The Competitiveness of Randomized Algorithms for On-Line Steiner Tree and On-Line Spanning Tree Problems

Abstract This paper considers a family of randomized on-line algorithms, Algorithm R ( m ), where 1 ⩽ m ⩽ n − 1 and n is the number of input points, for the on-line Steiner tree and on-line spanning tree problems on Euclidean space. Our main result is that if m is a fixed constant, the competitive ratios of Algorithm R ( m ) for the on-line Steiner tree and spanning tree problems are Θ( n ). We also show that the competitive ratio of Algorithm R ( n − 1), which is deterministic greedy algorithm, for the on-line spanning tree problem is the same as that for the on-line Steiner tree problem, which is O(log n ).