An Average Case Analysis of a Greedy Algorithm for the On-Line Steiner Tree Problem

This paper gives the average distance analysis for the Euclidean tree constructed by a simple greedy but efficient algorithm of the on-line Steiner tree problem. The algorithm accepts the data one by one following the order of input sequence. When a point arrives, the algorithm adds the shortest edge, between the new point and the points arriving already, to the previously constructed tree to form a new tree. We first show that, given n points uniformly on a unit disk in the plane, the ‘th expected Euclidean distance between a point and its 3 (1 5 j 5 n 1) nearest neighbor is less than or equal to (5/3)m h TZ w en n is large. Based upon this result, we show that the expected length of the tree constructed by the on-line algorithm is not greater than 4.34 times the expected length of the minimum Steiner tree when the number of input points is large. Keywords-Analysis of algorithms, On-line algorithms, Average case analysis, On-line Steiner tree problems, Euclidean space.