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Computing Klee's Measure of Grounded Boxes

Hakan Yıldız, Subhash Suri
2013 Algorithmica  
Our main result is an O(n (d−1)/2 log 2 n) time algorithm for computing Klee's measure for a set of n 2-grounded boxes.  ...  A well-known problem in computational geometry is Klee's measure problem, which asks for the volume of a union of axis-aligned boxes in d-space.  ...  the d-dimensional Klee's Measure Problem on 2-grounded boxes.  ... 
doi:10.1007/s00453-013-9797-9 fatcat:cs43h6lmdfezjiicjr4a35l3ti

On Klee's measure problem for grounded boxes

Hakan Yildiz, Subhash Suri
2012 Proceedings of the 2012 symposuim on Computational Geometry - SoCG '12  
Our main result is an O(n (d−1)/2 log 2 n) time algorithm for computing Klee's measure for a set of n 2-grounded boxes.  ...  A well-known problem in computational geometry is Klee's measure problem, which asks for the volume of a union of axis-aligned boxes in d-space.  ...  Klee's Measure Problem on 2-grounded boxes.  ... 
doi:10.1145/2261250.2261267 dblp:conf/compgeom/YildizS12 fatcat:pntxih3c2rgqri3kdxgo5lwf6m

Adaptive Computation of the Klee's Measure in High Dimensions [article]

Jérémy Barbay, Pablo Pérez-Lantero, Javiel Rojas-Ledesma
2015 arXiv   pre-print
The KLEE'S MESURE of n axis-parallel boxes in R^d is the volume of their union. It can be computed in time within O(n^d/2) in the worst case.  ...  We describe three techniques to boost its computation: one based on some type of "degeneracy" of the input, and two ones on the inherent "easiness" of the structure of the input.  ...  We thank Timothy Chan for his helpful comments, and one anonymous referee from ESA 2015 for pointing out the relation between our techniques of analysis and the treewidth of the intersection graph.  ... 
arXiv:1505.02855v2 fatcat:lpg4qriqnrddpkb2dm7ocmsamm

Bringing Order to Special Cases of Klee's Measure Problem [article]

Karl Bringmann
2013 arXiv   pre-print
Klee's Measure Problem (KMP) asks for the volume of the union of n axis-aligned boxes in d-space. Omitting logarithmic factors, the best algorithm has runtime O*(n^d/2) [Overmars,Yap'91].  ...  There are faster algorithms known for several special cases: Cube-KMP (where all boxes are cubes), Unitcube-KMP (where all boxes are cubes of equal side length), Hypervolume (where all boxes share a vertex  ...  Thus, we established some order among the special cases of Klee's measure problem.  ... 
arXiv:1301.7154v2 fatcat:h3qdi6rzczhzhpq7ub6pt76vli

Bringing Order to Special Cases of Klee's Measure Problem [chapter]

Karl Bringmann
2013 Lecture Notes in Computer Science  
Klee's Measure Problem (KMP) asks for the volume of the union of n axis-aligned boxes in R d . Omitting logarithmic factors, the best algorithm has runtime O * (n d/2 ) [Overmars,Yap'91].  ...  There are faster algorithms known for several special cases: Cube-KMP (where all boxes are cubes), Unitcube-KMP (where all boxes are cubes of equal side length), Hypervolume (where all boxes share a vertex  ...  Introduction Klee's measure problem (KMP) asks for the volume of the union of n axisaligned boxes in R d , where d is considered to be a constant.  ... 
doi:10.1007/978-3-642-40313-2_20 fatcat:xedlkxowwffvljfpfhtw7b5fi4

Klee's Measure Problem Made Easy

Timothy M. Chan
2013 2013 IEEE 54th Annual Symposium on Foundations of Computer Science  
We present a new algorithm for a classic problem in computational geometry, Klee's measure problem: given a set of n axis-parallel boxes in d-dimensional space, compute the volume of the union of the boxes  ...  O(n (d+1)/3 polylog n)-time algorithm for the case of arbitrary hypercubes or fat boxes, improving a previous O(n (d+2)/3 )-time algorithm  ...  Introduction Klee's measure problem is, without exaggeration, easy to state: Given a set B of n axis-parallel boxes (hyperrectangles) in IR d , compute the volume of the union of B.  ... 
doi:10.1109/focs.2013.51 dblp:conf/focs/Chan13 fatcat:c5st2mw2mbcglpxhoatw2aymzu

How to Make Analogies in a Digital Age

Whitney Davis
2006 October  
David Marr, Vision: A Computational Investigation into the Human Representation and Processing of Visual Cornell University Press, 2005)-this debate has yet to be joined, even though it must be by far  ...  the most important nexus for art history and criticism to address at the advent of the digital age.  ...  To be sure, and to modify Goodman's statement slightly, in the recent past "systems of such mongrel type seldom survive[d] long in computer practice" due in large measure to "the length of the message"  ... 
doi:10.1162/octo.2006.117.1.71 fatcat:znvef4upnndulprjdjft5hlhpq

The Hypervolume Indicator: Problems and Algorithms [article]

Andreia P. Guerreiro, Carlos M. Fonseca, Luís Paquete
2020 arXiv   pre-print
This paper discusses the computation of hypervolume-related problems, highlighting the relations between them, providing an overview of the paradigms and techniques used, a description of the main algorithms  ...  By providing a complete overview of the computational problems associated to the hypervolume indicator, this paper serves as the starting point for the development of new algorithms, and supports users  ...  The problem of computing the hypervolume indicator is known to be a special case of Klee's Measure Problem (KMP) [8] , which is the problem of measuring the region resulting from the union of axis-parallel  ... 
arXiv:2005.00515v1 fatcat:luy532oue5dh3csicpo42v2rpe

Design theory at Bauhaus: teaching "splitting" knowledge

Pascal Le Masson, Armand Hatchuel, Benoit Weil
2015 Research in Engineering Design  
to the structures of knowledge provided by teaching.  ...  We analyse teaching at Bauhaus by focusing on the courses of Itten and Klee.  ...  for instance, it still computes).  ... 
doi:10.1007/s00163-015-0206-z fatcat:n5tnqdoolvawte5q7e7jsposay

The Hypervolume Indicator

Andreia P. Guerreiro, Carlos M. Fonseca, Luís Paquete
2021 ACM Computing Surveys  
This article discusses the computation of hypervolume-related problems, highlighting the relations between them, providing an overview of the paradigms and techniques used, a description of the main algorithms  ...  By providing a complete overview of the computational problems associated to the hypervolume indicator, this article serves as the starting point for the development of new algorithms and supports users  ...  The problem of computing the hypervolume indicator is known to be a special case of Klee's Measure Problem (KMP) [8] , which is the problem of measuring the region resulting from the union of axis-parallel  ... 
doi:10.1145/3453474 fatcat:gkqz7wzplvhl3kit6nuhtd4ycu

December Variations (On A Theme By Earle Brown)

Richard Hoadley
2014 Proceedings of the SMC Conferences  
These are relative measurements only. The data are scaled between SuperCollider, INScore and the relevant computer screen; the scale of the original image is less relevant.  ...  used in visual and graphic arts, for instance Paul Klee's 1932 abstract painting Polyphony.  ... 
doi:10.5281/zenodo.850478 fatcat:dhghlajfanci5f6giedvixeslu

Walter Benjamin's Dionysian Adventures on Google Earth

Paul Kingsbury, John Paul Jones
2009 Geoforum  
empowermentwe undersell its capacities as an alluring digital peep-box, an uncertain orb spangled with vertiginous paranoia, frenzied navigation, jubilatory dissolution, and intoxicating giddiness.  ...  To do this, we examine Google Earth as a "digital peep-box" with an online collective that revels in its "Spot the Black Helicopter" competitions; illuminated profanities; alien and giant insect invaders  ...  One of his prized possessions was Paul Klee's painting, Angelus Novus (Fig. 8) .  ... 
doi:10.1016/j.geoforum.2008.10.002 fatcat:7o5xv424v5c6jhlqpqkend6pha

Joins via Geometric Resolutions: Worst-case and Beyond [article]

Mahmoud Abo Khamis, Hung Q. Ngo, Christopher Ré, Atri Rudra
2016 arXiv   pre-print
A key idea in our framework is formalizing the inference one does with an index as a type of geometric resolution; transforming the algorithmic problem of computing joins to a geometric problem.  ...  Our notion of geometric resolution can be viewed as a geometric analog of logical resolution.  ...  Acknowledgments We thank Paul Beame for clarifying the relation of our notion of geometric resolution with general resolution and we thank Javiel Rojas-Ledesma for bringing Klee's measure problem to our  ... 
arXiv:1404.0703v7 fatcat:g57exkfr7fekffnfwsx77e2xsm

Advanced statistical methods for eye movement analysis and modeling: a gentle introduction [article]

Giuseppe Boccignone
2017 arXiv   pre-print
After a brief, though critical, probabilistic tour of current computational models of eye movements and visual attention, we lay down the basis for gaze shift pattern analysis.  ...  Eventually, by resorting to machine learning techniques, we discuss how the analyses of movement patterns can develop into the inference of hidden patterns of the mind: inferring the observer's task, assessing  ...  We assume the task "look first at the left eye, then at the right eye" of Klee's drawing presented in Fig. 17 .  ... 
arXiv:1506.07194v4 fatcat:2lpo66aqqjhbfpi3hrm4szv75q

SAVIOR: Towards Bug-Driven Hybrid Testing

Yaohui Chen, Peng Li, Jun Xu, Shengjian Guo, Rundong Zhou, Yulong Zhang, Tao Wei, Long Lu
2020 2020 IEEE Symposium on Security and Privacy (SP)  
However, as statistics show, a large portion of the explored code is often bug-free. Therefore, giving equal attention to every part of the code during hybrid testing is a non-optimal strategy.  ...  It slows down the detection of real vulnerabilities by over 43%. Second, classic hybrid testing quickly moves on after reaching a chunk of code, rather than examining the hidden defects inside.  ...  Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the funding agency.  ... 
doi:10.1109/sp40000.2020.00002 dblp:conf/sp/ChenLXGZZWL20 fatcat:5kieeedzrzbetkohcjwwv3abqy
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