Consider a point-to-point message-passing network. We are interested in the asynchronous crash-tolerant consensus problem in incomplete networks. We study the feasibility and efficiency of approximate consensus under different restrictions on topology knowledge and the relay depth, i.e., the maximum number of hops any message can be relayed. These two constraints are common in large-scale networks, and are used to avoid memory overload and network congestion respectively. Specifically, for different values of integers k, k , we consider that each node knows all its neighbors of at most k-hop distance (k-hop topology knowledge), and the relay depth is k . We consider both directed and undirected graphs. More concretely, we answer the following main question in asynchronous systems: What is a tight condition on the underlying communication graphs for achieving approximate consensus if each node has only a k-hop topology knowledge and relay depth k? To prove that the necessary conditions presented in the paper are also sufficient, we have developed algorithms that achieve consensus in graphs satisfying those conditions: -The first class of algorithms requires k-hop topology knowledge and relay depth k. Unlike prior algorithms, these algorithms do not flood the network, and each node does not need the full topology knowledge. We show how the convergence time and the message complexity of those algorithms is affected by k, providing the respective upper bounds. -The second set of algorithms requires only one-hop neighborhood knowledge, i.e., immediate incoming and outgoing neighbors, but needs to flood the network (i.e., relay depth is n, where n is the number of nodes). One result that may be of independent interest is a topology discovery mechanism to learn and "estimate" the topology in asynchronous directed networks with crash faults.

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