Graph Spanners for Group Steiner Distances

TL;DR

The paper introduces and analyzes the group Steiner distance, a path-based metric requiring traversal through designated groups, and develops a comprehensive set of graph spanners and distance oracles for this metric. It reveals intricate connections to classical spanners and beer distance, proves fundamental hardness and tractable FPT results, and provides a spectrum of constructions for singleton and general-group instances, including $(1+ε)$- and α+1-stretch spanners with near-optimal sizes. For singleton groups, near-tight, scalable constructions are achieved, along with a constant-query-time distance oracle; for general group sizes, two robust constructions yield stretch $2α+1$ with complementary size trade-offs and corresponding oracles. The work also establishes single-source preservers, a stretch-3 tree spanner, and an α+1 spanner, all accompanied by efficient distance oracles and rigorous bounds, highlighting both theoretical insights and practical routing/data-structure implications. Overall, the results substantially extend spanner theory to group-based distance notions and open avenues for further tightening trade-offs and optimizing distance-query performance in related applications.

Abstract

A spanner is a sparse subgraph of a given graph $G$ which preserves distances, measured w.r.t.\ some distance metric, up to a multiplicative stretch factor. This paper addresses the problem of constructing graph spanners w.r.t.\ the group Steiner metric, which generalizes the recently introduced beer distance metric. In such a metric we are given a collection of groups of required vertices, and we measure the distance between two vertices as the length of the shortest path between them that traverses at least one required vertex from each group. We discuss the relation between group Steiner spanners and classic spanners and we show that they exhibit strong ties with sourcewise spanners w.r.t.\ the shortest path metric. Nevertheless, group Steiner spanners capture several interesting scenarios that are not encompassed by existing spanners. This happens, e.g., for the singleton case, in which each group consists of a single required vertex, thus modeling the setting in which routes need to traverse certain points of interests (in any order). We provide several constructions of group Steiner spanners for both the all-pairs and single-source case, which exhibit various size-stretch trade-offs. Notably, we provide spanners with almost-optimal trade-offs for the singleton case. Moreover, some of our spanners also yield novel trade-offs for classical sourcewise spanners. Finally, we also investigate the query times that can be achieved when our spanners are turned into group Steiner distance oracles with the same size, stretch, and building time.

Figures (6)

• Figure 1: A shortest group Steiner path from $s$ to $t$ with length $9$. The required vertices of the two groups $R_1$ and $R_2$ are depicted as squares and triangles, respectively.
• Figure 3: On the left: a tree with edge-weights, where unlabeled edges have weight $1$, and a possible decomposition into micro-trees as computed by our procedure with $W=6$. The edges $(v, u_i)$ are highlighted in red. On the right: a qualitative depiction of the paths used in the analysis of the stretch of our group Steiner $(1+\varepsilon)$-spanner. The shortest group Steiner path between $s$ and $t$ is shown in bold, while $\pi_s$ and $\pi_t$ are shown in blue. The white triangles are the trees in $F_i$ and $F_j$ rooted in $c_i$ and $c_j$, respectively.
• Figure 4: A qualitative depiction of the path constructed in the proof of \ref{['lemma:stretch_2_group_steiner_path_in_T']}. The path in bold is the unique path $\pi$ from $c_s$ to $c_t$ in $T$. Each $\pi_i$ is the Eulerian tour of the corresponding tree $T_i$, and the final path is in red.
• Figure 5: The lower bound constructions of \ref{['thm:singleton_lb_single_source']} (left) and \ref{['thm:singleton_preserver']} (right) with $k=10$ required vertices. The generic required vertex $r_i$ is depicted as a squared labelled with $i$.
• Figure 6: A qualitative depiction of analysis of the stretch $2 \alpha+1$. The shortest group Steiner path $\pi^*$ from $s$ to $t$ is in bold, while the path $\widetilde{\pi}$ in the spanner is in red.

Theorems & Definitions (15)

• Lemma 1
• Lemma 4
• Corollary 5
• Corollary 6
• Theorem 7
• Theorem 8
• Theorem 9
• Lemma 10
• Theorem 11
• Theorem 12