Abstract
This paper considers the problem of computing general commutative and associative aggregate functions (such as Sum) over distributed inputs held by nodes in a distributed system, while tolerating failures. Specifically, there are $N$ nodes in the system, and the topology among them is modeled as a general undirected graph. Whenever a node sends a message, the message is received by all of its neighbors in the graph. Each node has an input, and the goal is for a special root node (e.g., the base station in wireless sensor networks or the gateway node in wireless ad hoc networks) to learn a certain commutative and associate aggregate of all these inputs. All nodes in the system except the root node may experience crash failures, with the total number of edges incidental to failed nodes being upper bounded by f. The timing model is synchronous where protocols proceed in rounds. Within such a context, we focus on the following question: Under any given constraint on time complexity, what is the lowest communication complexity, in terms of the number of bits sent (i.e., locally broadcast) by each node, needed for computing general commutative and associate aggregate functions?
This work, for the first time, reduces the gap between the upper bound and the lower bound for the above question from polynomial to polylog. To achieve this reduction, we present significant improvements over both the existing upper bounds and the existing lower bounds on the problem.