Discrete & Computational Geometry

, Volume 6, Issue 3, pp 423–434

Small-dimensional linear programming and convex hulls made easy

  • Raimund Seidel


We present two randomized algorithms. One solves linear programs involvingm constraints ind variables in expected timeO(m). The other constructs convex hulls ofn points in ℝd,d>3, in expected timeO(n[d/2]). In both boundsd is considered to be a constant. In the linear programming algorithm the dependence of the time bound ond is of the formd!. The main virtue of our results lies in the utter simplicity of the algorithms as well as their analyses.


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  1. [AS] C. R. Aragon and R. G. Seidel, Randomized Search Trees,Proc. 30th IEEE Symp. on Foundations of Computer Science (1989), pp. 540–545.Google Scholar
  2. [CK] D. R. Chand and S. S. Kapur, An Algorithm for Convex Polytopes,J. Assoc. Comput. Mach. 17 (1970), 78–86.MathSciNetCrossRefMATHGoogle Scholar
  3. [C1] K. L. Clarkson, Linear Programming inO(n3d2) Time,Inform. Process. Lett. 22 (1986), 21–24.MathSciNetCrossRefGoogle Scholar
  4. [C2] K. L. Clarkson, Las Vegas Algorithms for Linear and Integer Programming when the Dimension is Small, Manuscript(Oct. 1989): a preliminary version appeared inProc. 29th IEEE Symp. on Foundations of Computer Science (1988), pp. 452–456.Google Scholar
  5. [CS] K. L. Clarkson and P. W. Shor, Applications of Random Sampling in Computational Geometry, II,Discrete Comput. Geom. 4 (1989), 387–422.MathSciNetCrossRefMATHGoogle Scholar
  6. [D1] M. E. Dyer, Linear Algorithms for Two- and Three-Variable Linear Programs,SIAM J. Comput. 13 (1984), 31–45.MathSciNetCrossRefMATHGoogle Scholar
  7. [D2] M. E. Dyer, On a Multidimensional Search Technique and Its Applications to the Euclidean One-Centre Problem,SIAM J. Comput. 15 (1986), 725–738.MathSciNetCrossRefMATHGoogle Scholar
  8. [DF] M. E. Dyer and A. M. Frieze, A Randomized Algorithm for Fixed-Dimensional Linear Programming,Math. Programming 44 (1989), 203–212.MathSciNetCrossRefMATHGoogle Scholar
  9. [E] H. Edelsbrunner,Algorithms in Combinatorial Geometry, Springer-Verlag, New York (1987).CrossRefMATHGoogle Scholar
  10. [G] R. L. Graham, An Efficient Algorithm for Constructing the Convex Hull of a Finite Planar Set,Inform. Process. Lett. 1 (1972), 132–133.CrossRefMATHGoogle Scholar
  11. [K] M. Kallay, Convex Hull Algorithms in Higher Dimensions, Manuscript (1981).Google Scholar
  12. [Mc] P. McMullen, The Maximum Number of Faces of a Convex Polytope,Mathematika 17 (1971), 179–184.MathSciNetCrossRefGoogle Scholar
  13. [M1] N. Megiddo, Linear-Time Algorithms for Linear Programming in ℝ3 and Related Problems,SIAM J. Comput. 12 (1983), 759–776.MathSciNetCrossRefMATHGoogle Scholar
  14. [M2] N. Megiddo, Linear Programming in Linear Time when the Dimension is Fixed,J. Assoc. Comput. Mach. 31 (1984), 114–127.MathSciNetCrossRefMATHGoogle Scholar
  15. [PH] F. P. Preparata and S. J. Hong, Convex Hulls of Finite Point Sets in Two and Three Dimensions,Comm. ACM 20 (1977), 87–93.MathSciNetCrossRefMATHGoogle Scholar
  16. [S1] R. Seidel, A Convex Hull Algorithm Optimal for Point Sets in Even Dimensions, Technical Report 81-14, Department of Computer Science, University of British Columbia (1981).Google Scholar
  17. [S2] R. Seidel, Constructing Higher-Dimensional Convex Hulls at Logarithmic Cost per Face,Proc. 18th ACM Symp. on Theory of Computing (1986), pp. 404–413.Google Scholar
  18. [S3] R. Seidel, Backwards Analysis of Randomized Geometric Algorithms (Manuscript).Google Scholar
  19. [Sw] G. Swart, Finding the Convex Hull Facet by Facet,J. Algorithms 6 (1985), 17–48.MathSciNetCrossRefMATHGoogle Scholar
  20. [T] R. E. Tarjan,Data Structures and Network Algorithm, Society for Industrial and Applied Mathematics, Philadelphia, PA (1983).CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc 1991

Authors and Affiliations

  • Raimund Seidel
    • 1
  1. 1.Computer Science DivisionUniversity of California at BerkeleyBerkeleyUSA

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