Dynamic Representations of Sparse Distributed Networks

Abstract

In 1999, Brodal and Fagerberg (BF) gave an algorithm for maintaining a low outdegree orientation of a dynamic uniformly sparse graph. Specifically, for a dynamic graph on n -vertices, with arboricity bounded by ɑ at all times, the BF algorithm supports edge updates in O (log n ) amortized update time, while keeping the maximum outdegree in the graph bounded by O ( ɑ ). Such an orientation provides a basic data structure for uniformly sparse graphs, which found applications to several dynamic graph algorithms, including adjacency queries and labeling schemes, maximal and approximate matching, approximate vertex cover, forest decomposition, and distance oracles. A significant weakness of the BF algorithm is the possible temporary blowup of the maximum outdegree, following edge insertions. Although BF eventually reduces all outdegrees to O ( ɑ ), some vertices may reach an outdegree of Ω( n ) during the process, and hence local memory usage at the vertices, which is an important quality measure in distributed systems, cannot be bounded. We show how to modify the BF algorithm to guarantee that the outdegrees of all vertices are bounded by O ( ɑ ) at all times, without hurting any of its other properties and present an efficient distributed implementation of the modified algorithm. This provides the first representation of distributed networks in which the local memory usage at all vertices is bounded by the arboricity (which is essentially the average degree of the densest subgraph) rather than the maximum degree. For settings where there is no strict limitation on the local memory, one may take the temporary outdegree blowup to the extreme and allow a permanent outdegree blowup. This allows us to address the second significant weakness of the BF algorithm—its inherently global nature: An insertion of an edge ( u,v ) may trigger changes in the orientations of edges that are arbitrarily far away from u and v . Such a non-local scheme may be prohibitively expensive in various practical applications. We suggest an alternative local scheme, which does not guarantee any outdegree bound on the vertices, yet is just as efficient as the BF scheme for some of the aforementioned applications. For example, we obtain a local dynamic algorithm for maintaining a maximal matching with sub-logarithmic update time in uniformly sparse networks, providing an exponential improvement over the state of the art in this context. We also present a distributed implementation of this scheme and some of its applications.