on Computational Geometry - SOCG'14  
However, for nonorthogonal problems, the definition of a "comparison" needs to be extended to include testing of certain predicates (e.g., determining whether three points are in clockwise order), and  ...  that require nontrivial interesting techniques to find the best constant factors, not just straightforward modifications of existing algorithms.  ...  The reduction from 3D maxima to 2D point location uses Clarkson-Shor-style sampling [17] .  ... 
doi:10.1145/2582112.2582166 dblp:conf/compgeom/ChanL14 fatcat:53usx2uf6jfoliwpuh6nyxkxd4

On tail estimates for Randomized Incremental Construction [article]

Sandeep Sen
2018 arXiv   pre-print
By combining several interesting applications of random sampling in geometric algorithms like point location, linear programming, segment intersections, binary space partitioning, Clarkson and Shor CS89  ...  In this paper we present some general techniques to obtain tail estimates for RIC and and provide applications to some fundamental problems like Delaunay triangulations and construction of Visibility maps  ...  Although our techniques can be applied to the latter work also, we will focus primarily on the Clarkson-Shor paradigm of a configuration space. 1 N = [x 1 , x 2 . . . x n ] : a random permutation of S.  ... 
arXiv:1808.02356v1 fatcat:rhz7im4bsvg5vkvokyzpjmqzae

Bottleneck partial-matching Voronoi diagrams and applications

Matthias Henze, Rafel Jaume
2016 Computational geometry  
Their complexity is shown to be polynomial in the sizes of the sets and algorithms for their construction are given.  ...  Given two point sets in the plane, we study the minimization of the (lexiocographic) bottleneck distance between the smaller set and an equally-sized subset of the larger set under translations.  ...  Before establishing the complexity bounds for the diagrams, we need the following technical lemma resulting from applying the Clarkson-Shor technique (see [5] ).  ... 
doi:10.1016/j.comgeo.2015.10.001 fatcat:xhwql2i2sfbwtg6yurrsreuofq

A Unified Approach to Tail Estimates for Randomized Incremental Construction

Sandeep Sen, Michael Wagner
2019 Symposium on Theoretical Aspects of Computer Science  
By combining several interesting applications of random sampling in geometric algorithms like point location, linear programming, segment intersections, binary space partitioning, Clarkson and Shor [4]  ...  In this paper we present a general technique to obtain tail estimates for RIC and and provide applications to some fundamental problems like Delaunay triangulations and construction of Visibility maps  ...  Although our techniques can be applied to the latter work also, we will focus primarily on the Clarkson-Shor incremental paradigm of a configuration space.  ... 
doi:10.4230/lipics.stacs.2019.58 dblp:conf/stacs/Sen19 fatcat:di4zukstbvdjff6zrswlzggwaq

Markov incremental constructions

Bernard Chazelle, Wolfgang Johann Heinrich Mulzer
2008 Proceedings of the twenty-fourth annual symposium on Computational geometry - SCG '08  
The main contribution of this work is threefold: (i) extending the theory of abstract configuration spaces to the Markov setting; (ii) proving Clarkson-Shor type bounds for this new model; (iii) applying  ...  We generalize Mulmuley's theory of Θ -series and prove that Markov incremental constructions with bounded spectral gap are optimal within polylog factors for trapezoidal maps, segment intersections, and  ...  We would also like to thank the anonymous referees for pointing us to additional references and providing numerous suggestions for improving the exposition of this paper.  ... 
doi:10.1145/1377676.1377701 dblp:conf/compgeom/ChazelleM08 fatcat:iazpl2ctfjbnfhya666bsg56we

Markov Incremental Constructions

Bernard Chazelle, Wolfgang Mulzer
2009 Discrete & Computational Geometry  
The main contribution of this work is threefold: (i) extending the theory of abstract configuration spaces to the Markov setting; (ii) proving Clarkson-Shor-type bounds for this new model; (iii) applying  ...  We generalize Mulmuley's theory of Θ-series and prove that Markov incremental constructions with bounded spectral gap are optimal within polylog factors for trapezoidal maps, segment intersections, and  ...  We would also like to thank the anonymous referees for pointing us to additional references and providing numerous suggestions for improving the exposition of this paper.  ... 
doi:10.1007/s00454-009-9170-6 fatcat:lgcbonywjjhf5bddgrfjbubwju

From Proximity to Utility: A Voronoi Partition of Pareto Optima [article]

Hsien-Chih Chang, Sariel Har-Peled, Benjamin Raichel
2016 arXiv   pre-print
To do so we revisit the classical backward analysis technique, both simplifying and improving relevant results in order to achieve the high-probability bounds.  ...  In particular, the precise site a client might use from this candidate set depends on parameters that might change between usages, and the candidate set lists all of the relevant sites.  ...  Acknowledgments The authors would like to thank Pankaj Agarwal, Ken Clarkson, Nirman Kumar, and Raimund Seidel for useful discussions related to this work.  ... 
arXiv:1404.3403v3 fatcat:b727gxjdbjhfrllm5rytc55upm
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