Balanced allocations: the weighted case

  • Authors:

  • Kunal Talwar;Udi Wieder

  • Affiliations:

  • Microsoft Research - Silicon Valley, Mountain View, CA;Microsoft Research - Silicon Valley, Mountain View, CA

  • Venue:
  • Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
  • Year:
  • 2007

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Abstract

We investigate balls-and-bins processes where m weighted balls areplaced into n bins using the "power of two choices" paradigm,whereby a ball is inserted into the less loaded of two randomly chosen bins. The case where each of the m balls has unit weight had been studied extensively. In a seminal paper Azar et.al. showed that when m=n the most loaded bin has Θ(log log n) balls with high probability. Surprisingly, thegap in load between the heaviest bin and the average bin does not increase with m and was shown by Berenbrink etal tobe Θ(log log n) with high probability for arbitrarily large m. We generalize this result to the weighted case where balls have weights drawn from an arbitrary weight distribution. We show that aslong as the weight distribution has finite second moment andsatisfies a mild technical condition, the gap between the weight of the heaviest bin and the weight of the average bin is independent ofthe number balls thrown. This is especially striking whenconsidering heavy tailed distributions such as Power-Law andLog-Normal distributions. In these cases, as more balls are thrown,heavier and heavier weights are encountered. Nevertheless with high probability, the imbalance in the load distribution does notincrease. Furthermore, if the fourth moment of the weight distribution is finite, the expected value of the gap is shown to beindependent of the number of balls.