## Abstract

In recent years, researchers have proven many theorems of the following form: given points distributed according to a Poisson process with intensity parameter*N* 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 with*N*.

## Key words

Spanning tree Steiner tree Heuristic algorithm Geometric probability Optimization Graph## References

- [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]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]F. R. K. Chung and R. L. Graham, On Steiner trees for bounded point sets,
*Geom. Dedicata*,**11**(1981), 353–361.zbMATHCrossRefMathSciNetGoogle Scholar - [4]F. R. K. Chung and F. K. Hwang, The largest minimal rectilinear Steiner trees for a set of
*n*points enclosed in a rectangle with given perimeter,*Networks*,**9**(1979), 19–36.zbMATHCrossRefMathSciNetGoogle Scholar - [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]M. Haimovich and A. H. G. Rinnooy Kan, Personal communication with A. H. G. Rinnooy Kan.Google Scholar
- [7]F. K. Hwang, On Steiner minimal trees with rectilinear distance,
*SIAM J. Appl. Math.*,**30**(1976), 104–114.zbMATHCrossRefMathSciNetGoogle Scholar - [8]F. K. Hwang, The rectilinear Steiner problem,
*Design Automat. Fault-Tolerant Comput.*,**2**(1978), 303–310.MathSciNetGoogle Scholar - [9]F. K. Hwang, An
*O(N*log*N)*algorithm for suboptimal rectilinear Steiner trees,*IEEE Trans. Circuits and Systems*,**26**(1979), 75–77.zbMATHCrossRefMathSciNetGoogle Scholar - [10]R. M. Karp and J. M. Steele, Probabilistic analysis of heuristics, in
*The 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]J. Komlós and M. T. Shing, Probabilistic partitioning algorithms for the rectilinear Steiner problem,
*Networks*,**15**(1985), 413–424.zbMATHCrossRefMathSciNetGoogle Scholar - [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]E. L. Lawler,
*Combinatorial Optimization: Networks and Matroids*, Holt, Rinehart, and Winston, New York, 1976, p. 290.zbMATHGoogle Scholar - [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]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]J. MacGregor-Smith, D. T. Lee, and J. S. Liebman, An
*O*(*N*log*N*) heuristic algorithm for the rectilinear Steiner problem,*Engrg. Optim.*,**4**(1980), 179–192.CrossRefGoogle Scholar - [17]A. Ng, Personal communication, University of California at Berkeley.Google Scholar
- [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]H. R. Pitt,
*Tauberian Theorems*, Oxford University Press, Oxford, 1958, pp. 37–42.zbMATHGoogle Scholar - [20]R. C. Prim, Shortest connecting networks and some generalizations,
*Bell System Tech. J.*,**36**(1957), 1389–1401.Google Scholar - [21]M. Servit, Heuristic algorithms for rectilinear Steiner trees,
*Digital Process.*,**7**(1981), 21–32.zbMATHGoogle Scholar - [22]J. M. Steele, Subadditive Euclidean functionals and nonlinear growth in geometric probability,
*Ann. Probab.*,**9**(1982), 365–376.CrossRefMathSciNetGoogle Scholar - [23]C. D. Thomborson, Personal communications, University of Minnesota at Duluth.Google Scholar
- [24]Y. Y. Yang and O. Wing, Suboptimal algorithm for a wire routing problem,
*IEEE Trans. Circuit Theory*,**19**(1972), 508–511.MathSciNetCrossRefGoogle Scholar