We give a survey up to 1989 on the Steiner tree problems which include the four important cases of euclidean, rectilinear, graphic, phylogenetic and some of their generalizations. We also provide a rather comprehensive and up-to-date bibliography which covers more than three hundred items.

Steiner tree problems

This paper has two purposes. The first is to present a new way to find a Steiner minimum tree (SMT) connectingN sites ind-space,d >- 2. We present (in Appendix 1) a computer code for this purpose. This is the only procedure known to the author for finding Steiner minimal trees ind-space ford > 2, and also the first one which fits naturally into the framework of “backtracking” and “branch-and-bound.” Finding SMTs of up toN = 12 general sites ind-space (for anyd) now appears feasible.

How to find Steiner minimal trees in euclideand-space

The history of the Euclidean Steiner tree problem, which is the problem of constructing a shortest possible network interconnecting a set of given points in the Euclidean plane, goes back to Gergonne in the early nineteenth century. We present a detailed account of the mathematical contributions of some of the earliest papers on the Euclidean Steiner tree problem. Furthermore, we link these initial contributions with results from the recent literature on the problem.

/pdf/on-the-history-of-the-euclidean-steiner-tree-problem-c00wielg1i.pdf

On the history of the Euclidean Steiner tree problem

We investigate the role convexity plays in the efficient solution to the Steiner tree problem. In general terms, we show that a Steiner tree problem is always computationally easier to solve when the points to be connected lie on the boundary of a “convex” region. For the Steiner tree problem on graphs and the rectilinear Steiner tree problem, we give definitions of “convexity” for which this condition is sufficient to allow a polynomial algorithm for finding the optimal Steiner tree. For the classical Steiner tree problem, we show that for the standard definition of convexity, this condition is sufficient to allow a fully polynomial approximation scheme.

Convexity and the Steiner tree problem

ASteiner Minimal Tree (SMT) for a given setA = {a
 1,...,a
 n
 } in the plane is a tree which interconnects these points and whose total length, i.e., the sum of lengths of the branches, is minimum. To achieve the minimum, the tree may contain other points (Steiner points) besidesa
 1,...,a
 n
 . Various improvements are presented to an earlier computer program of the authors for plane SMTs. These changes have radically reduced machine times. The existing program was limited in application to aboutn = 30, while the innovations have facilitated solution of many randomly generated 100-point problems in reasonable processing times.

/pdf/improved-computation-of-plane-steiner-minimal-trees-54tm6j28sx.pdf

Improved computation of plane Steiner Minimal Trees

LetS = {A, B, C, D} consist of the four corner points of a convex quadrilateral where diagonals [A, C] and [B, D] intersect at the pointO. There are two possible full Steiner trees forS, theAB-CD tree hasA andB adjacent to one Steiner point, andC andD to another; theAD-BC tree hasA andD adjacent to one Steiner point, andB andC to another. Pollak proved that if both full Steiner trees exist, then theAB-CD (AD-BC) tree is the Steiner minimal tree if[Figure not available: see fulltext.]AOD>3 (<) 90°, and both are Steiner minimal trees if[Figure not available: see fulltext.]AOD=90°. While the theorem has been crucially used in obtaining results on Steiner minimal trees in general, its applicability is sometimes restricted because of the condition that both full Steiner trees must exist. In this paper we remove this obstacle by showing: (i) Necessary and sufficient conditions for the existence of either full Steiner tree forS. (ii) If[Figure not available: see fulltext.]AOD?90°, then theAB-CD tree is the SMT even if theAD-BC tree does not exist. (iii) If[Figure not available: see fulltext.]AOD<90° but theAD-BC tree does not exist, then theAB-CD tree cannot be ruled out as a Steiner minimal tree, though under certain broad conditions it can.

/pdf/steiner-minimal-trees-on-sets-of-four-points-39feeb6gtz.pdf

Steiner minimal trees on sets of four points

The Euclidean Steiner minimal tree problem is known to be an NP-complete problem and current alogorithms cannot solve problems with more than 30 points. Thus decomposition theorems can be very helpful in extending the boundary of workable problems. There have been only two known decomposition theorems in the literature. This paper provides a 50% increase in the reservoir of decomposition theorems.

/pdf/a-decomposition-theorem-on-euclidean-steiner-minimal-trees-2ll8bhlaqk.pdf

G. Y. Ting

Papers

Steiner minimal trees on sets of four points

A decomposition theorem on Euclidean Steiner minimal trees