Temel Kavramlar
An asymptotically optimal algorithm is presented for efficiently generating the bin cardinalities in the balls-into-bins process, achieving optimal time complexity in expectation and with high probability.
Özet
The paper presents an efficient algorithm for generating the bin cardinalities in the balls-into-bins process, where n balls are thrown uniformly at random into n bins.
Key highlights:
The algorithm generates the vector of bin cardinalities (X0, X1, ..., XKn), where Xj is the number of bins with j balls, in optimal O(log n / log log n) time in expectation and with high probability.
This time complexity is asymptotically optimal, as the output vector has size Ω(log n / log log n).
The algorithm also achieves optimal performance for any m ∈ [n, n log n] balls.
The algorithm is used as a building block to efficiently simulate more involved load balancing processes, such as the Two-Choice algorithm, achieving a quadratic speedup over naive simulation.
The algorithm works by first generating a Poisson(m) distributed number of balls, and then using a recursive procedure to combine the bin cardinalities from two independent simulations. Careful adjustments are made to handle the case where the Poisson sample size differs from the target m.
İstatistikler
The number of balls, N, is Poisson distributed with parameter λ = m - m^(3/5).
The maximum occupancy, Kn, is bounded by (1 + o(1)) * log n / log log n with high probability.
Alıntılar
"The naïve way to generate the final load vector takes Θ(n) time. However, it is well-known that this load vector has with high probability bin cardinalities of size Θ(log n / log log n)."
"We present an algorithm in the RAM model that generates the bin cardinalities of the final load vector in the optimal Θ(log n / log log n) time in expectation and with high probability."