Networks cannot compute their diameter in sublinear time

Authors:
Silvio Frischknecht;Stephan Holzer;Roger Wattenhofer
Affiliations:
Computer Engineering and Networks Laboratory (TIK), ETH Zurich, Switzerland;Computer Engineering and Networks Laboratory (TIK), ETH Zurich, Switzerland;Computer Engineering and Networks Laboratory (TIK), ETH Zurich, Switzerland
Venue:
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Year:
2012

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Abstract

We study the problem of computing the diameter of a network in a distributed way. The model of distributed computation we consider is: in each synchronous round, each node can transmit a different (but short) message to each of its neighbors. We provide an Ω(n) lower bound for the number of communication rounds needed, where n denotes the number of nodes in the network. This lower bound is valid even if the diameter of the network is a small constant. We also show that a (3/2 − ε)-approximation of the diameter requires Ω (√n + D) rounds. Furthermore we use our new technique to prove an Ω (√n + D) lower bound on approximating the girth of a graph by a factor 2 − ε.