Streaming Algorithms for Geometric Steiner Forest

We consider a generalization of the Steiner tree problem, the Steiner forest problem, in the Euclidean plane: the input is a multiset (Formula presented), partitioned into color classes (Formula presented). The goal is to find a minimum-cost Euclidean graph such that every color class is connected in . We study this Steiner forest problem in the streaming setting, where the stream consists of insertions and deletions of points to . Each input point arrives with its color (Formula presented), and as usual for dynamic geometric streams, the input is restricted to the discrete grid (Formula presented). We design a single-pass streaming algorithm that uses space and time, and estimates the cost of an optimal Steiner forest solution within ratio arbitrarily close to the famous Euclidean Steiner ratio (currently) (Formula presented). This approximation guarantee matches the state-of-the-art bound for streaming Steiner tree, i.e., when , and it is a major open question to improve the ratio to even for this special case. Our approach relies on a novel combination of streaming techniques, like sampling and linear sketching, with the classical Arora-style dynamic-programming framework for geometric optimization problems, which usually requires large memory and so far has not been applied in the streaming setting.We complement our streaming algorithm for the Steiner forest problem with simple arguments showing that any finite multiplicative approximation requires ω(k) bits of space.