Steiner Minimal Trees: An Introduction, Parallel Computation, and Future Work

• Frederick C. HarrisJr.
Chapter

Abstract

Minimizing a network’s length is one of the oldest optimization problems in mathematics and, consequently, it has been worked on by many of the leading mathematicians in history. In the mid-seventeenth century a simple problem was posed: Find the point P that minimizes the sum of the distances from P to each of three given points in the plane. Solutions to this problem were derived independently by Fermat, Torricelli, and Cavaliers. They all deduced that either P is inside the triangle formed by the given points and that the angles at P formed by the lines joining P to the three points are all 120°, or P is one of the three vertices and the angle at P formed by the lines joining P to the other two points is greater than or equal to 120°.

Keywords

Simulated Annealing Parallel Computation Parallel Algorithm Point Problem Steiner Point
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

1. [1]
A. Aggarwal, B. Chazelle, L. Guibas, C. O’Dunlaing, and C. Yap. Parallel computational geometry. Algorithmica, 3(3):293–327, 1988.
2. [2]
M.J. Atallah and M.T. Goodrich. Parallel algorithms for some functions of two convex polygons. Algorithmica, 3(4):535–548, 1988.
3. [3]
M.W. Bern and R.L. Graham. The shortest-network problem. Sci. Am., 260(1):84–89, January 1989.
4. [4]
W.M. Boyce and J.R. Seery. STEINER 72 - an improved version of Cockayne and Schiller’s program STEINER for the minimal network problem. Technical Report 35, Bell Labs., Dept. of Computer Science, 1975.Google Scholar
5. [5]
G. X. Chen. The shortest path between two points with a (linear) constraint [in Chinese]. Knowledge and Appl. of Math., 4:1–8, 1980.Google Scholar
6. [6]
A. Chow. Parallel Algorithms for Geometric Problems. PhD thesis, University of Illinois, Urbana-Champaign, IL, 1980.Google Scholar
7. [7]
F.R.K. Chung, M. Gardner, and R.L. Graham. Steiner trees on a checkerboard. Math. Mag., 62:83–96, 1989.
8. [8]
F.R.K. Chung and R.L. Graham. Steiner trees for ladders. In B. Alspach, P. Hell, and D.J. Miller, editors, Annals of Discrete Mathematics:2, pages 173–200. North-Holland Publishing Company, 1978.Google Scholar
9. [9]
E.J. Cockayne. On the Steiner problem. Canad. Math. Bull., 10(3):431–450, 1967.
10. [10]
E.J. Cockayne. On the efficiency of the algorithm for Steiner minimal trees. SIAM J. Appl. Math., 18(1):150–159, January 1970.
11. [11]
E.J. Cockayne and D.E. Hewgill. Exact computation of Steiner minimal trees in the plane. Info. Proccess. Lett., 22(3):151–156, March 1986.
12. [12]
E.J. Cockayne and D.E. Hewgill. Improved computation of plane Steiner minimal trees. Algorithmica, 7(2/3):219–229, 1992.
13. [13]
E.J. Cockayne and D.G. Schiller. Computation of Steiner minimal trees. In D.J.A. Welsh and D.R. Woodall, editors, Combinatorics, pages 5271, Maitland House, Warrior Square, Southend-on-Sea, Essex SS1 2J4, 1972. Mathematical Institute, Oxford, Inst. Math. Appl.Google Scholar
14. [14]
R. Courant and H. Robbins. What is Mathematics? an elementary approach to ideas and methods. Oxford University Press, London, 1941.
15. [15]
D.Z. Du and F.H. Hwang. A proof of the Gilbert-Pollak conjecture on the Steiner ratio. Algorithmica, 7(2/3):121–135, 1992.
16. [16]
M.R. Garey, R.L. Graham, and D.S Johnson. The complexity of computing Steiner minimal trees. SIAM J. Appl. Math.,32(4):835–859, June 1977.
17. [17]
Al Geist, Adam Beguelin, Jack Dongarra, Weicheng Jiang, Robert Manchek, and Vaidy Sunderam. PVM: Parallel Virtual Machine - A User’s guide and tutorial for networked parallel computing. MIT Press, Cambridge, MA, 1994.Google Scholar
18. [18]
R. Geist, R. Reynolds, and C. Dove. Context sensitive color quantization. Technical Report 91–120, Dept. of Comp. Sci., Clemson Univ., Clemson, SC 296–34, July 1991.Google Scholar
19. [19]
R. Geist, R. Reynolds, and D. Suggs. A markovian framework for digital halftoning. ACM Trans. Graphics,12(2):136–159, April 1993.
20. [20]
R. Geist and D. Suggs. Neural networks for the design of distributed, fault-tolerant, computing environments. In Proc. 11th IEEE Symp. on Reliable Distributed Systems (SRDS), pages 189–195, Houston, Texas, October 1992.Google Scholar
21. [21]
R. Geist, D. Suggs, and R. Reynolds. Minimizing mean seek distance in mirrored disk systems by cylinder remapping. In Proc. 16th IFIP Int. Symp. on Computer Performance Modeling Measurement, and Evaluation (PERFORMANCE `93), pages 91–108, Rome, Italy, September 1993.Google Scholar
22. [22]
R. Geist, D. Suggs, R. Reynolds, S. Divatia, F. Harris, E. Foster, and P. Kolte. Disk performance enhancement through Markov-based cylinder remapping. In Cherri M. Pancake and Douglas S. Reeves, editors, Proc. of the ACM Southeastern Regional Conf., pages 23–28, Raleigh, North Carolina, April 1992. The Association for Computing Machinery, Inc.
23. [23]
G. Georgakopoulos and C. Papadimitriou. A 1-steiner tree problem. J. Algorithms, 8(1):122–130, Mar 1987.
24. [24]
E.N. Gilbert and H.O. Pollak. Steiner minimal trees. SIAM J. Appl. Math.,16(1):1–29, January 1968.
25. [25]
26. [26]
S. Grossberg. Nonlinear neural networks: Principles, mechanisms, and architectures. Neural Networks, 1:17–61, 1988.
27. [27]
F.C. Harris, Jr. Parallel Computation of Steiner Minimal Trees. PhD thesis, Clemson, University, Clemson, SC 296–34, May 1994.Google Scholar
28. [28]
F.C. Harris, Jr. A stochastic optimization algorithm for steiner minimal trees. Congr. Numer., 105:54–64, 1994.
29. [29]
F.C. Harris, Jr. An introduction to steiner minimal trees on grids. Congr. Numer., 111:3–17, 1995.
30. [30]
F.C. Harris, Jr. Parallel computation of steiner minimal trees. In David H. Bailey, Petter E. Bjorstad, John R. Gilbert, Michael V. Mascagni, Robert S. Schreiber, Horst D. Simon, Virgia J. Torczan, and Layne T. Watson, editors, Proc. Of the 7th SIAM Conf. on Parallel Process. for Sci Comput., pages 267–272, San Francisco, California, February 1995. SIAM.Google Scholar
31. [31]
S. Hedetniemi. Characterizations and constructions of minimally 2-connected graphs and minimally strong digraphs. In Proc. 2nd Louisiana Conf. on Combinatorics, Graph Theory, and Computing, pages 257–282, Louisiana State Univ., Baton Rouge, Louisiana, March 1971.Google Scholar
32. [32]
J.J. Hopfield. Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Nat. Acad. Sci., 81:3088–3092, 1984.
33. [33]
F. K. Hwang and J. F. Weng. The shortest network under a given topology. J. Algorithms, 13(3):468–488, Sept. 1992.
34. [34]
F.K. Hwang and D.S. Richards. Steiner tree problems. Networks, 22(1):55–89, January 1992.
35. [35]
F.K. Hwang, D.S. Richards, and P. Winter. The Steiner Tree Problem, volume 53 of Ann. Discrete Math. North-Holland, Amsterdam, 1992.Google Scholar
36. [36]
F.K. Hwang, G.D. Song, G.Y. Ting, and D.Z. Du. A decomposition theorem on Euclidian Steiner minimal trees. Disc. Comput. Geom., 3(4):367–382, 1988.
37. [37]
J. JáJá. An Introduction to Parallel Algorithms. Addison-Wesley Publishing Company, Reading, MA, 1992.
38. [38]
V. Jarnik and O. Kössler. O minimâlnich gratech obsahujicich n danÿch bodu [in Czech]. Casopis Pesk. Mat. Fyr., 63:223–235, 1934.Google Scholar
39. [39]
S. Kirkpatrick, C. Gelatt, and M. Vecchi. Optimization by simulated annealing. Science, 220(13):671–680, May 1983.
40. [40]
V. Kumar, A. Grama, A. Gupta, and G. Karypis. Introduction to Parallel Computing: Design and Analysis of Algorithms. The Benjamin/Cummings Publishing Company, Inc., Redwood City, CA, 1994.
41. [41]
Z.A. Melzak. On the problem of Steiner. Canad. Math. Bull., 4(2):143–150, 1961.
42. [42]
Michael K. Molloy. Performance analysis using stochastic Petri nets. IEEE Trans. Comput.,C-31(9):913–917, September 1982.
43. [43]
J.L. Peterson. Petri Net Theory and the Modeling of Systems. Prentice-Hall, Englewood Cliffs, NJ, 1981.Google Scholar
44. [44]
F.P. Preparata and M.I. Shamos. Computational Geometry: an introduction. Springer-Verlag, New York, NY, 1988.Google Scholar
45. [45]
Michael J. Quinn. Parallel Computing: Theory and Practice. McGraw-Hill Inc., New York, NY, 1994.Google Scholar
46. [46]
M.J. Quinn and N. Deo. An upper bound for the speedup of parallel best-bound branch-and-bound algorithms. BIT, 26(1):35–43, 1986.
47. [47]
W.R. Reynolds. A Markov Random Field Approach to Large Combinatorial Optimization Problems. PhD thesis, Clemson, University, Clemson, SC 296–34, August 1993.Google Scholar
48. [48]
M.I. Shamos. Computational Geometry. PhD thesis, Department of Computer Science, Yale University, New Haven, CT, 1978.Google Scholar
49. [49]
Justin R. Smith. The Design and Analysis of Parallel Algorithms. Oxford University Press, Inc., New York, NY, 1993.Google Scholar
50. [50]
D. Trietsch. Augmenting Euclidean networks — the Steiner case. SIAM J. Appl. Math., 45:855–860, 1985.
51. [51]
D. Trietsch and F. K. Hwang. An improved algorithm for Steiner trees. SIAM J. Appl. Math., 50:244–263, 1990.
52. [52]
P. Winter. An algorithm for the Steiner problem in the Euclidian plane. Networks, 15(3):323–345, Fall 1985.