Mathematical systems theory

, Volume 24, Issue 1, pp 201–220 | Cite as

Selection and sorting in totally monotone arrays

  • Dina Kravets
  • James K. Park
Article

Abstract

A two-dimensional arrayA={a[i, j]} is calledtotally monotone if, for alli1<i2 andj1<j2,a[i1,j1]<a[i1,j2] impliesa[i2,j1]<a[i2,j2]. Totally monotone arrays were introduced in 1987 by Aggarwal, Klawe, Moran, Shor, and Wilber, who showed that several problems in computational geometry and VLSI river routing could be reduced to the problem of finding a maximum entry in each row of a totally monotone array. In this paper we consider several selection and sorting problems involving totally monotone arrays and give a number of applications of solutions for these problems. In particular, we obtain the following results for anm × n totally monotone arrayA:
  1. 1.

    Thek largest (ork smallest) entries in each row ofA can be computed inO(k(m + n)) time. This result allows us to determine thek farthest (ork nearest) neighbors of each vertex of a convexn-gon inO(kn) time.

     
  2. 2.

    Provided the transpose ofA is also totally monotone, thek largest (ork smallest) entries overall inA can be computed inO(m + n + k lg(mn/k)) time. This result allows us to find thek farthest (ork nearest) pairs of vertices from a convexn-gon inO(n + k lg(n2/k)) time.

     
  3. 3.

    The rows ofA can be sorted inO(mn) time whenmn and inO(mn(1 + lg(n/m))) time whenm < n. This result allows us to solve the of Ω(S) on the number of combinations of row permutations possible for a totally monotone array would imply an Ω(lgS) lower bound on the time necessary to sort the array's rows in a linear decision tree model.)

     
  4. 4.

    In Subsection 4.2 we applied our algorithm for sorting the rows of a totally monotone array to the neighbor-ranking problem for the vertices of a convex polygonP. We then extended this technique to arbitrary point sets. It remains open whether our two selection algorithms for totally monotone arrays, which we also applied to the vertices of a convex polygon, can likewise be applied to arbitrary point sets.

     

Keywords

Decision Tree Computational Mathematic Tree Model Selection Algorithm Arbitrary Point 

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References

  1. [AALM]
    A. Apostolico, M. J. Atallah, L. L. Larmore, and H. S. McFaddin. Efficient parallel algorithms for string editing and related problems.SIAM Journal on Computing, 19(5):968–988, 1990.Google Scholar
  2. [AGSS]
    A. Aggarwal, L. J. Guibas, J. Saxe, and P. W. Shor. A linear-time algorithm for computing the Voronoi diagram of a convex polygon.Discrete and Computational Geometry, 4(6):591–604, 1989.Google Scholar
  3. [AKM+]
    A. Aggarwal, M. M. Klawe, S. Moran, P. Shor, and R. Wilber. Geometric applications of a matrix-searching algorithm.Algorithmica, 2(2):195–208, 1987.Google Scholar
  4. [AK]
    A. Aggarwal and D. Kravets. A linear-time algorithm for finding all farthest neighbors in a convex polygon.Information Processing Letters, 31(1):17–20, 1989.Google Scholar
  5. [AP1]
    A. Aggarwal and J. Park. Parallel searching in multidimensional monotone arrays. Research Report RC 14826, IBM T. J. Watson Research Center, August 1989. Submitted toJournal of Algorithms. Portions of this paper appeared inProceedings of the 29th Annual IEEE Symposium on Foundations of Computer Science, pages 497–512, 1988.Google Scholar
  6. [AP2]
    A. Aggarwal and J. Park. Sequential searching in multidimensional monotone arrays. Research Report RC 15128, IBM T. J. Watson Research Center, November 1989. Submitted toJournal of Algorithms. Portions of this paper appeared inProceedings of the 29th Annual IEEE Symposium on Foundations of Computer Science, pages 497–512, 1988.Google Scholar
  7. [A]
    M. J. Atallah. A faster parallel algorithm for a matrix searching problem. In G. Goos and J. Hartmanis, editors,Proceedings of the 2nd Scandinavian Workshop on Algorithm Theory, pages 192–200, Springer-Verlag, New York, 1990. Submitted toAlgorithmica.Google Scholar
  8. [BFP+]
    M. Blum, R. W. Floyd, V. Pratt, R. L. Rivest, and R. E. Tarjan. Time bounds for selection.Journal of Computer and System Sciences, 7(4):448–461, 1973.Google Scholar
  9. [C1]
    B. M. Chazelle. On the convex layers of a planar set.IEEE Transactions on Information Theory, 31(4):509–517, 1985.Google Scholar
  10. [C2]
    B. M. Chazelle. Some techniques for geometric searching with implicit set representations.Acta Informatica, 24(5):565–582, 1987.Google Scholar
  11. [Ed]
    H. Edelsbrunner. Edge skeletons in arrangements with applications.Algorithmica, 1(1):93–109, 1986.Google Scholar
  12. [Ep]
    D. Eppstein. Sequence comparison with mixed convex and concave costs.Journal of Algorithms, 11(1):85–101, 1990.Google Scholar
  13. [FJ]
    G. N. Frederickson and D. B. Johnson. The complexity of selection and ranking inX + Y and matrices with sorted columns.Journal of Computer and System Sciences, 24(4):197–208, 1982.Google Scholar
  14. [F]
    M. L. Fredman. How good is the information theory bound in sorting?Theoretical Computer Science, 1:355–361, 1976.Google Scholar
  15. [GP]
    Z. Galil and K. Park. A linear-time algorithm for concave one-dimensional dynamic programming.Information Processing Letters, 33(6):309–311, 1990.Google Scholar
  16. [G]
    L. J. Guibas. Personal communication (via Alok Aggarwal), 1988.Google Scholar
  17. [Kl]
    M. M. Klawe. A simple linear-time algorithm for concave one-dimensional dynamic programming. Technical Report 89-16, University of British Columbia, Vancouver, 1989.Google Scholar
  18. [Kn]
    D. E. Knuth.The Art of Computer Programming, Vol. 3. Addison-Wesley, Reading, MA, 1973.Google Scholar
  19. [Kr]
    D. Kravets. Finding farthest neighbors in a convex polygon and related problems. Master's thesis, Massachusetts Institute of Technology, December 1988. Published as Technical Report MIT/LCS/TR-437, Laboratory for Computer Science, Massachusetts Institute of Technology, January, 1989.Google Scholar
  20. [L]
    J.-L. Lambert. Sorting the sums (x i +y j) inO(n 2) comparisons. InProceedings of the 7th Annual Symposium on Theoretical Aspects of Computer Science, pages 195–206, 1990.Google Scholar
  21. [LS]
    L. L. Larmore and B. Schieber. On-line dynamic programming with applications to the prediction of RNA secondary structure.Journal of Algorithms, 1991. To appear. An earlier version of this paper appeared inProceedings of the 1st Annual ACM-SIAM Symposium on Discrete Algorithms, pages 503–512, 1990.Google Scholar
  22. [LP]
    D. T. Lee and F. P. Preparata. The all nearest-neighbor problem for convex polygons.Information Processing Letters, 7(4):189–192, 1978.Google Scholar
  23. [MSS]
    Y. Mansour, B. Schieber, and S. Sen. Personal communication (via Yishay Mansour), 1989.Google Scholar
  24. [W]
    R. Wilber. The concave least-weight subsequence problem revisited.Journal of Algorithms, 9(3):418–425, 1988.Google Scholar
  25. [YL]
    C.C. Yang and D. T. Lee. A note on the all nearest-neighbor problem for convex polygons.Information Processing Letters, 8(4):193–194, 1979.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1991

Authors and Affiliations

  • Dina Kravets
    • 1
  • James K. Park
    • 1
  1. 1.Laboratory for Computer ScienceMassachusetts Institute of TechnologyCambridgeUSA

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