Tight Analyses of Two Local Load Balancing Algorithms

  • Authors:
  • Bhaskar Ghosh;F. T. Leighton;Bruce M. Maggs;S. Muthukrishnan;C. Greg Plaxton;R. Rajaraman;Andréa W. Richa;Robert E. Tarjan;David Zuckerman

  • Affiliations:
  • -;-;-;-;-;-;-;-;-

  • Venue:
  • SIAM Journal on Computing
  • Year:
  • 1999

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Abstract

This paper presents an analysis of the following load balancing algorithm. At each step, each node in a network examines the number of tokens at each of its neighbors and sends a token to each neighbor with at least 2d+1 fewer tokens, where d is the maximum degree of any node in the network. We show that within $O(\Delta / \alpha)$ steps, the algorithm reduces the maximum difference in tokens between any two nodes to at most $O((d^2 \log n)/\alpha)$, where $\Delta$ is the global imbalance in tokens (i.e., the maximum difference between the number of tokens at any node initially and the average number of tokens), n is the number of nodes in the network, and $\alpha$ is the edge expansion of the network. The time bound is tight in the sense that for any graph with edge expansion $\alpha$, and for any value $\Delta$, there exists an initial distribution of tokens with imbalance $\Delta$ for which the time to reduce the imbalance to even $\Delta/2$ is at least $\Omega(\Delta/\alpha)$. The bound on the final imbalance is tight in the sense that there exists a class of networks that can be locally balanced everywhere (i.e., the maximum difference in tokens between any two neighbors is at most 2d), while the global imbalance remains $\Omega((d^2 \log n) / \alpha)$. Furthermore, we show that upon reaching a state with a global imbalance of $O((d^2 \log n)/\alpha)$, the time for this algorithm to locally balance the network can be as large as $\Omega(n^{1/2})$. We extend our analysis to a variant of this algorithm for dynamic and asynchronous networks. We also present tight bounds for a randomized algorithm in which each node sends at most one token in each step.