# On Balls and Bins with Deletions

## Abstract

We consider the problem of extending the analysis of balls and bins processes where a ball is placed in the least loaded of *d* randomly chosen bins to cover deletions. In particular, we are interested in the case where the system maintains a fixed load, and deletions are determined by an adversary before the process begins. We show that with high probability the load in any bin is O(log log *n*). In fact, this result follows from recent work by Cole et al. concerning a more difficult problem of routing in a butterfly network.

The main contribution of this paper is to give a different proof of this bound, which follows the lines of the analysis of Azar, Broder, Karlin, and Upfal for the corresponding static load balancing problem. We also give a specialized (and hence simpler) version of the argument from the paper by Cole et al. for the balls and bins scenario. Finally, we provide an alternative analysis also based on the approach of Azar, Broder, Karlin, and Upfal for the special case where items are deleted according to their age. Although this analysis does not yield better bounds than our argument for the general case, it is interesting because it utilizes a two-dimensional family of random variables in order to account for the age of the items. This technique may be of more general use.

## Preview

Unable to display preview. Download preview PDF.

### References

- 1.M. Adler, P. Berenbrink, and K. Schróder. Analyzing an infinite parallel job allocation process. To appear in
*ESA 98*.Google Scholar - 2.M. Adler, S. Chakrabarti, M. Mitzenmacher, and L. Rasmussen. Parallel randomized load balancing. In
*Proceedings of the 27th Annual ACM Symposium on Theory of Computing*, 1995, pp. 238–247.Google Scholar - 3.N. Alon and J. H. Spencer.
**The Probabilistic Method**. John Wiley and Sons, 1992.Google Scholar - 4.Y. Azar, A. Z. Broder, A. R. Karlin, and E. Upfal. Balanced allocations. In
*Proceedings of the 26th Annual ACM Symposium on Theory of Computing*, 1994, pp. 593–602.Google Scholar - 5.P. Berenbrink, F. Meyer auf der Heide, and K. Schróder. Allocating weighted jobs in parallel. In
*Proceedings of the 9th Annual ACM Symposium on Parallel Algorithms and Architectures*, 1997, pp. 302–310.Google Scholar - 6.R. Cole, B. M. Maggs, F. Meyer auf der Heide, M. Mitzenmacher, A. W. Richa, K. Schröder, R. K. Sitaraman, and B. Vöcking. Randomized Protocols for Low-Congestion Circuit Routing inMultistage Interconnection Networks. In
*Proceedings of the 30th Annual ACM Symposium on Theory of Computing*, 1998, pp. 378–388.Google Scholar - 7.A. Czumaj and V. Stemann. Randomized allocation processes. In
*Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science*, 1997, pp. 194–203.Google Scholar - 8.M. Mitzenmacher. Density dependent jump markov processes and applications to load balancing. In
*Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science*, 1996, pp. 213–223.Google Scholar - 9.M. Mitzenmacher. On the analysis of randomized load balancing schemes. In
*Proceedings of the 9th Annual ACM Symposium on Parallel Algorithms and Architectures*, 1997, pp. 292–301.Google Scholar - 10.M. Mitzenmacher. Studying balanced allocations with diffierential equations. Technical Note 1997-024, Digital Equipment Corporation Systems Research Center, Palo Alto, CA, October 1997.Google Scholar
- 11.M. Mitzenmacher, How useful is old information? In
*Proceedings of the 16th Annual ACM Symposium on Principles of Distributed Computing*, 1997, pp. 83–91. Extended version available as Digital Systems Research Center Technical Note 1998-003.Google Scholar - 12.V. Stemann. Parallel balanced allocations. In
*Proceedings of the 8th Annual ACM Symposium on Parallel Algorithms and Architectures*, 1996, pp. 261–269.Google Scholar