This thesis examines the (geometric) Steiner tree problem: Given a set of points P in the plane, find a shortest tree interconnecting all points in P, with the possibility of adding points outside P, called the Steiner points, as additional vertices of the tree. The Steiner tree problem has been studied in different metric spaces. In this thesis, we study the problem in Euclidean and rectilinear metrics. One of the most natural heuristics for the Steiner tree problem is to use a minimum spanning tree, which can be found in O(nlogn) time . The performance ratio of this heuristic is given by the Steiner ratio, which is defined as the minimum possible ratio between the lengths of a minimum Steiner tree and a minimum spanning tree. We survey the background literature on the Steiner ratio and study the generalization of the Steiner ratio to the case of obstacles. We introduce the concept of an anchored Steiner tree: an obstacle-avoiding Steiner tree in which the Steiner points are only allowed at obstacle corners. We define the obstacle-avoiding Steiner ratio as the ratio of the length of an obstacle-avoiding minimum Steiner tree to that of an anchored obstacle-avoiding minimum Steiner tree. We prove that, for the rectilinear metric, the obstacle-avoiding Steiner ratio is equal to the traditional (obstacle-free) Steiner ratio. We conjecture that this is also the case for the Euclidean metric and we prove this conjecture for three points and any number of obstacles.

This thesis examines the (geometric) Steiner tree problem: Given a set of points P in the plane, find a shortest tree interconnecting all points in P, with the possibility of adding points outside P, called the Steiner points, as additional vertices of the tree. The Steiner tree problem has been studied in different metric spaces. In this thesis, we study the problem in Euclidean and rectilinear metrics. One of the most natural heuristics for the Steiner tree problem is to use a minimum spanning tree, which can be found in O(nlogn) time . The performance ratio of this heuristic is given by the Steiner ratio, which is defined as the minimum possible ratio between the lengths of a minimum Steiner tree and a minimum spanning tree. We survey the background literature on the Steiner ratio and study the generalization of the Steiner ratio to the case of obstacles. We introduce the concept of an anchored Steiner tree: an obstacle-avoiding Steiner tree in which the Steiner points are only allowed at obstacle corners. We define the obstacle-avoiding Steiner ratio as the ratio of the length of an obstacle-avoiding minimum Steiner tree to that of an anchored obstacle-avoiding minimum Steiner tree. We prove that, for the rectilinear metric, the obstacle-avoiding Steiner ratio is equal to the traditional (obstacle-free) Steiner ratio. We conjecture that this is also the case for the Euclidean metric and we prove this conjecture for three points and any number of obstacles.

In 1992 Du and Hwang published a paper confirming the correctness of a well known 1968 conjecture of Gilbert and Pollak suggesting that the Euclidean Steiner ratio for the plane is 2/3. The original objective of this thesis was to adapt the technique used in this proof to obtain results for other Minkowski spaces. In an attempt to create a rigorous and complete version of the proof, some known results were given new proofs (results for hexagonal trees and for the rectilinear Steiner ratio) and some new results were obtained (on approximation of Steiner ratios and on transforming Steiner trees). The most surprising result, however, was the discovery of a fundamental gap in the proof of Du and Hwang. We give counter examples demonstrating that a statement made about inner spanning trees, which plays an important role in the proof, is not correct. There seems to be no simple way out of this dilemma, and whether the Gilbert-Pollak conjecture is true or not for any number of points seems once again to be an open question. Finally we consider the question of whether Du and Hwang's strategy can be used for cases where the number of points is restricted. After introducing some extra lemmas, we are able to show that the Gilbert-Pollak conjecture is true for 7 or fewer points. This is an improvement on the 1991 proof for 6 points of Rubinstein and Thomas. / Mathematical Sciences / Ph. D. (Mathematics)

In 1992 Du and Hwang published a paper confirming the correctness of a well known 1968 conjecture of Gilbert and Pollak suggesting that the Euclidean Steiner ratio for the plane is 2/3. The original objective of this thesis was to adapt the technique used in this proof to obtain results for other Minkowski spaces. In an attempt to create a rigorous and complete version of the proof, some known results were given new proofs (results for hexagonal trees and for the rectilinear Steiner ratio) and some new results were obtained (on approximation of Steiner ratios and on transforming Steiner trees). The most surprising result, however, was the discovery of a fundamental gap in the proof of Du and Hwang. We give counter examples demonstrating that a statement made about inner spanning trees, which plays an important role in the proof, is not correct. There seems to be no simple way out of this dilemma, and whether the Gilbert-Pollak conjecture is true or not for any number of points seems once again to be an open question. Finally we consider the question of whether Du and Hwang's strategy can be used for cases where the number of points is restricted. After introducing some extra lemmas, we are able to show that the Gilbert-Pollak conjecture is true for 7 or fewer points. This is an improvement on the 1991 proof for 6 points of Rubinstein and Thomas. / Mathematical Sciences / Ph. D. (Mathematics)