On-line steiner trees in the Euclidean plane

Suppose we are given a sequence of <italic>n</italic> points <italic>v</italic><subscrpt>1</subscrpt>,…,<italic>v<subscrpt>n</subscrpt></italic> in the Euclidean plane, and our objective is to construct, on-line, a connected graph that connects all of them, trying to minimize the total sum of lengths of its edges. We assume that the points appear one at a time, <italic>v<subscrpt>i</subscrpt></italic> arriving at step <italic>i</italic>. At the end of step <italic>i</italic>, the on-line algorithm must construct a connected graph <italic>T</italic><subscrpt><italic>i</italic>-1</subscrpt>. This can be done by joining <italic>v<subscrpt>i</subscrpt></italic> (not necessarily by a straight line) to any point of <italic>T<subscrpt>i</subscrpt></italic>-1, which need not necessarily be one of the previously given points <italic>v<subscrpt>j</subscrpt></italic>. The performance of our algorithm is measured by its competitive ratio: the supremum, over all sequences <italic>v</italic><subscrpt>1</subscrpt>,…,<italic>v<subscrpt>n</subscrpt></italic> as above, of the ratio between the total length of the graph constructed by our algorithm and the total length of the best Steiner tree that connects all the points <italic>v</italic><subscrpt>1</subscrpt>,…, <italic>v<subscrpt>n</subscrpt></italic>. There are known on-line algorithms whose competitive ratio is <italic>O</italic>(log <italic>n</italic>), but there is no known nontrivial lower bound for the best possible competitive ratio. Here we prove that the upper bound is almost tight by establishing an &OHgr;(log <italic>n</italic>/log log <italic>n</italic>) lower bound for the competitive ratio of any on-line algorithm. The lower bound holds for deterministic algorithms as well as for randomized ones, and obviously holds in any Euclidean space of dimension greater than 2 as well.