Time-Optimal Construction of Overlay Networks

Abstract

We show how to construct an overlay network of constant degree and diameter O(log n) in time O(log n) starting from an arbitrary weakly connected graph. We assume a synchronous communication network in which nodes can send messages to nodes they know the identifier of, and new connections can be established by sending node identifiers. If the initial network's graph is weakly connected and has constant degree, then our algorithm constructs the desired topology with each node sending and receiving only O(log n) messages in each round in time O(log n), w.h.p., which beats the currently best O(log 3/2 n) time algorithm of [Götte et al., SIROCCO'19]. Since the problem cannot be solved faster than by using pointer jumping for O(log n) rounds (which would even require each node to communicate Ω(n) bits), our algorithm is asymptotically optimal. We achieve this speedup by using short random walks to repeatedly establish random connections between the nodes that quickly reduce the conductance of the graph using an observation of [Kwok and Lau, APPROX'14].Additionally, we show how our algorithm can be used to efficiently solve graph problems in hybrid networks [Augustine et al., SODA'20]. Motivated by the idea that nodes possess two different modes of communication, we assume that communication of the initial edges is unrestricted, whereas only polylogarithmically many messages can be communicated over edges that have been established throughout an algorithm's execution. For an (undirected) graph G with arbitrary degree, we show how to compute connected components, a spanning tree, and biconnected components in time O(log n), w.h.p. Furthermore, we show how to compute an MIS in time O(log d+log log n), w.h.p., where d is the initial degree of G.

Deterministic Distributed Algorithms and Lower Bounds in the Hybrid Model

“…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.…”

“…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].…”