Handbook of Combinatorial Optimization pp 851-903 | Cite as

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

## 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## Preview

Unable to display preview. Download preview PDF.

## References

- [1]A. Aggarwal, B. Chazelle, L. Guibas, C. O’Dunlaing, and C. Yap. Parallel computational geometry.
*Algorithmica*,**3**(3):293–327, 1988.CrossRefzbMATHMathSciNetGoogle Scholar - [2]M.J. Atallah and M.T. Goodrich. Parallel algorithms for some functions of two convex polygons.
*Algorithmica*,**3**(4):535–548, 1988.CrossRefzbMATHMathSciNetGoogle Scholar - [3]M.W. Bern and R.L. Graham. The shortest-network problem.
*Sci.*Am.,**260**(1):84–89, January 1989.CrossRefGoogle Scholar - [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]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]A. Chow.
*Parallel Algorithms for Geometric Problems.*PhD thesis, University of Illinois, Urbana-Champaign, IL, 1980.Google Scholar - [7]F.R.K. Chung, M. Gardner, and R.L. Graham. Steiner trees on a checkerboard.
*Math. Mag.*,**62**:83–96, 1989.CrossRefzbMATHMathSciNetGoogle Scholar - [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]E.J. Cockayne. On the Steiner problem.
*Canad. Math. Bull.*,**10**(3):431–450, 1967.CrossRefzbMATHMathSciNetGoogle Scholar - [10]E.J. Cockayne. On the efficiency of the algorithm for Steiner minimal trees.
*SIAM J. Appl. Math.*,**18**(1):150–159, January 1970.CrossRefzbMATHMathSciNetGoogle Scholar - [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.CrossRefzbMATHMathSciNetGoogle Scholar - [12]E.J. Cockayne and D.E. Hewgill. Improved computation of plane Steiner minimal trees.
*Algorithmica*,**7**(2/3):219–229, 1992.CrossRefzbMATHMathSciNetGoogle Scholar - [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]R. Courant and H. Robbins.
*What is Mathematics? an elementary approach to ideas and methods.*Oxford University Press, London, 1941.zbMATHGoogle Scholar - [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.CrossRefzbMATHMathSciNetGoogle Scholar - [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.CrossRefzbMATHMathSciNetGoogle Scholar - [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]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]R. Geist, R. Reynolds, and D. Suggs. A markovian framework for digital halftoning.
*ACM Trans. Graphics*,12(2):136–159, April 1993.CrossRefzbMATHGoogle Scholar - [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]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]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.CrossRefGoogle Scholar - [23]G. Georgakopoulos and C. Papadimitriou. A 1-steiner tree problem.
*J. Algorithms*, 8(1):122–130, Mar 1987.CrossRefzbMATHMathSciNetGoogle Scholar - [24]E.N. Gilbert and H.O. Pollak. Steiner minimal trees.
*SIAM J. Appl. Math.*,**16**(1):1–29, January 1968.CrossRefzbMATHMathSciNetGoogle Scholar - [25]R.L. Graham. Private Communication.Google Scholar
- [26]S. Grossberg. Nonlinear neural networks: Principles, mechanisms, and architectures.
*Neural Networks*, 1:17–61, 1988.CrossRefGoogle Scholar - [27]F.C. Harris, Jr.
*Parallel Computation of Steiner Minimal Trees.*PhD thesis, Clemson, University, Clemson, SC 296–34, May 1994.Google Scholar - [28]F.C. Harris, Jr. A stochastic optimization algorithm for steiner minimal trees.
*Congr. Numer.*,**105**:54–64, 1994.MathSciNetGoogle Scholar - [29]F.C. Harris, Jr. An introduction to steiner minimal trees on grids.
*Congr. Numer.*,**111**:3–17, 1995.zbMATHMathSciNetGoogle Scholar - [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]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]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.CrossRefGoogle Scholar - [33]F. K. Hwang and J. F. Weng. The shortest network under a given topology.
*J. Algorithms*,**13**(3):468–488, Sept. 1992.CrossRefzbMATHMathSciNetGoogle Scholar - [34]F.K. Hwang and D.S. Richards. Steiner tree problems.
*Networks*,**22**(1):55–89, January 1992.CrossRefzbMATHMathSciNetGoogle Scholar - [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]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.CrossRefzbMATHMathSciNetGoogle Scholar - [37]J. JáJá
*. An Introduction to Parallel Algorithms.*Addison-Wesley Publishing Company, Reading, MA, 1992.zbMATHGoogle Scholar - [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]S. Kirkpatrick, C. Gelatt, and M. Vecchi. Optimization by simulated annealing.
*Science*, 220(13):671–680, May 1983.CrossRefzbMATHMathSciNetGoogle Scholar - [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.zbMATHGoogle Scholar - [41]Z.A. Melzak. On the problem of Steiner.
*Canad. Math. Bull.*,**4**(2):143–150, 1961.CrossRefzbMATHMathSciNetGoogle Scholar - [42]Michael K. Molloy. Performance analysis using stochastic Petri nets.
*IEEE Trans. Comput.*,C-31(9):913–917, September 1982.CrossRefGoogle Scholar - [43]J.L. Peterson. Petri Net Theory and the Modeling of Systems. Prentice-Hall, Englewood Cliffs, NJ, 1981.Google Scholar
- [44]F.P. Preparata and M.I. Shamos. Computational Geometry: an introduction. Springer-Verlag, New York, NY, 1988.Google Scholar
- [45]Michael J. Quinn.
*Parallel Computing: Theory and Practice.*McGraw-Hill Inc., New York, NY, 1994.Google Scholar - [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.CrossRefzbMATHMathSciNetGoogle Scholar - [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]M.I. Shamos.
*Computational Geometry.*PhD thesis, Department of Computer Science, Yale University, New Haven, CT, 1978.Google Scholar - [49]Justin R. Smith. The Design and Analysis of Parallel Algorithms. Oxford University Press, Inc., New York, NY, 1993.Google Scholar
- [50]D. Trietsch. Augmenting Euclidean networks — the Steiner case. SIAM J. Appl. Math.,
**45**:855–860, 1985.CrossRefzbMATHMathSciNetGoogle Scholar - [51]D. Trietsch and F. K. Hwang. An improved algorithm for Steiner trees. SIAM J. Appl. Math.,
**50**:244–263, 1990.CrossRefzbMATHMathSciNetGoogle Scholar - [52]P. Winter. An algorithm for the Steiner problem in the Euclidian plane. Networks,
**15**(3):323–345, Fall 1985.CrossRefzbMATHMathSciNetGoogle Scholar