Algorithmica

, Volume 7, Issue 1–6, pp 137–177 | Cite as

How to find Steiner minimal trees in euclideand-space

  • Warren D. Smith
Article

Abstract

This paper has two purposes. The first is to present a new way to find a Steiner minimum tree (SMT) connectingN sites ind-space,d >- 2. We present (in Appendix 1) a computer code for this purpose. This is the only procedure known to the author for finding Steiner minimal trees ind-space ford > 2, and also the first one which fits naturally into the framework of “backtracking” and “branch-and-bound.” Finding SMTs of up toN = 12 general sites ind-space (for anyd) now appears feasible.

We tabulate Steiner minimal trees for many point sets, including the vertices of most of the regular and Archimedeand-polytopes with <- 16 vertices. As a consequence of these tables, the Gilbert-Pollak conjecture is shown to be false in dimensions 3–9. (The conjecture remains open in other dimensions; it is probably false in all dimensionsd withd ≥ 3, but it is probably true whend = 2.)

The second purpose is to present some new theoretical results regarding the asymptotic computational complexity of finding SMTs to precision ɛ.

We show that in two-dimensions, Steiner minimum trees may be found exactly in exponential time O(CN) on a real RAM. (All previous provable time bounds were superexponential.) If the tree is only wanted to precision ɛ, then there is an (N/ɛ)O(√N)-time algorithm, which is subexponential if 1/ɛ grows only polynomially withN. Also, therectilinear Steiner minimal tree ofN points in the plane may be found inNO(√N) time.

J. S. Provan devised an O(N64)-time algorithm for finding the SMT of a convexN-point set in the plane. (Also the rectilinear SMT of such a set may be found in O(N6) time.) One therefore suspects that this problem may be solved exactly in polynomial time. We show that this suspicion is in fact true—if a certain conjecture about the size of “Steiner sensitivity diagrams” is correct.

All of these algorithms are for a “real RAM” model of computation allowing infinite precision arithmetic. They make no probabilistic or other assumptions about the input; the time bounds are valid in the worst case; and all our algorithms may be implemented with a polynomial amount of space. Only algorithms yielding theexact optimum SMT, or trees with lengths ≤ (1 + ɛ) × optimum, where ɛ is arbitrarily small, are considered here.

Key words

Steiner trees Gilbert-Pollak conjecture Subexponential algorithms Regular polytopes Sensitivity diagrams 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    M. Ajtai, V. Chvatal, M. M. Newborn, and E. Szemeredi: Crossing free subgraphs,Ann. Discrete Math.,12 (1982), 9–12.MATHMathSciNetGoogle Scholar
  3. [3]
    S. K. Chang: The generation of minimal trees with a Steiner topology,J. Assoc. Comput. Math.,19 (1972), 699–711.MATHGoogle Scholar
  4. [4]
    F. R. K. Chung and E. N. Gilbert: Steiner trees for the regular simplex,Bull. Inst. Math. Acad. Sinica,4 (1976), 313–325.MATHMathSciNetGoogle Scholar
  5. [5]
    F. R. K. Chung and R. L. Graham: A new bound for Euclidean Steiner trees,Ann. N. Y. Acad. Sci,440 (1985), 328–346.CrossRefMathSciNetGoogle Scholar
  6. [6]
    E. J. Cockayne: On Fermat's problem on the surface of a sphere,Math. Mag.,45 (1972), 216–219.MATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    E. J. Cockayne and D. E. Hewgill: Exact computation of Steiner minimal trees in the plane,Inform. Process. Lett.,22 (1986), 151–156.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    U. Derigs: A shortest augmenting path method for solving minimum perfect matching problems,Networks,11 (1981), 379–390.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    D. Z. Du: On Steiner ratio conjectures, manuscript, Inst. Appl. Math. Academia Sinica, Beijing, China, 1989.Google Scholar
  10. [10]
    D. Z. Du: The Steiner ratio conjecture is true for six points, to be published.Google Scholar
  11. [11]
    D. Z. Du, F. K. Hwang, and J. F. Weng: Steiner minimal trees for regular polygons,Discrete Comput. Geom.,2, 1 (1987), 65–87.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    H. Edelsbrunner:Algorithms in Combinatorial Geometry, EATCS Monographs in Theoretical Computer Science, Springer-Verlag, Berlin, 1987.Google Scholar
  13. [13]
    J. H. Friedman, J. L. Bentley, and R. A. Finkel: An algorithm for performing best matches in logarithmic expected time,ACMTOMS,3, 3 (1977), 209–226.MATHGoogle Scholar
  14. [14]
    M. R. Garey and D. S. Johnson: The complexity of computing Steiner minimum trees,SIAM J. Algebraic Discrete Methods,32 (1977), 835–859.MATHMathSciNetGoogle Scholar
  15. [15]
    M. R. Garey and D. S. Johnson: The rectilinear Steiner minimum tree problem is NP-completeSIAM J. Algebraic Discrete Methods,32 (1977), 826–834.MATHMathSciNetGoogle Scholar
  16. [16]
    E. Gekeler: On the solution of systems of equations by the epsilon algorithm of Wynn,Math. Comp.,26, 118 (1972), 427–436.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    G. Georgakopoulos and C. H. Papadimitriou: The 1-Steiner tree problem,J. Algorithms, 8 (1987), 122–130.MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    E. N. Gilbert and H. O. Pollak: Steiner minimal trees,SIAM J. Appl. Math.,16 (1968), 1–29.MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    R. L. Graham and F. K. Hwang: Remarks on Steiner minimal trees I,Bull. Inst. Math. Acad. Sinica,4 (1976), 177–182.MATHMathSciNetGoogle Scholar
  20. [20]
    M. Hanan: On Steiner's problem with rectilinear distance,SIAM J. Appl. Math.,14 (1966), 255–265.MATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    K. Heibig-Hansen and J. Krarup: Improvements of the Held-Karp algorithm for the symmetric TSP,Math. Programming,7 (1974), 87–96.CrossRefMathSciNetGoogle Scholar
  22. [22]
    M. Held and R. M. Karp: The traveling salesman problem and minimum spanning trees,Oper. Res.,18 (1970), 1138–1162.MATHMathSciNetCrossRefGoogle Scholar
  23. [23]
    M. Held and R. M. Karp: The traveling salesman problem and minimum spanning trees, part II,Math. Programming,1 (1971), 6–25.MATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    J. E. Hopcroft and R. E. Tarjan: Efficient planarity testing,J. Assoc. Comput. Math.,21 (1974), 549–558.MATHMathSciNetGoogle Scholar
  25. [25]
    F. K. Hwang: A linear time algorithm for full Steiner trees,Oper. Res. Lett.,5 (1986), 235–237.CrossRefGoogle Scholar
  26. [26]
    F. K. Hwang and D. S. Richards: Steiner tree problems, Bell Laboratories, Murray Hill, NJ, Technical Memorandum 1989. To be published inNetworks. Google Scholar
  27. [27]
    F. K. Hwang, G. D. Song, G. Y. Ting, and D. Z. Du: A decomposition theorem on Euclidean Steiner minimal trees,Discrete Comput. Geom.,3, 4 (1988), 367–392.MATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    F. K. Hwang and J. F. Weng: Hexagonal coordinate systems and Steiner minimal trees,Discrete Math.,62 (1986), 49–57.MATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    F. K. Hwang and J. F. Weng: The shortest network with a given topology, Bell Laboratories, Murray Hill, NJ, Technical Memorandum 1988.Google Scholar
  30. [30]
    A. N. C. Kang and D. A. Ault: Some properties of the centroid of a free tree,Inform. Process. Lett,4, 1 (1975), 18–20.MATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    B. Kernighan and D. Ritchie:The C Programming Language, Prentice-Hall, Englewood Cliffs, NJ, 1978.Google Scholar
  32. [32]
    J. R. Kruskal Jr.: On the shortest spanning subtree of a graph and the traveling salesman problem,Proc. Amer. Math. Soc.,7, 1 (1956), 48–50.CrossRefMathSciNetGoogle Scholar
  33. [33]
    H. W. Kuhn; On a pair of dual nonlinear programs, in:Methods of Nonlinear Programming, pp. 38–54, Ed. J. Abadie, North-Holland, Amsterdam, 1967.Google Scholar
  34. [34]
    H. W. Kuhn: A note on Fermat's problem,Math. Programming,4 (1973), 98–107.MATHCrossRefMathSciNetGoogle Scholar
  35. [35]
    E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, and D. B. Schmoys (eds.)The TST, A Guided Tour of Combinatoral Optimization, Wiley-Interscience, New York, 1985.Google Scholar
  36. [36]
    Z. A. Melzak: On the problem of Steiner,Canad. Math. Bull.,4 (1961), 143–148.MATHMathSciNetGoogle Scholar
  37. [37]
    G. L. Miller: Finding small simple cycle separators for 2-connected planar graphs,J. Comput. System Sci.,32 (1986), 265–179.MATHCrossRefMathSciNetGoogle Scholar
  38. [38]
    J. Milnor: On the Betti numbers of real algebraic varieties,Amer. Math. Soc.,15 (1964), 275–280.MATHCrossRefMathSciNetGoogle Scholar
  39. [39]
    C. Monma, M. Paterson, S. Suri, and F. Yao: Computing Euclidean maximum spanning trees, ACM Computational Geometry Conference 1988, pp. 241–251.Google Scholar
  40. [40]
    F. P. Preparata and M. I. Shamos:Computational Geometry: An Introduction, Springer-Verlag, New York, 1985.Google Scholar
  41. [41]
    R. C. Prim: Shortest Connection Networks and Some Generalizations,Bell System Tech. J.,36 (1957), 1389–1401.Google Scholar
  42. [42]
    J. S. Provan: Convexity and the Steiner tree problem,Networks,18 (1988), 55–72.MATHCrossRefMathSciNetGoogle Scholar
  43. [43]
    P. W. Shor and W. D. Smith: Steiner hulls and θ-hulls, manuscript, 1989.Google Scholar
  44. [44]
    W. D. Smith: Studies in Discrete and Computational Geometry, Ph.D. Thesis, Program in Applied and Computational Mathematics, Princeton University, September 1988.Google Scholar
  45. [45]
    D. Smith: Finding the optimum traveling salesman tour forN sites in the Euclidean plane in subexponential time and polynomial space,SIAM J. Comput., submitted.Google Scholar
  46. [46]
    J. M. Smith, D. T. Lee, and J. S. Liebman: A O(N 1gN) algorithm for Steiner minimal tree problems in the Euclidean metric,Networks,11 (1981), 23–29.MATHCrossRefMathSciNetGoogle Scholar
  47. [47]
    J. Soukup: Minimum Steiner trees, roots of a polynomial, and other magic,ACM SIGMAP Newsletter,22 (1977), 37–51.Google Scholar
  48. [48]
    R. P. Stanley: The number of faces of simplicial poly topes and spheres, in: Discrete Geometry and Convexity,Proc. N. Y. Acad. Sci., (1986).Google Scholar
  49. [49]
    R. E. Tarjan: Data structures and network algorithms, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 44, Society of Industrial and Applied Mathematics, Philadelphia, PA, 1983.Google Scholar
  50. [50]
    J. V. Uspensky:Theory of Equations, McGraw-Hill, New York, 1948.Google Scholar
  51. [51]
    B. L. van der Waerden:Algebra, Ungar, New York, 1970.Google Scholar
  52. [52]
    H. E. Warren: Lower Bounds for approximation by nonlinear manifolds,Trans. Amer. Math. Soc., 133 (1968), 167–178.MATHCrossRefMathSciNetGoogle Scholar
  53. [53]
    P. Winter: An algorithm for the Steiner problem in the Euclidean Plane,Networks,15 (1985), 323–345.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1992

Authors and Affiliations

  • Warren D. Smith
    • 1
  1. 1.AT&T Bell LaboratoriesMurray HillUSA

Personalised recommendations