Streaming Algorithms for Geometric Steiner Forest

TL;DR

This paper introduces a streaming algorithm for the Euclidean Steiner forest problem that approximates the optimal solution within a ratio close to the Euclidean Steiner ratio, using poly(k, log Δ) space, combining sampling, sketching, and dynamic programming.

Contribution

It presents the first single-pass streaming algorithm for the Steiner forest problem with near-optimal approximation guarantees and novel techniques combining streaming methods with classical geometric optimization.

Findings

Achieves approximation ratio close to Euclidean Steiner ratio α₂.

Uses poly(k, log Δ) space and time in the streaming model.

Establishes space lower bounds for finite approximation of Steiner forest.

Abstract

We consider an important generalization of the Steiner tree problem, the \emph{Steiner forest problem}, in the Euclidean plane: the input is a multiset $X \subseteq \mathbb{R}^2$, partitioned into $k$ color classes $C_1, C_2, \ldots, C_k \subseteq X$. The goal is to find a minimum-cost Euclidean graph $G$ such that every color class $C_i$ is connected in $G$. We study this Steiner forest problem in the streaming setting, where the stream consists of insertions and deletions of points to $X$. Each input point $x\in X$ arrives with its color $\textsf{color}(x) \in [k]$, and as usual for dynamic geometric streams, the input points are restricted to the discrete grid $\{0, \ldots, \Delta\}^2$. We design a single-pass streaming algorithm that uses $\mathrm{poly}(k \cdot \log\Delta)$ space and time, and estimates the cost of an optimal Steiner forest solution within ratio arbitrarily close to the famous Euclidean Steiner ratio $\alpha_2$ (currently $1.1547 \le \alpha_2 \le 1.214$). This approximation guarantee matches the state-of-the-art bound for streaming Steiner tree, i.e., when $k=1$, and it is a major open question to improve the ratio to $1 + \epsilon$ 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 has so far not been applied in the streaming setting. We complement our streaming algorithm for the Steiner forest problem with simple arguments showing that any finite approximation requires $\Omega(k)$ bits of space.