Fast Hybrid Network Algorithms for Shortest Paths in Sparse Graphs

Abstract

We consider the problem of computing shortest paths in hybrid networks, in which nodes can make use of different communication modes. For example, mobile phones may use ad-hoc connections via Bluetooth or Wi-Fi in addition to the cellular network to solve tasks more efficiently. Like in this case, the different communication modes may differ considerably in range, bandwidth, and flexibility. We build upon the model of Augustine et al. [SODA '20], which captures these differences by a local and a global mode. Specifically, the local edges model a fixed communication network in which O(1) messages of size O(log n) can be sent over every edge in each synchronous round. The global edges form a clique, but nodes are only allowed to send and receive a total of at most O(log n) messages over global edges, which restricts the nodes to use these edges only very sparsely.We demonstrate the power of hybrid networks by presenting algorithms to compute Single-Source Shortest Paths and the diameter very efficiently in sparse graphs. Specifically, we present exact O(log n) time algorithms for pseudotrees (i.e., graphs that contain at most one cycle), and 3-approximations for graphs that have at most n + O(n 1/3 ) edges and arboricity O(log n). For these graph classes, our algorithms provide exponentially faster solutions than the best known algorithms for general graphs in this model.

Deterministic Distributed Algorithms and Lower Bounds in the Hybrid Model

“…More restricted families of graphs (e.g. very sparse graphs or cactus graphs) were considered by Feldmann et al [23], establishing an exponential speedup over some of the previous results even though they modeled the local network via CONGEST, which is of course substantially weaker than LOCAL. Finally, Götte et al [32] provided several fast hybrid algorithms for problems such as connected components, spanning tree, and the maximal independent set.…”

“…We refer to Appendix B for the detailed implementation. It should be noted that a deterministic O(log 2 n) algorithm in CONGEST + NCC for the MST problem was developed in [23] with very different techniques. More importantly, Ghaffari and Haeupler [26] managed to establish the following: ▶ Theorem 27 ([26]).…”

“…cables or optical fibers) offer large bandwidth and high efficiency, but lack flexibility as they cannot be dynamically adapted by the nodes. In contrast, the global mode relates to logical networks, which are formed as an overlay over a shared physical network [23]; here the feature of all-to-all communication comes at the cost of providing very limited throughput.…”

“…In this work we follow the recent line of research [6,41,23,11,32,12] which endeavors to explore from a theoretical standpoint the power of hybrid models in distributed computing; specifically, the main issue that arises is whether combining two different communication modes offers a substantial improvement over each mode separately. This question is answered in the affirmative for a series of fundamental problems in distributed algorithms, while we also provide some hardness results mainly based on well-established communication-complexity lower bounds.…”

Almost Universally Optimal Distributed Laplacian Solvers via Low-Congestion Shortcuts

“…In our work we consider a substantially weaker model, imposing a severe limitation on the communication over the "local edges". This particular variant has been already studied in some recent works for a variety of fundamental problems [FHS20;Göt+20].…”