Universally Optimal Information Dissemination and Shortest Paths in the HYBRID Distributed Model

TL;DR

This work studies universal optimality in the HYBRID distributed model, introducing neighborhood quality $\mathcal{NQ}_{k}$ to capture the inherent complexity of global tasks under a dual communication regime. It develops universally optimal algorithms for information dissemination (including $k$-dissemination, $k$-aggregation, and $(k,\ell)$-routing) and for shortest-path problems (including $(k,\ell)$-SP and APSP), with runtimes that scale as $\widetilde{O}(\mathcal{NQ}_{k})$ and, in weighted cases, via refined techniques such as skeleton graphs and spanners. The paper also presents existentially optimal SSP subroutines that feed into the universal results and proves universal lower bounds, showing that the proposed algorithms are near-optimal for broad graph classes. By combining clustering, overlays, and adaptive helper sets, the authors demonstrate that many fundamental problems can be solved efficiently in HYBRID, with strong implications for edge computing and heterogeneous networks. Overall, the results significantly advance understanding of universal optimality in hybrid networks and provide versatile subroutines for a range of graph problems.

Abstract

In this work we consider the HYBRID model of distributed computing, introduced recently by Augustine, Hinnenthal, Kuhn, Scheideler, and Schneider (SODA 2020), where nodes have access to two different communication modes: high-bandwidth local communication along the edges of the graph and low-bandwidth all-to-all communication, capturing the non-uniform nature of modern communication networks. Prior work in HYBRID has focused on showing existentially optimal algorithms, meaning there exists a pathological family of instances on which no algorithm can do better. This neglects the fact that such worst-case instances often do not appear or can be actively avoided in practice. In this work, we focus on the notion of universal optimality, first raised by Garay, Kutten, and Peleg (FOCS 1993). Roughly speaking, a universally optimal algorithm is one that, given any input graph, runs as fast as the best algorithm designed specifically for that graph. We show the first universally optimal algorithms in HYBRID. We present universally optimal solutions for fundamental information dissemination tasks, such as broadcasting and unicasting multiple messages in HYBRID. Furthermore, we apply these tools to obtain universally optimal solutions for various shortest paths problems in HYBRID. A main conceptual contribution of this work is the conception of a new graph parameter called neighborhood quality that captures the inherent complexity of many fundamental graph problems in HYBRID. We also show new existentially optimal shortest paths algorithms in HYBRID, which are utilized as key subroutines in our universally optimal algorithms and are of independent interest. Our new algorithms for $k$-source shortest paths match the existing $\tildeΩ(\sqrt{k})$ lower bound for all $k$. Previously, the lower bound was only known to be tight when $k \in \tildeΩ(n^{2/3})$.

Figures (2)

• Figure 1: (Existential) complexity landscape of the $k$-SSP problem with the number of sources on the horizontal and the round complexity on the vertical axis. Circles or bars denote known upper bounds (ours in gray). The gray shaded area denotes the lower bound. References are as follows $*$: augustine2020shortest, $\dagger$: Kuhn2020, $\ddagger$: CensorHillel2021, §: this work.
• Figure 2: Overview of the proof of \ref{['thm:optimal_dissemination']}. We first create clusters with $*{\mathcal{NQ}_{k}}$ weak diameter and roughly the same number of nodes, and then we construct a logical tree of the clusters, with logarithmic maximum degree and depth. Inside each cluster, we create a logical binary tree over the nodes of the cluster. As the clusters are roughly of the same size, we can ensure that the trees inside the clusters have the exact same shape. We ensure that for any two neighboring clusters in the cluster tree, the nodes in their internal trees know their respective nodes in the tree of the other cluster -- e.g., node $3$ in cluster $C_i$ knows the identifier of nodes $3$ in $C_j, C_\ell$ and can communicate directly with them using global communication. Once we are done constructing all of these trees, we propagate all the $k$ messages in the graph up to the top of the cluster tree, and then we propagate them back down to all the clusters to ensure that every node in the graph receives all the messages.

Theorems & Definitions (131)

• Definition 1.1: $k$-dissemination
• Definition 1.2: $k$-aggregation
• Definition 1.3: $(k,\ell)$-routing
• Definition 1.4: Universal Optimality, see Garay1998haeupler2021universally
• Theorem 1: Broadcast
• Theorem 2: Aggregation
• Theorem 3: Unicast
• Theorem 4: Information dissemination lower bounds
• Corollary 2.1: Implied by \ref{['thm:optimal_dissemination', 'thm:broadcast_unicast_LB']}
• Theorem 5: $(k,\ell)$-SP