Distributed Computation in Node-Capacitated Networks

Shortest Paths in a Hybrid Network Model

“…For example, the work of Gmyr et al [20] implies that making use of global edges with the same constraints as our model significantly improves the ability to monitor properties of the network formed by the local edges. Furthermore, Augustine et al [4] propose the so-called node-capacitated clique model, which is identical to our model for the global edges (but which has no local edges). They present distributed algorithms for various fundamental graph problems, including problems such as computing an MST, a BFS tree, a maximal independent set, a maximal matching, or a vertex coloring of the given input graph.…”

“…This modeling choice is motivated by the idea that local communication rather relates to physical networks, where an edge corresponds to a physical connection (e.g., cable-or ad-hoc networks), whereas global communication primarily captures aspects of logical networks that are formed as an overlay on top of some shared physical infrastructure. For appropriate choices of λ and γ, our model captures various established network models: LOCAL (λ = ∞, γ = 0), CONGEST (λ = O(1), γ = 0), Congested Clique 5 (λ = O(1), γ = 0 and G is a clique), and the recently introduced Node-Capacitated Clique model (λ = 0, γ = O(log n)) [4].…”

“…For (2), v does the following: It first sends a message (u, d + d i (v, x)) to x. More precisely, u does not send the message directly, but participates in an aggregation in a simulated butterfly network as described in [4]. In the aggregation procedure x does not receive all messages, but only the one that contains the minimum distance value.…”

“…This modeling choice is motivated by the idea that local communication rather relates to physical networks, where an edge corresponds to a physical connection (e.g., cable-or ad-hoc networks), whereas global communication primarily captures aspects of logical networks that are formed as an overlay on top of some shared physical infrastructure. For appropriate choices of λ and γ, our model captures various established network models: LOCAL (λ = ∞, γ = 0), CONGEST (λ = O(1), γ = 0), Congested Clique 5 (λ = O(1), γ = 0 and G is a clique), and the recently introduced Node-Capacitated Clique model (λ = 0, γ = O(log n)) [4].In order to demonstrate the power of combining local and global communication, we will focus the variant of the model in which local edges are fully uncapacitated (λ = ∞), and global communication is heavily restricted (γ = O(log n)). Thus, our model is a combination of the most permissive LOCAL model for local edges and the very restrictive node-capacitated clique model for the global edges, which makes it particularly clean and well-suited to investigate the power of hybrid networks from a theoretical perspective.…”

“…This knowledge can be used to distribute distance information to the source s in an iterative fashion over the global network. In iteration i, all nodes that have a shortest path to s with O(i 2 ) hops learn their distance to s. An iteration takes only O(1) rounds, leveraging a divide and conquer approach on the aforementioned trees and the aggregation protocol of [4]. Note that on unweighted graphs using only either local or global edges, the best known algorithms require Ω(SPD) rounds (which is tight for local edges as D = SPD on unweighted graphs).…”