
The power of thinning in balanced allocation
Balls are sequentially allocated into n bins as follows: for each ball, ...
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Optimal Ball Recycling
Ballsandbins games have been a wildly successful tool for modeling loa...
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Balanced Allocation on Dynamic Hypergraphs
The ballsintobins model randomly allocates n sequential balls into n b...
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Longterm balanced allocation via thinning
We study the longterm behavior of the twothinning variant of the class...
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Balanced Allocations: Caching and Packing, Twinning and Thinning
We consider the sequential allocation of m balls (jobs) into n bins (ser...
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Power of d Choices with Simple Tabulation
Suppose that we are to place m balls into n bins sequentially using the ...
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Balanced Allocation with Random Walk Based Sampling
In the standard ballinbins experiment, a wellknown scheme is to sampl...
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Load Balancing with Dynamic Set of Balls and Bins
In dynamic load balancing, we wish to distribute balls into bins in an environment where both balls and bins can be added and removed. We want to minimize the maximum load of any bin but we also want to minimize the number of balls and bins affected when adding or removing a ball or a bin. We want a hashingstyle solution where we given the ID of a ball can find its bin efficiently. We are given a balancing parameter c=1+ϵ, where ϵ∈ (0,1). With n and m the current numbers of balls and bins, we want no bin with load above C=⌈ c n/m⌉, referred to as the capacity of the bins. We present a scheme where we can locate a ball checking 1+O(log 1/ϵ) bins in expectation. When inserting or deleting a ball, we expect to move O(1/ϵ) balls, and when inserting or deleting a bin, we expect to move O(C/ϵ) balls. Previous bounds were off by a factor 1/ϵ. These bounds are best possible when C=O(1) but for larger C, we can do much better: Let f=ϵ C if C≤log 1/ϵ, f=ϵ√(C)·√(log(1/(ϵ√(C)))) if log 1/ϵ≤ C<12ϵ^2, and C=1 if C≥12ϵ^2. We show that we expect to move O(1/f) balls when inserting or deleting a ball, and O(C/f) balls when inserting or deleting a bin. For the bounds with larger C, we first have to resolve a much simpler probabilistic problem. Place n balls in m bins of capacity C, one ball at the time. Each ball picks a uniformly random nonfull bin. We show that in expectation and with high probability, the fraction of nonfull bins is Θ(f). Then the expected number of bins that a new ball would have to visit to find one that is not full is Θ(1/f). As it turns out, we obtain the same complexity in our more complicated scheme where both balls and bins can be added and removed.
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