Global Computation in a Poorly Connected World: Fast Rumor Spreading with No Dependence on Conductance

In this paper, we study the question of how efficiently a collection of interconnected nodes can perform a global computation in the widely studied GOSSIP model of communication. In this model, nodes do not know the global topology of the network, and they may only initiate contact with a single neighbor in each round. This model contrasts with the much less restrictive LOCAL model, where a node may simultaneously communicate with all of its neighbors in a single round. A basic question in this setting is how many rounds of communication are required for the information dissemination problem, in which each node has some piece of information and is required to collect all others. In this paper, we give an algorithm that solves the information dissemination problem in at most $O(D+\text{polylog}{(n)})$ rounds in a network of diameter $D$, withno dependence on the conductance. This is at most an additive polylogarithmic factor from the trivial lower bound of $D$, which applies even in the LOCAL model. In fact, we prove that something stronger is true: any algorithm that requires $T$ rounds in the LOCAL model can be simulated in $O(T +\mathrm{polylog}(n))$ rounds in the GOSSIP model. We thus prove that these two models of distributed computation are essentially equivalent

An FPT Algorithm for Minimum Additive Spanner Problem

For a positive integer t and a graph G, an additive t-spanner of G is a spanning subgraph in which the distance between every pair of vertices is at most the original distance plus t. The Minimum Additive t-Spanner Problem is to find an additive t-spanner with the minimum number of edges in a given graph, which is known to be NP-hard. Since we need to care about global properties of graphs when we deal with additive t-spanners, the Minimum Additive t-Spanner Problem is hard to handle and hence only few results are known for it. In this paper, we study the Minimum Additive t-Spanner Problem from the viewpoint of parameterized complexity. We formulate a parameterized version of the problem in which the number of removed edges is regarded as a parameter, and give a fixed-parameter algorithm for it. We also extend our result to the case with both a multiplicative approximation factor ? and an additive approximation parameter ?, which we call (?, ?)-spanners

Improved Parallel Algorithms for Spanners and Hopsets

We use exponential start time clustering to design faster and more work-efficient parallel graph algorithms involving distances. Previous algorithms usually rely on graph decomposition routines with strict restrictions on the diameters of the decomposed pieces. We weaken these bounds in favor of stronger local probabilistic guarantees. This allows more direct analyses of the overall process, giving: * Linear work parallel algorithms that construct spanners with $O(k)$ stretch and size $O(n^{1+1/k})$ in unweighted graphs, and size $O(n^{1+1/k} \log k)$ in weighted graphs. * Hopsets that lead to the first parallel algorithm for approximating shortest paths in undirected graphs with $O(m\;\mathrm{polylog}\;n)$ work

Message Reduction in the LOCAL Model Is a Free Lunch

A new spanner construction algorithm is presented, working under the LOCAL model with unique edge IDs. Given an n-node communication graph, a spanner with a constant stretch and O(n^{1 + epsilon}) edges (for an arbitrarily small constant epsilon > 0) is constructed in a constant number of rounds sending O(n^{1 + epsilon}) messages whp. Consequently, we conclude that every t-round LOCAL algorithm can be transformed into an O(t)-round LOCAL algorithm that sends O(t * n^{1 + epsilon}) messages whp. This improves upon all previous message-reduction schemes for LOCAL algorithms that incur a log^{Omega (1)} n blow-up of the round complexity

Rumor Spreading with No Dependence on Conductance

National Science Foundation (U.S.) (CCF-0843915