Steiner minimal trees for three points with one convex polygonal obstacle

Abstract

The problem of constructing Steiner minimal trees in the Euclidean plane is NP-hard. When in addition obstacles are present, difficulties of constructing obstacle-avoiding Steiner minimal trees are compounded. This problem, which has many obvious practical applications when designing complex transportation and distribution systems, has received very little attention in the literature. The construction of Steiner minimal trees for three terminal points in the Euclidean plane (without obstacles) has been completely solved (among others by Fermat, Torricelli, Cavallieri, Simpson, Heinen) during the span of the last three centuries. This construction is a cornerstone for both exact algorithms and heuristics for the Euclidean Steiner tree problem with arbitrarily many terminal points. An algorithm for three terminal points in the presence of one polygonal convex obstacle is given. It is shown that this algorithm has the worst-case time complexityO(n), wheren is the number of extreme points on the obstacle. As an extension to the underlying algorithm, if the obstacle is appropriately preprocessed inO(n) time, we can solve any problem instance with three arbitrary terminal points and the preprocessed convex polygonal obstacle inO(logn) time. We believe that the three terminal points algorithm will play a critical role in the development of heuristics for problem instances with arbitrarily many terminal points and obstacles.