Skip to main content
SpringerLink
Log in

Algorithmic techniques for geometric optimization

  • Chapter
  • First Online:
Computer Science Today

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1000))

Abstract

We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. The emphasis in this survey is on the techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, prune-and-search techniques for linear programming and related problems, and LP-type problems and their efficient solution.

Pankaj Agarwal has been supported by National Science Foundation Grant CCR-93-01259, an NYI award, and by matching funds from Xerox Corp. Micha Sharir has been supported by NSF Grants CCR-91-22103 and CCR-93-11127, by a Max-Planck Research Award, and by grants from the U.S.-Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. P. Agarwal, B. Aronov, and M. Sharir, Computing lower envelopes in four dimensions with applications, Proc. 10th ACM Symp. Comput. Geom., 1994, pp. 348–358.

    Google Scholar 

  2. P. Agarwal, B. Aronov, M. Sharir, and S. Suri, Selecting distances in the plane, Algorithmica 9 (1993), 495–514.

    Article  Google Scholar 

  3. P. Agarwal, A. Efrat, and M. Sharir, Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications, Proc. 11th ACM Symp. Comput. Geom., 1995, 39–50.

    Google Scholar 

  4. P. Agarwal, A. Efrat, M. Sharir, and S. Toledo, Computing a segment-center for a planar point set, J. Algorithms 15 (1993), 314–323.

    Article  Google Scholar 

  5. P. Agarwal and J. Matoušek, Ray shooting and parametric search, SIAM J. Comput. 22 (1993), 794–806.

    Article  Google Scholar 

  6. P. Agarwal and J. Matoušek, Range searching with semialgebraic sets, Discr. Comput. Geom. 11 (1994), 393–418.

    Google Scholar 

  7. P. Agarwal and J. Matoušek, Dynamic half-space range searching and its applications, Algorithmica 14 (1995), 325–345.

    Article  Google Scholar 

  8. P. Agarwal and M. Sharir, Off line dynamic maintenance of the width of a planar point set, Comput. Geom. Theory Appls. 1 (1991), 65–78.

    Google Scholar 

  9. P. Agarwal and M. Sharir, Ray shooting among convex polytopes in three dimensions, SIAM J. Comput., to appear.

    Google Scholar 

  10. P. Agarwal and M. Sharir, Planar geometric location problems, Algorithmica 11 (1994), 185–195.

    Article  Google Scholar 

  11. P. Agarwal and M. Sharir, Efficient randomized algorithms for some geometric optimization problems, Proc. 11th ACM Symp. Comput. Geom., 1995, pp. 326–335.

    Google Scholar 

  12. P. Agarwal, M. Sharir, and S. Toledo, New applications of parametric searching in computational geometry, J. Algorithms 17 (1994), 292–318.

    Article  Google Scholar 

  13. P. Agarwal, M. Sharir, and S. Toledo, An efficient multi-dimensional searching technique and its applications, Tech. Rept. CS-1993-20, Dept. Comp. Sci., Duke University, 1993.

    Google Scholar 

  14. R. Agarwala and D. Fernández-Baca, Weighted multidimensional search and its applications to convex optimization, SIAM J. Comput., to appear.

    Google Scholar 

  15. M. Ajtai, J. Komlós, and E. Szemerédi, Sorting in c log n parallel steps, Combinatorica, 3 (1983), 1–19.

    Google Scholar 

  16. M. Ajtai and N. Megiddo, A deterministic poly(loglog N)-time N-processor algorithm for linear programming in fixed dimension, Proc. 24th ACM Symp. Theory of Comput., 1992, pp. 327–338.

    Google Scholar 

  17. H. Alt, B. Behrends, and J. Blömer, Approximate matching of polygonal shapes, Proc. 7th ACM Symp. Comput. Geom., 1991, pp. 186–193.

    Google Scholar 

  18. N. Amenta, Finding a line transversal of axial objects in three dimensions, Proc. 3rd ACM-SIAM Symp. Discr. Algo., 1992, pp. 66–71.

    Google Scholar 

  19. N. Amenta, Helly-type theorems and generalized linear programming, Discr. Comput. Geom. 12 (1994), 241–261.

    Google Scholar 

  20. N. Amenta, Bounded boxes, Hausdorff distance, and a new proof of an interesting Helly-type theorem, Proc. 10th ACM Symp. Comput. Geom., 1994, pp. 340–347.

    Google Scholar 

  21. R. Bar-Yehuda, A. Efrat, and A. Itai, A simple algorithm for maintaining the center of a planar point-set, Proc. 5th Canad. Conf. Comput. Geom., 1993, pp. 252–257.

    Google Scholar 

  22. H. Brönnimann and B. Chazelle, Optimal slope selection via cuttings, Proc. 6th Canad. Conf. Comput. Geom., 1994, pp. 99–103.

    Google Scholar 

  23. H. Brönnimann, B. Chazelle, and J. Matoušek, Product range spaces, sensitive sampling, and derandomization, Proc. 34th IEEE Symp. Found. of Comp. Sci., 1993, pp. 400–409.

    Google Scholar 

  24. B. Chazelle, H. Edelsbrunner, L. Guibas, and M. Sharir, Algorithms for bichromatic line segment problems and polyhedral terrains, Algorithmica 11 (1994), 116–132.

    Article  Google Scholar 

  25. B. Chazelle, H. Edelsbrunner, L. Guibas, and M. Sharir, Diameter, width, closest line pair, and parametric searching, Discr. Comput. Geom. 10 (1993), 183–196.

    Google Scholar 

  26. B. Chazelle and J. Matoušek, On linear-time deterministic algorithms for optimization problems in fixed dimensions, Proc. 4th ACM-SIAM Symp. Discr. Algo., 1993, pp. 281–290.

    Google Scholar 

  27. P. Chew, D. Dor, A. Efrat, and K. Kedem, Geometric pattern matching in d-dimensional space, Proc. 3rd European Symp. Algo., 1995, to appear.

    Google Scholar 

  28. P. Chew and K. Kedem, Improvements on geometric pattern matching problems, Proc. 3rd Scand. Work. Algo. Theory, 1992, pp. 318–325.

    Google Scholar 

  29. P. Chew and K. Kedem, A convex polygon among polygonal obstacles: Placement and high-clearance motion, Comput. Geom. Theory Appls. 3 (1993), 59–89.

    Google Scholar 

  30. K. Clarkson, Linear Programming in O(n · \(3^{d^2 }\)) time, Inform. Process. Lett. 22 (1986), 21–24.

    Article  Google Scholar 

  31. K. Clarkson, Las Vegas algorithms for linear and integer programming when the dimension is small, J. ACM 45 (1995), 488–499.

    Article  Google Scholar 

  32. K. Clarkson, D. Eppstein, G. Miller, C. Sturtivant, and S.-H. Teng, Approximating center points with iterated Radon points, Proc. 9th ACM Symp. Comput. Geom., 1993, pp. 91–98.

    Google Scholar 

  33. K. Clarkson and P. Shor, Applications of random sampling in computational geometry, II, Discr. Comput. Geom. 4 (1989), 387–421.

    Article  Google Scholar 

  34. E. Cohen and N. Megiddo, Maximizing concave functions in fixed dimension, in Complexity in Numeric Computation (P. Pardalos, ed.), World Scientific, 1993.

    Google Scholar 

  35. R. Cole, Slowing down sorting networks to obtain faster sorting algorithms, J. ACM 31 (1984), 200–208.

    Google Scholar 

  36. R. Cole, J. Salowe, W. Steiger, and E. Szemerédi, Optimal slope selection, SIAM J. Comput. 18 (1989), 792–810.

    Article  Google Scholar 

  37. R. Cole, M. Sharir, and C. Yap, On k-hulls and related problems, SIAM J. Comput. 16 (1987), 61–77.

    Article  Google Scholar 

  38. G. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decomposition, in Second GI Conf. on Automata Theory and Formal Languages, Lecture Notes in Comp. Sci., 33, Springer-Verlag, 1975, pp. 134–183.

    Google Scholar 

  39. Z. Drezner, On the rectangular p-center problem, Naval Res. Logist. Quart., 34 (1987), 229–234.

    Google Scholar 

  40. Z. Drezner, The P-center problem: Heuristics and optimal algorithms, J. Oper. Res. Soc. 35 (1984), 741–748.

    Google Scholar 

  41. M. Dyer, On a multidimensional search technique and its application to the Euclidean one-center problem, SIAM J. Comput. 15 (1986), 725–738.

    Article  Google Scholar 

  42. M. Dyer, A class of convex programs with applications to computational geometry, Proc. 8th ACM Symp. Comput. Geom., 1992, pp. 9–15.

    Google Scholar 

  43. M. Dyer and A. Frieze, A randomized algorithm for fixed-dimensional linear programming, Math. Prog. 44 (1989), 203–212.

    Article  Google Scholar 

  44. H. Ebara, N. Fukuyama, H. Nakano and Y. Nakanishi, Roundness algorithms using the Voronoi diagrams, First Canad. Conf. Comput. Geom., 1989.

    Google Scholar 

  45. H. Edelsbrunner, Algorithms in Combinatorial Geometry, Springer-Verlag, Heidelberg, 1987.

    Google Scholar 

  46. H. Edelsbrunner and R. Waupotitsch, Computing a ham-sandwich cut in two dimensions, J. Symb. Comput. 2 (1986), 171–178.

    Google Scholar 

  47. A. Efrat and M. Sharir, A near-linear algorithm for the planar segment center problem, Discr. Comput. Geom., to appear.

    Google Scholar 

  48. A. Efrat, M. Sharir and A. Ziv, Computing the smallest k-enclosing circle and related problems, Comput. Geom. Theory Appls. 4 (1994), 119–136.

    Google Scholar 

  49. M. Eisner and D. Severance, Mathematical techniques for efficient record segmentation in large shared databases, JACM 23 (1976), 619–635.

    Article  Google Scholar 

  50. G. Frederickson, Optimal algorithms for tree partitioning, Proc. 2nd ACM-SIAM Symp. Discr. Algo., 1991, pp. 168–177.

    Google Scholar 

  51. G. Frederickson and D. Johnson, The complexity of selection and ranking in X+Y and matrices with sorted columns, J. Comp. Syst. Sci. 24 (1982), 197–208.

    Article  Google Scholar 

  52. G. Frederickson and D. Johnson, Finding the k-th shortest paths and p-centers by generating and searching good data structures, J. Algorithms 4 (1983), 61–80.

    Article  Google Scholar 

  53. G. Frederickson and D. Johnson, Generalized selection and ranking: sorted matrices, SIAM J. Comput. 13 (1984), 14–30.

    Article  Google Scholar 

  54. B. Gärtner, A subexponential algorithm for abstract optimization problems, Proc. 33rd IEEE Symp. Found. of Comp. Sci., 1992, pp. 464–72.

    Google Scholar 

  55. A. Glozman, K. Kedem, and G. Shpitalnik, On some geometric selection and optimization problems via sorted matrices, Proc. 4th Workshop Algo. and Data Struct., 1995, to appear.

    Google Scholar 

  56. M. Goodrich, Geometric partitioning made easier, even in parallel, Proc. 9th ACM Symp. Comput. Geom., 1993, pp. 73–82.

    Google Scholar 

  57. D. Gusfield, Sensitivity analysis for combinatorial optimization, Tech. Rept. UCB/ERLM80/22, Univ. of California, Berkeley, 1980.

    Google Scholar 

  58. D. Gusfield, Parametric combinatorial computing and a problem in program module allocation, JACM 30 (1983), 551–563.

    Article  Google Scholar 

  59. D. Gusfield, K. Balasubramanian, and D. Naor, Parametric optimization of sequence alignment, Algorithmic 12 (1994), 312–326.

    Article  Google Scholar 

  60. E. Helly, über systeme von abgeschlossenen Mengen mit gemeinschaftlichen Punkten, Monaths. Math. und Physik 37 (1930), 281–302.

    Article  Google Scholar 

  61. J. Hershberger, A faster algorithm for the two-center decision problem, Inform. Process. Lett. 47 (1993), 23–29.

    Article  Google Scholar 

  62. M. Houle and G. Toussaint, Computing the width of a set, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-10 (1988), 761–765.

    Article  Google Scholar 

  63. D. Huttenlocher and K. Kedem, Efficiently computing the Hausdorff distance for point sets under translation, Proc. 6th ACM Symp. Comput. Geom., 1990, pp. 340–349.

    Google Scholar 

  64. H. Imai, D.T. Lee, and C. Yang, 1-Segment center covering problems, ORSA J. Comput. 4 (1992), 426–434.

    MathSciNet  Google Scholar 

  65. S. Jadhav and A. Mukhopadhyay, Computing a centerpoint of a finite planar set of points in linear time, Proc. 9th ACM Symp. Comput. Geom., 1993, pp. 83–90.

    Google Scholar 

  66. J. Jaromczyk and M. Kowaluk, An efficient algorithm for the Euclidean two-center problem, Proc. 10th ACM Symp. Comput. Geom., 1994, pp. 303–311.

    Google Scholar 

  67. J. Jaromczyk and M. Kowaluk, The two-line center problem from a polar view: A new algorithm, Proc. 4th Workshop Algo. Data Struct., 1995.

    Google Scholar 

  68. G. Kalai, A subexponential randomized simplex algorithm, Proc. 24th ACM Symp. Theory of Comput., 1992, pp. 475–482.

    Google Scholar 

  69. N. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica 4 (1984), 373–395.

    Google Scholar 

  70. M. Katz, Improved algorithms in geometric optimization via expanders, Proc. 3rd Israeli Symp. Theory of Comput. & Syst., 1995.

    Google Scholar 

  71. M. Katz and M. Sharir, Optimal slope selection via expanders, Inform. Process. Lett. 47 (1993), 115–122.

    Article  Google Scholar 

  72. M. Katz and M. Sharir, An expander-based approach to geometric optimization, Proc. 9th ACM Symp. Comput. Geom., 1993, pp. 198–207.

    Google Scholar 

  73. L.G. Khachiyan, Polynomial algorithm in linear programming, U.S.S.R. Comput. Math. and Math. Phys. 20 (1980), 53–72.

    Article  Google Scholar 

  74. M. Ko and R. Lee, On weighted rectilinear 2-center and 3-center problems, Info. Sci. 54 (1991), 169–190.

    Article  Google Scholar 

  75. N. Korneenko and H. Martini, Hyperplane approximation and related topics, in New Trends in Discrete and Computational Geometry, (J. Pach, ed.), Springer-Verlag, 1993, 163–198.

    Google Scholar 

  76. C.-Y. Lo, J. Matoušek, and W. Steiger, Algorithms for ham-sandwich cuts, Discr. Comput. Geom. 11 (1994), 433–452.

    Google Scholar 

  77. C.-Y. Lo and W. Steiger, An optimal-time algorithm for ham-sandwich cuts in the plane, Proc. 2nd Canad. Conf. Comput. Geom., 1990, pp. 5–9.

    Google Scholar 

  78. D. Knuth, The Art of Computer Programming, Vol. 3: Sorting and Searching, Addison-Wesley, Reading, MA, 1973.

    Google Scholar 

  79. J. Matoušek, Randomized optimal algorithm for slope selection, Inform. Process. Lett. 39 (1991), 183–187.

    Article  Google Scholar 

  80. J. Matoušek, Computing the center of planar point sets, in Computational Geometry: Papers from the DIMACS Special Year (J. Goodman, et al., eds.), AMS, Providence, RI, 1991, 221–230.

    Google Scholar 

  81. J. Matoušek, Linear optimization queries, J. Algorithms 14 (1993), 432–448.

    Article  Google Scholar 

  82. J. Matoušek, Lower bounds for a subexponential optimization algorithm, Random Struct. & Algo. 5 (1994), 591–607.

    Google Scholar 

  83. J. Matoušek, On geometric optimization with few violated constraints, Proc. 10th ACM Symp. Comput. Geom., 1994, pp. 312–321.

    Google Scholar 

  84. J. Matoušek, D. Mount, and N. Netanyahu, Efficient randomized algorithms for the repeated median line estimator, Proc. 4th ACM-SIAM Symp. Discr. Algo., 1993, pp. 74–82.

    Google Scholar 

  85. J. Matoušek and O. Schwarzkopf, A deterministic algorithm for the three-dimensional diameter problem, Proc. 25th ACM Symp. Theory of Comput., 1993, pp. 478–484.

    Google Scholar 

  86. J. Matoušek, M. Sharir, and E. Welzl, A subexponential bound for linear programming and related problems, Algorithmica, to appear.

    Google Scholar 

  87. N. Megiddo, Combinatorial optimization with rational objective functions, Math. Oper. Res. 4 (1979), 414–424.

    Google Scholar 

  88. N. Megiddo, Applying parallel computation algorithms in the design of serial algorithms, J. ACM 30 (1983), 852–865.

    Article  Google Scholar 

  89. N. Megiddo, Linear time algorithms for linear time programming in R 3 and related problems, SIAM J. Comput. 12 (1983), 759–776.

    Article  Google Scholar 

  90. N. Megiddo, Linear programming in linear time when the dimension is fixed, J. ACM 31 (1984), 114–127.

    Article  Google Scholar 

  91. N. Megiddo, Partitioning with two lines in the plane, J. Algorithms 6 (1985), 403–433.

    Article  Google Scholar 

  92. N. Megiddo, The weighted Euclidean 1-center problem, Math. Oper. Res. 8 (1983), 498–504.

    Google Scholar 

  93. N. Megiddo, On the ball spanned by balls, Discr. Comput. Geom. 4 (1989), 605–610.

    Article  Google Scholar 

  94. N. Megiddo and K. Supowit, On the complexity of some common geometric location problems, SIAM J. Comput. 13 (1984), 182–196.

    Article  Google Scholar 

  95. R. Motwani and P. Raghavan, Randomized Algorithms, Cambridge University Press, New York, 1995.

    Google Scholar 

  96. K. Mulmuley, Computational Geometry: An Introduction Through Randomized Algorithms, Prentice Hall, New York, 1993.

    Google Scholar 

  97. N. Naor and M. Sharir, Computing the center of a point set in three dimensions, Proc. 2nd Canad. Conf. Comput. Geom., 1990, pp. 10–13.

    Google Scholar 

  98. C. Norton, S. Plotkin, and E. Tárdos, Using separation algorithms in fixed dimensions, J. Algorithms 13 (1992), 79–98.

    Article  Google Scholar 

  99. M. Pellegrini, Incidence and nearest-neighbor problems for lines in 3-space, Proc. 8th ACM Symp. Comput. Geom., 1992, pp. 130–137.

    Google Scholar 

  100. M. Pellegrini, Repetitive hidden-surface-removal for polyhedral scenes, Proc. 3rd Workshop Algo. Data Struct., 1993, pp. 541–552.

    Google Scholar 

  101. M. Post, Minimum spanning ellipsoids, Proc. 16th ACM Symp. Theory of Comput, 1984, pp. 108–116.

    Google Scholar 

  102. J. Salowe, L ∞-interdistance selection by parametric search, Inform. Process. Lett. 30 (1989), 9–14.

    Article  Google Scholar 

  103. R. Seidel, Low dimensional linear programming and convex hulls made easy, Discr. Comput. Geom. 6 (1991), 423–434.

    Google Scholar 

  104. E. Ramos, Intersection of unit-balls and diameter of a point set in ℝ3, manuscript, 1993.

    Google Scholar 

  105. M. Reichling, On the detection of a common intersection of k convex objects in the plane, Inform. Process. Lett., 29 (1988), 25–29.

    Article  Google Scholar 

  106. M. Reichling, On the detection of a common intersection of k convex polyhedra, in Comput. Geom. & its Appl., Lecture Notes in Comp. Sci., 333, Springer-Verlag, 1988, pp. 180–186.

    Google Scholar 

  107. M. Sharir, A near-linear algorithm for the planar 2-center problem, submitted to Discr. Comput. Geom.

    Google Scholar 

  108. M. Sharir and P. Agarwal, Davenport-Schinzel Sequences and Their Geometric Applications, Cambridge University Press, Cambridge-New York-Melbourne, 1995.

    Google Scholar 

  109. M. Sharir and S. Toledo, Extremal polygon containment problems, Comput. Geom. Theory Appls. 4 (1994), 99–118.

    Google Scholar 

  110. M. Sharir and E. Welzl, A combinatorial bound for linear programming and related problems, Proc. 9th Symp. Theo. Asp. Comp. Sci., (1992), pp. 569–579.

    Google Scholar 

  111. A. Stein and M. Werman, Finding the repeated median regression line, Proc. 3rd ACM-SIAM Symp. Discr. Algo., 1992, pp. 409–413.

    Google Scholar 

  112. S. Toledo, Extremal Polygon Containment Problems and Other Issues in Parametric Searching, M.Sc. Thesis, Tel Aviv University, Tel Aviv, 1991.

    Google Scholar 

  113. S. Toledo, Approximate parametric search, Inform. Process. Lett. 47 (1993), 1–4.

    Article  Google Scholar 

  114. S. Toledo, Maximizing non-linear convex functions in fixed dimensions, in Complexity of Numerical Computations (P. Pardalos, ed.), World Scientific, 1993.

    Google Scholar 

  115. L. Valiant, Parallelism in comparison problems, SIAM J. Comput. 4 (1975), 348–355.

    Article  Google Scholar 

  116. K. Vardarajan and P. Agarwal, Linear approximation of simple objects, Proc. 7th Canad. Conf. Comput. Geom., 1995, to appear.

    Google Scholar 

  117. E. Welzl, Smallest enclosing disks (balls and ellipsoids), in New Results and New Trends in Computer Science (H. Maurer, ed.), Lecture Notes in Comp. Sci., 555, 1991, Springer-Verlag, pp. 359–370.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Duke University, Box 90129, 27708-0129, Durham, NC, USA

    Pankaj K. Agarwal

  2. School of Mathematical Sciences, Tel Aviv University, 69978, Tel Aviv, Israel

    Micha Sharir

  3. Courant Institute of Mathematical Sciences, New York University, 10012, New York, NY, USA

    Micha Sharir

Authors
  1. Pankaj K. Agarwal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Micha Sharir
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Jan van Leeuwen

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Agarwal, P.K., Sharir, M. (1995). Algorithmic techniques for geometric optimization. In: van Leeuwen, J. (eds) Computer Science Today. Lecture Notes in Computer Science, vol 1000. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0015247

Download citation

  • DOI: https://doi.org/10.1007/BFb0015247

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-60105-0

  • Online ISBN: 978-3-540-49435-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation