Unexplored Steiner Ratios in Geometric Networks

Abstract

Abstract. In this paper we extend the context of Steiner ratio and examine the influence of Steiner points on the weight of a graph in two generalizations of the Euclidean minimum weight connected graph (MST). The studied generalizations are with respect to the weight function and the connectivity condition. First, we consider the Steiner ratio of Euclidean minimum weight connected graph under the budget allocation model. The budget allocation model is a geometric version of a new model for weighted graphs introduced by . It is known that adding auxiliary points, called Steiner points, to the initial point set may result in a lighter Euclidean minimum spanning tree. We show that this behavior changes under the budget allocation model. Apparently, Steiner points are not helpful in weight reduction of the geometric minimum spanning trees under the budget allocation model (BMST), as opposed to the traditional model. An interesting relation between the BMST and the Euclidean square root metric reveals a somewhat surprising result: Steiner points are also redundant in weight reduction of the minimum spanning tree in the Euclidean square root metric. Finally, we consider the Steiner ratio of geometric t-spanners. A geometric t-spanner is a geometric connected graph with the following strengthened connectivity requirement: any two nodes p and q are connected with a path of length at most t × |pq|, where |pq| is the Euclidean distance between p and q. Surprisingly, the contribution of Steiner points to the weight of geometric t-spanners has not been a subject of research so far (to the best of our knowledge). Here, we show that the influence of Steiner points on reducing the weight of geometric spanner networks goes much further than what is known for trees.