Hierarchical Steiner tree construction in uniform orientations

A hierarchical approach to Steiner tree construction in lambda -geometry is proposed. The algorithm runs in time O(n log n) and the length of the constructed tree is at most ( sigma /cos( pi /2 lambda )) times (for lambda =2, 3/2 times) the length of the optimal Steiner tree where n is the cardinality of the point set and it was recently proved that sigma is (2/ square root 3). How to trade off between the running time of the algorithm and the length of the produced Steiner tree is shown. Given enough time, an optimal Steiner tree will be obtained. The algorithm is extended to construct a Steiner tree of a set of subtrees (i.e., partial trees) and runs in O( lambda N log N) time, where N is the total number of edges of the subtrees.<
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