Distributed Graph Realizations

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. (1) A $\tilde O(\min \{ \sqrt m ,\Delta \} )$ time algorithm for implicit realization of a degree sequence. Here, Δ = max
v
d(v) is the maximum degree and m = (1/2) v d(v) is the number of edges in the final realization. (2) A $\tilde O\left( \Delta \right)$ 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 $\tilde O\left( \Delta \right)$ additional rounds. (3) A $\tilde O\left( \Delta \right)$ time algorithm for the threshold connectivity problem that obtains an explicit solution and an improved $\tilde O\left( 1 \right)$ algorithm for implicit realization when all nodes know each other’s IDs. These algorithms are 2-approximations w.r.t. the number of edges. Our algorithms are complemented by lower bounds showing tightness up to log n factors. Additionally, we provide algorithms for realizing trees and an $\tilde O\left( 1 \right)$ round algorithm for approximate degree sequence realization.