Streaming Algorithms for Geometric Steiner Forest

TL;DR

This work tackles the Euclidean Steiner Forest Problem (SFP) in dynamic geometric streams, where points with color labels arrive and depart on a grid in the plane. The authors adapt Arora’s offline dynamic-programming framework to the streaming setting by introducing a small, carefully chosen set of base subproblems (maximal simple squares) and developing streaming techniques—sampling, linear sketching, and subset-MST queries—to estimate DP subproblem costs efficiently. They prove a single-pass streaming algorithm achieving a $(α_2+ε)$-approximation with space and update time poly$(k \, \text{log} \, Δ)$ and query time poly$(k, \log k, \varepsilon^{-1}, \log Δ)$, matching the state-of-the-art streaming bound for Steiner tree when $k=1$ and providing a near-optimal space bound up to a proven Ω$(k)$ lower bound. The method leverages a fusion of sampling-based MST sketches, delta-coverings, and a streaming-enabled Arora-style DP to deliver a practical, theoretically-grounded approach to connectivity-constrained geometric streaming problems, with potential extensions to related objectives and higher dimensions.

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 $\{0, \ldots, Δ\}^2$. We design a single-pass streaming algorithm that uses $\mathrm{poly}(k \cdot \logΔ)$ space and time, and estimates the cost of an optimal Steiner forest solution within ratio arbitrarily close to the famous Euclidean Steiner ratio $α_2$ (currently $1.1547 \le α_2 \le 1.214$). 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 $1 + ε$ 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 $Ω(k)$ bits of space.

Figures (5)

• Figure 1: An SFP instance with $k=2$ color classes (blue squares and red disks). The left figure shows a solution (straight lines) that forms a single connected component, which is cheaper than a separate component for each color. The right figure shows an even cheaper solution using Steiner points (black circles).
• Figure 2: A hierarchical geometric decomposition with four levels (on the left) and its corresponding quadtree (on the right). In particular, the shaded square on the left corresponds to the shaded node in the quadtree.
• Figure 3: A structured solution inside a square $R$ that has portals (hollowed squares) on its boundary and is split into $4 \times 4$ cells (depicted by dashed lines). The solution inside $R$ is portal-respecting with active portals $a_1, a_2, a_3, b$. It has three local components; two of them are connected to the boundary, and then by the cell property, points of same color (say red disks) in the same cell must be in the same local component. A third local component (whose points are depicted as hollowed circles) is entirely inside $R$, so the cell property does not apply to it. For simplicity, local components are depicted without Steiner points (i.e., using MST).
• Figure 4: An input with three colors that has $13$ maximal simple squares (counting also the $4$ empty squares).
• Figure 5: Approximately testing which color classes intersect every cell. Input points (red disks) are covered by nearby grid points (crosses), and the artificial points added in close-enough cells (black circles), shown for simplicity only for cells containing no input points.

Theorems & Definitions (33)

• Remark
• Theorem 1.1
• Remark
• Theorem 3.2: DBLP:journals/algorithmica/BateniH12
• Theorem 4.1
• Definition 4.1: Simple squares
• Lemma 4.2
• proof
• Lemma 4.3
• proof