Certified Approximation Algorithms for the Fermat Point and n-Ellipses

Authors Kolja Junginger, Ioannis Mantas , Evanthia Papadopoulou , Martin Suderland , Chee Yap



PDF
Thumbnail PDF

File

LIPIcs.ESA.2021.54.pdf
  • Filesize: 4.02 MB
  • 19 pages

Document Identifiers

Author Details

Kolja Junginger
  • Faculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland
Ioannis Mantas
  • Faculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland
Evanthia Papadopoulou
  • Faculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland
Martin Suderland
  • Faculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland
Chee Yap
  • Courant Institute, New York University, NY, USA

Cite AsGet BibTex

Kolja Junginger, Ioannis Mantas, Evanthia Papadopoulou, Martin Suderland, and Chee Yap. Certified Approximation Algorithms for the Fermat Point and n-Ellipses. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 54:1-54:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ESA.2021.54

Abstract

Given a set A of n points in ℝ^d with weight function w: A→ℝ_{> 0}, the Fermat distance function is φ(x): = ∑_{a∈A}w(a)‖x-a‖. A classic problem in facility location dating back to 1643, is to find the Fermat point x*, the point that minimizes the function φ. We consider the problem of computing a point x̃* that is an ε-approximation of x* in the sense that ‖x̃*-x*‖<ε. The algorithmic literature has so far used a different notion based on ε-approximation of the value φ(x*). We devise a certified subdivision algorithm for computing x̃*, enhanced by Newton operator techniques. We also revisit the classic Weiszfeld-Kuhn iteration scheme for x*, turning it into an ε-approximate Fermat point algorithm. Our second problem is the certified construction of ε-isotopic approximations of n-ellipses. These are the level sets φ^{-1}(r) for r > φ(x*) and d = 2. Finally, all our planar (d = 2) algorithms are implemented in order to experimentally evaluate them, using both synthetic as well as real world datasets. These experiments show the practicality of our techniques.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Fermat point
  • n-ellipse
  • subdivision
  • approximation
  • certified
  • algorithms

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. A. Karim Abu-Affash and Matthew J. Katz. Improved bounds on the average distance to the Fermat-Weber center of a convex object. Information Processing Letters, 109(6):329-333, 2009. Google Scholar
  2. Mihai Badoiu, Sariel Har-Peled, and Piotr Indyk. Approximate clustering via core-sets. In Proc. Symposium on Theory of Computing, pages 250-257. ACM, 2002. URL: https://doi.org/10.1145/509907.509947.
  3. Chanderjit Bajaj. The algebraic degree of geometric optimization problems. Discrete & Computational Geometry, 3(2):177-191, 1988. Google Scholar
  4. Amir Beck and Shoham Sabach. Weiszfeld’s method: Old and new results. Journal of Optimization Theory and Applications, 164(1):1-40, 2015. Google Scholar
  5. Huck Bennett, Evanthia Papadopoulou, and Chee Yap. Planar minimization diagrams via subdivision with applications to anisotropic Voronoi diagrams. Computer Graphics Forum, 35(5):229-247, 2016. URL: https://doi.org/10.1111/cgf.12979.
  6. Huck Bennett and Chee Yap. Amortized analysis of smooth quadtrees in all dimensions. Computational Geometry, 63:20-39, 2017. URL: https://doi.org/10.1016/j.comgeo.2017.02.001.
  7. Bhaswar B. Bhattacharya. On the Fermat-Weber point of a polygonal chain and its generalizations. Fundamenta Informaticae, 107(4):331-343, 2011. Google Scholar
  8. Prosenjit Bose, Anil Maheshwari, and Pat Morin. Fast approximations for sums of distances, clustering and the Fermat-Weber problem. Computational Geometry, 24(3):135-146, 2003. Google Scholar
  9. Luitzen Egbertus Jan Brouwer. Über Abbildung von Mannigfaltigkeiten. Mathematische Annalen, 71(1):97-115, 1911. Google Scholar
  10. Michael Burr, Felix Krahmer, and Chee Yap. Continuous amortization: A non-probabilistic adaptive analysis technique. Electronic Colloquium on Computational Complexity, TR09(136), 2009. Google Scholar
  11. Michael A. Burr. Continuous amortization and extensions: With applications to bisection-based root isolation. Journal of Symbolic Computation, 77:78-126, 2016. URL: https://doi.org/10.1016/j.jsc.2016.01.007.
  12. Paz Carmi, Sariel Har-Peled, and Matthew J. Katz. On the Fermat-Weber center of a convex object. Computational Geometry, 32(3):188-195, 2005. URL: https://doi.org/10.1016/j.comgeo.2005.01.002.
  13. Hui Han Chin, Aleksander Madry, Gary L. Miller, and Richard Peng. Runtime guarantees for regression problems. In Proc. Innovations in Theoretical Computer Science, pages 269-282. ACM, 2013. Google Scholar
  14. Dietmar Cieslik. Steiner minimal trees, volume 23. Springer Science & Business Media, 2013. Google Scholar
  15. Ernest J. Cockayne and Zdzislaw A. Melzak. Euclidean constructibility in graph-minimization problems. Mathematics Magazine, 42(4):206-208, 1969. Google Scholar
  16. Michael B. Cohen, Yin Tat Lee, Gary L. Miller, Jakub Pachocki, and Aaron Sidford. Geometric median in nearly linear time. In Proc. Symposium on Theory of Computing, pages 9-21. ACM, 2016. URL: https://doi.org/10.1145/2897518.2897647.
  17. Theodorus Jozef Dekker. Finding a zero by means of successive linear interpolation. In Constructive Aspects of the Fundamental Theorem of Algebra, pages 37-48. Wiley Interscience, 1967. Google Scholar
  18. Adrian Dumitrescu, Minghui Jiang, and Csaba D. Tóth. New bounds on the average distance from the Fermat-Weber center of a planar convex body. Discrete Optimization, 8(3):417-427, 2011. URL: https://doi.org/10.1016/j.disopt.2011.02.004.
  19. Ioannis Z. Emiris, Elias P. Tsigaridas, and George M. Tzoumas. The predicates for the Voronoi diagram of ellipses. In Proc. Symposium on Computational Geometry, pages 227-236. ACM, 2006. Google Scholar
  20. Giovanni Francesco Fagnano. Problemata quaedam ad methodum maximorum et minimorum spectantia. Nova Acta Eruditorum, pages 281-303, 1775. Google Scholar
  21. Sándor P. Fekete, Joseph S.B. Mitchell, and Karin Beurer. On the continuous Fermat-Weber problem. Operations Research, 53(1):61-76, 2005. Google Scholar
  22. Dan Feldman and Michael Langberg. A unified framework for approximating and clustering data. In Proc. Symposium on Theory of Computing, pages 569-578. ACM, 2011. Google Scholar
  23. Horst Hamacher and Zvi Drezner. Facility location: applications and theory. Science & Business Media: Springer, 2002. Google Scholar
  24. Eldon R. Hansen. A multidimensional interval newton method. Reliable Computing, 12(4):253-272, 2006. URL: https://doi.org/10.1007/s11155-006-9000-y.
  25. Eldon R. Hansen and Saumyendra Sengupta. Bounding solutions of systems of equations using interval analysis. BIT, 21:203-211, 1981. Google Scholar
  26. Sariel Har-Peled and Akash Kushal. Smaller coresets for k-median and k-means clustering. Discrete & Computational Geometry, 37(1):3-19, 2007. Google Scholar
  27. Sariel Har-Peled and Soham Mazumdar. On coresets for k-means and k-median clustering. In Proc. 36th Annual ACM Symposium on Theory of computing, pages 291-300. ACM, 2004. Google Scholar
  28. Daniel P. Huttenlocher, Klara Kedem, and Micha Sharir. The upper envelope of Voronoi surfaces and its applications. Discrete & Computational Geometry, 9(3):267-291, 1993. Google Scholar
  29. I. Norman Katz. Local convergence in Fermat’s problem. Mathematical Programming, 6(1):89-104, 1974. Google Scholar
  30. Jakob Krarup and Steven Vajda. On Torricelli’s geometrical solution to a problem of Fermat. Journal of Management Mathematics, 8(3):215-224, 1997. Google Scholar
  31. Rudolf Krawczyk. Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing, 4(3):187-201, 1969. URL: https://doi.org/10.1007/BF02234767.
  32. Harold W. Kuhn. A note on Fermat’s problem. Mathematical programming, 4(1):98-107, 1973. Google Scholar
  33. Long Lin and Chee Yap. Adaptive isotopic approximation of nonsingular curves: the parameterizability and nonlocal isotopy approach. Discrete & Computational Geometry, 45(4):760-795, 2011. URL: https://doi.org/10.1007/s00454-011-9345-9.
  34. Long Lin, Chee Yap, and Jihun Yu. Non-local isotopic approximation of nonsingular surfaces. Computer-Aided Design, 45(2):451-462, 2012. Google Scholar
  35. Luis Fernando Mello and Lucas Ruiz dos Santos. On the location of the minimum point in the Euclidean distance sum problem. São Paulo Journal of Mathematical Sciences, 12:108-120, 2018. Google Scholar
  36. Ramon E. Moore. Interval Analysis, volume 4. Prentice-Hall Englewood Cliffs, NJ, 1966. Google Scholar
  37. Kent E Morrison. The fedex problem. The College Mathematics Journal, 41(3):222-232, 2010. Google Scholar
  38. Gyula Sz Nagy. Tschirnhaus’sche Eiflächen und Eikurven. Acta Mathematica Academiae Scientiarum Hungarica, 1(1):36-45, 1950. Google Scholar
  39. Nguyen Mau Nam. The Fermat-Torricelli problem in the light of convex analysis. ArXiv e-prints, 2013. URL: http://arxiv.org/abs/1302.5244v3.
  40. Karl Nickel. Triplex-algol and applications. Interner Bericht des Instituts für Informatik der Universität Karlsruhe, 1969. Google Scholar
  41. Karl Nickel. On the Newton method in interval analysis. Technical report, Wisconsin University-Madison Mathematics Research Center, 1971. Google Scholar
  42. Jiawang Nie, Pablo A. Parrilo, and Bernd Sturmfels. Semidefinite representation of the k-ellipse. In Algorithms in algebraic geometry, pages 117-132. Springer, 2008. Google Scholar
  43. Lawrence M. Ostresh Jr. Convergence and descent in the Fermat location problem. Transportation Science, 12(2):153-164, 1978. Google Scholar
  44. Evanthia Papadopoulou. The Hausdorff Voronoi diagram of point clusters in the plane. Algorithmica, 40(2):63-82, 2004. Google Scholar
  45. Pablo A. Parrilo and Bernd Sturmfels. Minimizing polynomial functions. Algorithmic and quantitative real algebraic geometry, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 60:83-99, 2003. Google Scholar
  46. Maja Petrović, Bojan Banjac, and Branko Malešević. The geometry of trifocal curves with applications in architecture, urban and spatial planning. Spatium, pages 28-33, 2014. Google Scholar
  47. Simon Plantinga and Gert Vegter. Isotopic approximation of implicit curves and surfaces. In Proc. of the Eurographics/ACM SIGGRAPH Symposium on Geometry Processing, pages 245-254. ACM, 2004. Google Scholar
  48. Helmut Ratschek and Jon Rokne. Computer methods for the range of functions. Horwood, 1984. Google Scholar
  49. Gerhard Reinelt. TSPLIB - A traveling salesman problem library. ORSA Journal on Computing, 3(4):376-384, 1991. Google Scholar
  50. Hanan Samet. The Design and Analysis of Spatial Data Structures. Addison-Wesley, 1990. Google Scholar
  51. Junpei Sekino. n-ellipses and the minimum distance sum problem. The American mathematical monthly, 106(3):193-202, 1999. Google Scholar
  52. Sergey P Shary. Krawczyk operator revised. Novosibirsk, Institute of Computational Technologies, Rússia, 2004. Google Scholar
  53. Rudolf Sturm. Über den Punkt kleinster Entfernungssumme von gegebenen Punkten. Journal für die reine und angewandte Mathematik, 97:49-61, 1884. Google Scholar
  54. Warwick Tucker. Validated Numerics: A short intro to rigorous computations. Princeton Press, 2011. Google Scholar
  55. Ehrenfried Walther von Tschirnhaus. Medicina Mentis Et Corporis. Fritsch, Lipsiae, 1695. URL: http://mdz-nbn-resolving.de/urn:nbn:de:bvb:12-bsb10008248-3.
  56. Cong Wang, Yi-Jen Chiang, and Chee Yap. On soft predicates in subdivision motion planning. Computational Geometry: Theory and Applications., 48(8):589-605, 2015. Google Scholar
  57. Alfred Weber. Über den Standort der Industrien. English translation by CJ Friedrich (1929) Theory of the Location of Industries, 1909. Google Scholar
  58. Endre Weiszfeld. Sur le point pour lequel la somme des distances de n points donnés est minimum. Tohoku Mathematical Journal, First Series, 43:355-386, 1937. Google Scholar
  59. Juan Xu and Chee Yap. Effective subdivision algorithm for isolating zeros of real systems of equations, with complexity analysis. In Proc. International Symposium on Symbolic and Algebraic Computation, pages 399-406. ACM, 2019. Google Scholar
  60. Guoliang Xue and Yinyu Ye. An efficient algorithm for minimizing a sum of Euclidean norms with applications. SIAM Journal on Optimization, 7(4):1017-1036, 1997. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail