The (1 + β)-Choice Process and Weighted Balls-into-Bins

by Yuval Peres , Kunal Talwar , Udi Wieder

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Datum	Value	Source
TITLE	The (1 + β)-Choice Process and Weighted Balls-into-Bins	SVM HeaderParse 0.2
AUTHOR NAME	Yuval Peres	SVM HeaderParse 0.2
AUTHOR NAME	Kunal Talwar	SVM HeaderParse 0.2
AUTHOR NAME	Udi Wieder	SVM HeaderParse 0.2
ABSTRACT	Suppose m balls are sequentially thrown into n bins where each ball goes into a random bin. It is well-known that the gap between the load of the most loaded bin m log n and the average is Θ( n), for large m. If each ball goes to the lesser loaded of two random bins, this gap dramatically reduces to Θ(log log n) independent of m. Consider now the following “(1 + β)-choice ” process for some parameter β ∈ (0, 1): each ball goes to a random bin with probability (1−β) and the lesser loaded of two random bins with probability β. How does the gap for such a process behave? Suppose that the weight of each ball was drawn from a geometric distribution. How is the gap (now defined in terms of weight) affected? In this work, we develop general techniques for analyzing such balls-into-bins processes. Specifically, we show that for the (1 + β)-choice process above, the gap is Θ(log n/β), irrespective of m. Moreover the gap stays at Θ(log n/β) in the weighted case for a large class of weight distributions. No non-trivial explicit bounds were previously known in the weighted case, even for the 2-choice paradigm. 1	SVM HeaderParse 0.2
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