A lower bound for distributed averaging algorithms on the line graph

We derive lower bounds on the convergence speed of a widely used class of distributed averaging algorithms. In particular, we prove that any distributed averaging algorithm whose state consists of a single real number and whose (possibly nonlinear) update function satisfies a natural smoothness condition has a worst case running time of at least on the order of n2 on a line network of n nodes. Our results suggest that increased memory or expansion of the state space is crucial for improving the running times of distributed averaging algorithms.

[1]  Stephen P. Boyd,et al.  Fast linear iterations for distributed averaging , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[2]  Qun Li,et al.  Global clock synchronization in sensor networks , 2006, IEEE Transactions on Computers.

[3]  Stephen P. Boyd,et al.  Fastest Mixing Markov Chain on a Path , 2006, Am. Math. Mon..

[4]  John N. Tsitsiklis,et al.  On distributed averaging algorithms and quantization effects , 2007, 2008 47th IEEE Conference on Decision and Control.

[5]  Jorge Cortes,et al.  Notes on averaging over acyclic digraphs and discrete coverage control , 2006, CDC.

[6]  Naoki Hayashi,et al.  Application of a consensus problem to fair multi-resource allocation in real-time systems , 2008, 2008 47th IEEE Conference on Decision and Control.

[7]  John N. Tsitsiklis,et al.  Distributed Asynchronous Deterministic and Stochastic Gradient Optimization Algorithms , 1984, 1984 American Control Conference.

[8]  Benjamin Van Roy,et al.  Consensus Propagation , 2005, IEEE Transactions on Information Theory.

[9]  Mac Schwager,et al.  Consensus learning for distributed coverage control , 2008, 2008 IEEE International Conference on Robotics and Automation.

[10]  Han-Lim Choi,et al.  Consensus-Based Auction Approaches for Decentralized Task Assignment , 2008 .

[11]  Jie Lin,et al.  Coordination of groups of mobile autonomous agents using nearest neighbor rules , 2003, IEEE Trans. Autom. Control..

[12]  Jorge Cortés,et al.  Finite-time convergent gradient flows with applications to network consensus , 2006, Autom..

[13]  Kyomin Jung,et al.  Distributed Averaging Via Lifted Markov Chains , 2009, IEEE Transactions on Information Theory.

[14]  C.N. Hadjicostis,et al.  Finite-Time Distributed Consensus in Graphs with Time-Invariant Topologies , 2007, 2007 American Control Conference.

[15]  J.N. Tsitsiklis,et al.  Convergence in Multiagent Coordination, Consensus, and Flocking , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[16]  Brian D. O. Anderson,et al.  Reaching a Consensus in a Dynamically Changing Environment: Convergence Rates, Measurement Delays, and Asynchronous Events , 2008, SIAM J. Control. Optim..

[17]  Huaiqing Wang,et al.  Multi-agent coordination using nearest neighbor rules: revisiting the Vicsek model , 2004, ArXiv.

[18]  Rolf Stadler,et al.  Gossiping for threshold detection , 2009, 2009 IFIP/IEEE International Symposium on Integrated Network Management.

[19]  Richard M. Murray,et al.  Consensus problems in networks of agents with switching topology and time-delays , 2004, IEEE Transactions on Automatic Control.

[20]  Luc Moreau,et al.  Stability of multiagent systems with time-dependent communication links , 2005, IEEE Transactions on Automatic Control.

[21]  Ruggero Carli,et al.  Distributed Kalman filtering based on consensus strategies , 2008, IEEE Journal on Selected Areas in Communications.

[22]  Jonathan P. How,et al.  An unbiased Kalman consensus algorithm , 2006 .

[23]  John N. Tsitsiklis,et al.  Convergence Speed in Distributed Consensus and Averaging , 2009, SIAM J. Control. Optim..

[24]  Stephen P. Boyd,et al.  A scheme for robust distributed sensor fusion based on average consensus , 2005, IPSN 2005. Fourth International Symposium on Information Processing in Sensor Networks, 2005..

[25]  Sonia Martínez,et al.  On the Convergence Time of Asynchronous Distributed Quantized Averaging Algorithms , 2010, IEEE Transactions on Automatic Control.

[26]  R. Srikant,et al.  Quantized Consensus , 2006, 2006 IEEE International Symposium on Information Theory.