Euclidean Steiner Minimal Tree Inside Simple Polygon Avoiding Obstacles

The Steiner problem leads to solutions in several scientific and business applications, computer networks' routing and electronic integrated circuits are few examples. The computational features of this problem make it an important research subject in computational geometry. Assuming some points in the Euclidean plane, we can construct a minimum spanning tree connecting these (terminal) nodes. It is possible to add some extra points (Steiner Points) in order to decrease the length of this tree which would in turn lead to Euclidean Steiner minimal tree (ESMT). This problem is considered as a NP-hard problem, as it may contain some nodes that are not in the set of given nodes. Assuming a simple polygon P with m vertices and n terminals in it, we try to find a Euclidean Steiner minimal tree connecting all these n terminals in P. In this paper we propose solutions for any number of terminals in a simple polygon. Proposed algorithms can also solve the problem in a simple polygon with some obstacles inside. These algorithms are simple to implement and lead to good results.

1. Introduction

The Steiner problem is applied in several scientific and business applications, such as computer networks' routing, electronic integrated circuits, petroleum shaft and post networks. The computational features of this problem make it an important research subject in computational geometry.