Time-Optimal Construction of Overlay Networks

Abstract

We show how to construct an overlay network of constant degree and diameter (log ) in time (log ) 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 establish new connections 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 (log ) messages in each round in time (log ), w.h.p., which beats the currently best (log 3/2 ) time algorithm of [Götte et al., SIROCCO'19]. Since the problem cannot be solved faster than by using pointer jumping for (log ) rounds (which would even require each node to communicate Ω( ) 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. In contrast, only polylogarithmically many messages can be communicated over edges that have been established throughout an algorithm's execution. For an (undirected) graph with arbitrary degree, we show how to compute connected components, a spanning tree, and biconnected components in time (log ), w.h.p. Furthermore, we show how to compute an MIS in time (log + log log ), w.h.p., where is the initial degree of .

Applications and Implications of a General Framework for Self-Stabilizing Overlay Networks

“…Said in another way, DConstructor assumes an arbitrary initial topology, but not an arbitrary initial state. This is also the same assumption in the work of Götte et al [5], who presented an algorithm for transforming a constant-degree network into a tree in O(log n) rounds.…”

Near-Shortest Path Routing in Hybrid Communication Networks

“…The use of the NCC 0 model in the global communication mode is motivated by the fact that nodes can communicate with any other node in the world via the cellular infrastructure given its ID (e.g., its phone number). Note that NCC 0 is weaker than NCC, which assumes a clique from the start, but it is known that once the right topology has been set up in NCC 0 (which can be done in O(log n) rounds [16]), any communication round in NCC can be simulated by O(log n) communication rounds in NCC 0 [25].…”

“…We give an overview on some techniques for hybrid networks that are used by our tree labelling algorithm in Section 4. A more detailed description of these techniques can be found in [11,16,15]. Performing pointer jumping on each of our portals in the portal tree, the grid nodes within each portal P are able to set up a structure on which they can quickly broadcast information to all grid nodes within P. By doing so, we immediately obtain the following lemma.…”

“…We compute the line L via the Euler-tour described earlier on the rooted tree T and apply Lemma 34 on L to obtain the structure L + . 9 A more general approach to this problem is presented in [16,Lemma 4.12], where the goal is to compute the value of a distributive aggregate function for each node v's own subtree. An aggregate function f is called distributive if there is an aggregate function g such that for any multiset S and any partition S1, .…”

Network Scaffolding for Efficient Stabilization of the Chord Overlay Network

“…More recently, Gilbert et al presented DConstructor [6] which is able to build a correct topology from any initial topology and maintain this in the face of some joins and leaves. Götte et al [8] also presented an algorithm for transforming a constant-degree network into a tree in O(log n) rounds. The key difference between these works and ours, however, is we use a stronger fault model, requiring our algorithm to build the correct configuration regardless of the initial topology or the initial state of the nodes.…”