Steiner minimal trees in Lp2

Abstract For a finite set of points in a metric space a Steiner Minimal Tree (SMT) is a shortest tree which interconnects these points. We also consider a relative of this problem allowing at most k additional points in the tree ( k -SMT), where k is a given number. We intend to discuss these problems for all planes with p -norm, i.e. the affine plane with norm |( t 1 , t 2 )| p = (| t 1 | p + | t 2 | p ) 1/ p for 1 ⩽ p t 1 , t 2 ) | ∞ = max { | t 1 |, | t 2 |}. We give a survey of results for the combinatorial structure of SMT and k -SMT and show the consequences for the methods to construct such trees.

[1]  Ronald L. Graham,et al.  On the History of the Minimum Spanning Tree Problem , 1985, Annals of the History of Computing.

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

[3]  George O. Wesolowsky,et al.  FACILITIES LOCATION: MODELS AND METHODS , 1988 .

[4]  Dietmar Cleslik.  The Fermat-Steiner-Weber-problem in Minkowski spaces , 1988 .

[5]  David S. Johnson,et al.  The Complexity of Computing Steiner Minimal Trees , 1977 .

[6]  Ulrich Eckhardt,et al.  Weber's problem and weiszfeld's algorithm in general spaces , 1980, Math. Program..

[7]  Robert E. Tarjan,et al.  Finding Minimum Spanning Trees , 1976, SIAM J. Comput..

[8]  R. Graham,et al.  The Shortest-Network Problem , 1989 .

[9]  Ding-Zhu Du,et al.  An approach for proving lower bounds: solution of Gilbert-Pollak's conjecture on Steiner ratio , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[10]  Chuan Yi Tang,et al.  An Optimal Algorithm for Constructing Oriented Voronoi Diagrams and Geographic Neighborhood Graphs , 1990, Inf. Process. Lett..

[11]  Rainer Bodendiek,et al.  Contemporary methods in graph theory , 1990 .

[12]  D. T. Lee,et al.  An O(n log n) heuristic for steiner minimal tree problems on the euclidean metric , 1981, Networks.

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

[14]  D. T. Lee,et al.  Two-Dimensional Voronoi Diagrams in the Lp-Metric , 1980, J. ACM.

[15]  J. Krarup,et al.  Selected Families of Location Problems , 1979 .

[16]  D. Cieslik.  The Vertex-Degrees of Steiner Minimal Trees in Minkowski Planes , 1990 .

[17]  Rainer Bodendiek,et al.  Topics In Combinatorics and Graph Theory , 1992 .

[18]  R. Love,et al.  Modelling Inter-city Road Distances by Mathematical Functions , 1972 .

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

[20]  D. Cieslik.  The 1-Steiner-Minimal-Tree problem in Minkowski-spaces , 1991 .

[21]  Ronald L. Graham,et al.  A NEW BOUND FOR EUCLIDEAN STEINER MINIMAL TREES , 1985 .

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

[23]  Christos H. Papadimitriou,et al.  The 1-Steiner Tree Problem , 1987, J. Algorithms.

[24]  Ding-Zhu Du,et al.  On Steiner minimal trees withLp distance , 2005, Algorithmica.