Distributed Graph Realizations

Abstract

We study graph realization problems from a distributed perspective. The problem is naturally applicable to the distributed construction of overlay networks that must satisfy certain degree or connectivity properties, and we study it in the node capacitated clique (NCC) model of distributed computing, recently introduced for representing peer-to-peer networks.We focus on two central variants, degree-sequence realization and minimum threshold-connectivity realization. In the degree sequence problem, each node v is associated with a degree d(v), and the resulting degree sequence is realizable if it is possible to construct an overlay network in which the degree of each node v is d(v). The minimum threshold-connectivity problem requires us to construct an overlay network that satisfies connectivity constraints specified between every pair of nodes. Overlay network realizations can be either explicit or implicit. Explicit realizations require both endpoints of any edge in the realized graph to be aware of the edge. In implicit realizations, on the other hand, at least one endpoint of each edge of the realized graph needs to be aware of the edge. The main realization algorithms we present are the following.• An Õ(min{ √ m, ∆}) time algorithm for implicit realization of a degree sequence. Here, ∆ = max v d(v) is the maximum degree and m = (1/2) v d(v) is the number of edges in the final realization.• Õ(∆) time algorithm for an explicit realization of a degree sequence. We first compute an implicit realization and then transform it into an explicit one in Õ(∆) additional rounds.• An Õ(∆) time algorithm for the threshold connectivity problem that obtains an explicit solution and an improved Õ(1) algorithm for implicit realization when all nodes know each other's IDs. These algorithms are 2-approximations w.r.t. the number of edges.We complement our upper bounds with lower bounds to show that the above algorithms are tight up to factors of log n. Additionally, we provide algorithms for realizing trees and an Õ(1) round algorithm for approximate degree sequence realization.

Time-Optimal Construction of Overlay Networks

“…2. For [29,8,17,9,5], the assumption that the graph starts in well-initialized overlay can be dropped.…”

“…This corresponds to the so-called NCC 0 model [5], which is a variant of the general Node-Capacitated Clique (NCC) model for overlay networks [6]. The bound of O(log n) is argued as a natural choice, preventing algorithms from being needlessly complicated while still ensuring scalability.…”

“…The authors present O(a) algorithms (where O hides polylogarithmic factors, and a is the arboricity 4 of G) for local problems such as MIS, matching, or coloring, a O(D + a) algorithm for BFS tree, and a O(1) algorithm for the minimum spanning tree (MST) problem. Augustine et al [5], who consider the NCC 0 model, consider O(d)-time algorithms for so-called graph realization problems. They assume, however, that the networks starts as a line, which makes overlay construction very easy.…”

Fast Hybrid Network Algorithms for Shortest Paths in Sparse Graphs

“…The authors present O(a) algorithms for local problems such as MIS, matching, or coloring, a O(D + a) algorithm for BFS tree, and a O(1) algorithm for MST. Recently, O(∆)-time algorithms for graph realization problems have been presented [6], where ∆ is the maximum node degree; notably, most of the algorithms work in the NCC 0 variant. Furthermore, Robinson [48] investigates the information the nodes need to learn to jointly solve graph problems and derives a lower bound for constructing spanners in the NCC.…”

“…Remark 4. Let H = (V, E) be a forest in which every node v ∈ V stores some value p v , and let f be a distributive aggregate function 6…”