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Algorithmica

, 3:191 | Cite as

Two probabilistic results on rectilinear steiner trees

  • Marshall W. Bern
Article

Abstract

In recent years, researchers have proven many theorems of the following form: given points distributed according to a Poisson process with intensity parameterN on the unit square, the length of the shortest spanning tree (rectilinear Steiner tree, traveling salesman tour, or some other functional) on these points is, with probability one, asymptotic to β√N for some constant β (which is presumably different for different functionals). Though these theorems are well understood, very little is known about the constants β. In this paper we prove that the constants in the cases of rectilinear spanning tree and rectilinear Steiner tree do, indeed, differ. This proof is constructive in the sense that we give a polynomial-time heuristic algorithm that produces a Steiner tree of expected length some fraction shorter than a minimum spanning tree. We continue the analysis to prove a second result: the expected value of the minimum number of Steiner points in a shortest rectilinear Steiner tree grows linearly withN.

Key words

Spanning tree Steiner tree Heuristic algorithm Geometric probability Optimization Graph 

References

  1. [1]
    A. V. Aho, M. R. Garey, and F. K. Hwang, Rectilinear Steiner trees: efficient special case algorithms,Networks,7 (1977), 37–58.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    M. W. Bern and M. de Carvalho. A greedy heuristic for the rectilinear Steiner tree problem, Technical Report UCB/CSD 87/306, University of California at Berkeley, 1986.Google Scholar
  3. [3]
    F. R. K. Chung and R. L. Graham, On Steiner trees for bounded point sets,Geom. Dedicata,11 (1981), 353–361.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    F. R. K. Chung and F. K. Hwang, The largest minimal rectilinear Steiner trees for a set ofn points enclosed in a rectangle with given perimeter,Networks,9 (1979), 19–36.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    M. R. Garey and D. S. Johnson, The rectilinear Steiner tree problem is NP-complete,SIAM J. Appl. Math.,32 (1977), 826–834.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    M. Haimovich and A. H. G. Rinnooy Kan, Personal communication with A. H. G. Rinnooy Kan.Google Scholar
  7. [7]
    F. K. Hwang, On Steiner minimal trees with rectilinear distance,SIAM J. Appl. Math.,30 (1976), 104–114.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    F. K. Hwang, The rectilinear Steiner problem,Design Automat. Fault-Tolerant Comput.,2 (1978), 303–310.MathSciNetGoogle Scholar
  9. [9]
    F. K. Hwang, AnO(N logN) algorithm for suboptimal rectilinear Steiner trees,IEEE Trans. Circuits and Systems,26 (1979), 75–77.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    R. M. Karp and J. M. Steele, Probabilistic analysis of heuristics, inThe Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization (E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, and D. B. Shmoys, eds.), Wiley, New York, 1985, Chapter 6.Google Scholar
  11. [11]
    J. Komlós and M. T. Shing, Probabilistic partitioning algorithms for the rectilinear Steiner problem,Networks,15 (1985), 413–424.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    J. B. Kruskal, On the shortest spanning subtree of a graph and the traveling salesman problem,Proc. Amer. Math. Soc.,7 (1956), 48–50.CrossRefMathSciNetGoogle Scholar
  13. [13]
    E. L. Lawler,Combinatorial Optimization: Networks and Matroids, Holt, Rinehart, and Winston, New York, 1976, p. 290.zbMATHGoogle Scholar
  14. [14]
    J. H. Lee, N. K. Bose, and F. K. Hwang, Use of Steiner's problem in suboptimal routing in rectilinear metric,IEEE Trans. Circuits and Systems,23 (1976), 470–476.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    J. MacGregor-Smith and J. S. Liebman, Steiner trees, Steiner circuits, and the interference problem in building design,Engrg. Optim.,4 (1979), 15–36.CrossRefGoogle Scholar
  16. [16]
    J. MacGregor-Smith, D. T. Lee, and J. S. Liebman, AnO(N logN) heuristic algorithm for the rectilinear Steiner problem,Engrg. Optim.,4 (1980), 179–192.CrossRefGoogle Scholar
  17. [17]
    A. Ng, Personal communication, University of California at Berkeley.Google Scholar
  18. [18]
    C. H. Papadimitriou, The probabilistic analysis of matching heuristics,Proceedings of the 15th Annual Allerton Conference on Communication, Control, and Computing, 1977, pp. 368–378.Google Scholar
  19. [19]
    H. R. Pitt,Tauberian Theorems, Oxford University Press, Oxford, 1958, pp. 37–42.zbMATHGoogle Scholar
  20. [20]
    R. C. Prim, Shortest connecting networks and some generalizations,Bell System Tech. J.,36 (1957), 1389–1401.Google Scholar
  21. [21]
    M. Servit, Heuristic algorithms for rectilinear Steiner trees,Digital Process.,7 (1981), 21–32.zbMATHGoogle Scholar
  22. [22]
    J. M. Steele, Subadditive Euclidean functionals and nonlinear growth in geometric probability,Ann. Probab.,9 (1982), 365–376.CrossRefMathSciNetGoogle Scholar
  23. [23]
    C. D. Thomborson, Personal communications, University of Minnesota at Duluth.Google Scholar
  24. [24]
    Y. Y. Yang and O. Wing, Suboptimal algorithm for a wire routing problem,IEEE Trans. Circuit Theory,19 (1972), 508–511.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • Marshall W. Bern
    • 1
  1. 1.Computer Science DivisionUniversity of CaliforniaBerkeleyUSA

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