On Exact Solutions for the Rectilinear Steiner Tree Problem Part I: Theoretical Results

Abstract. The rectilinear Steiner tree problem asks for a shortest tree connecting given points in the plane with rectilinear distance. The best theoretically analyzed algorithms for this problem are based on dynamic programming and have a running time of O(n2 . . . 2.62n) (Ganley and Cohoon), resp. $n^{O(\sqrt{n})}$ (Smith). The first algorithm can solve problems of size 27, the second one is highly impractical because of the large constant in the exponent. The best implementations perform poorly even on small problem instances; the best practical results can be reached using a Branch \& Bound approach (Salowe and Warme); this implementation can solve random problems of size 35 within a day, while the dynamic programming approach of Ganley and Cohoon can handle only 27 point examples. In this paper we improve the theoretical worst-case time bound to O(n2 . . . 2.38n) , for random problem instances we prove a running time of αn with a constant α < 2 . We have implemented our algorithms and can now solve problems of 40 points in a day using a provably good dynamic programming approach, and can solve problems of 55 points with a Branch \& Bound strategy. For exponential-time algorithms, this is an enormous improvement.

[1]  Jeffrey S. Salowe,et al.  Thirty-five-point rectilinear steiner minimal trees in a day , 1995, Networks.

[2]  Kurt Mehlhorn,et al.  A Faster Approximation Algorithm for the Steiner Problem in Graphs , 1988, Inf. Process. Lett..

[3]  Pawel Winter,et al.  An algorithm for the steiner problem in the euclidean plane , 1985, Networks.

[4]  Rajeev Motwani,et al.  Randomized algorithms , 1996, CSUR.

[5]  F. Hwang.  On Steiner Minimal Trees with Rectilinear Distance , 1976 .

[6]  Hans Jürgen Prömel,et al.  The Steiner Tree Problem , 2002 .

[7]  Piotr Berman,et al.  Improved approximations for the Steiner tree problem , 1992, SODA '92.

[8]  Sanjeev Arora,et al.  Polynomial time approximation schemes for Euclidean TSP and other geometric problems , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[9]  Alexander Zelikovsky.  An 11/6-Approximation Algorithm for the Steiner Problem on Graphs , 1992 .

[10]  Thomas Lengauer,et al.  Combinatorial algorithms for integrated circuit layout , 1990, Applicable theory in computer science.

[11]  Michael Kaufmann,et al.  Faster Approximation Algorithms for the Rectilinear Steiner Tree Problem , 1993, ISAAC.

[12]  Russ Bubley,et al.  Randomized algorithms , 1995, CSUR.

[13]  Joseph L. Ganley,et al.  Optimal Rectilinear Steiner Minimal Trees in O (n22.62n) Time , 1994, CCCG.

[14]  David S. Johnson,et al.  The Rectilinear Steiner Problem is NP-Complete , 1977 .

[15]  M. Hanan,et al.  On Steiner’s Problem with Rectilinear Distance , 1966 .

[16]  Q. Feng,et al.  On better heuristic for Euclidean Steiner minimum trees , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[17]  H. Pollak,et al.  Steiner Minimal Trees , 1968 .