Stability of load balancing algorithms in dynamic adversarial systems

  • Authors:

  • Elliot Anshelevich;David Kempe;Jon Kleinberg

  • Affiliations:

  • Cornell University, Ithaca NY;Cornell University, Ithaca NY;Cornell University, Ithaca NY

  • Venue:
  • STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
  • Year:
  • 2002

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Abstract

In the dynamic load balancing problem, we seek to keep the job load roughly evenly distributed among the processors of a given network. The arrival and departure of jobs is modeled by an adversary restricted in its power. Muthukrishnan and Rajaraman (1998) gave a clean characterization of a restriction on the adversary that can be considered the natural analogue of a cut condition. They proved that a simple local balancing algorithm proposed by Aiello et. al. (1993) is stable against such an adversary if the insertion rate is restricted to a (1—&egr;) fraction of the cut size. They left as an open question whether the algorithm is stable at rate 1.In this paper, we resolve this question positively, by proving stability of the local algorithm at rate 1. Our proof techniques are very different from the ones used by Muthukrishnan and Rajaraman, and yield a simpler proof and tighter bounds on the difference in loads.In addition, we introduce a multi-commodity version of this load balancing model, and show how to extend the result to the case of balancing two different kinds of loads at once (obtaining as a corollary a new proof of the 2-commodity Max-Flow Min-Cut Theorem). We also show how to apply the proof techniques to the problem of routing packets in adversarial systems. Awerbuch et. al. (2001) showed that the same load balancing algorithm is stable against an adversary inserting packets at rate 1 with a single destination, in dynamically changing networks. Our techniques give a much simpler proof for a different model of adversarially changing networks.