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).