 
Summary: Online Steiner Trees in the Euclidean Plane
Noga Alon
Yossi Azar
Abstract
Suppose we are given a sequence of n points in the Euclidean plane, and
our objective is to construct, online, a connected graph that connects all of
them, trying to minimize the total sum of lengths of its edges. The points
appear one at a time, and at each step the online algorithm must construct a
connected graph that contains all current points by connecting the new point
to the previously constructed graph. This can be done by joining the new
point (not necessarily by a straight line) to any point of the previous graph,
(not necessarily one of the given points). The performance of our algorithm is
measured by its competitive ratio: the supremum, over all sequences of points,
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. There
are known online algorithms whose competitive ratio is O(log n) even for all
metric spaces, but the only lower bound known is of [IW] for some contrived
discrete metric space. Moreover, for the plane, online algorithms could have
been more powerful and achieve a better competitive ratio, and no nontrivial
lower bounds for the best possible competitive ratio were known. Here we prove
