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\( \newcommand {\multicolumn }[3]{#3}\)
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\(\let \Grave \grave \)
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The Mean-Thinnning and Relative-Threshold Processes
The \(\MeanThinning \) process is the process which samples a bin \(i_1\) and allocates to it iff \(x_{i_1}^t \leq \frac {t}{n}\); otherwise it allocates to another bin sampled uniformly at random.
The \(\MeanThinning \) process:
Iteration: At each step \(t = 1, 2, \ldots \),
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• Sample two bins \(i_1 = i_1^t\) and \(i_2 = i_2^t\) uniformly at random.
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• If \(x_{i_1}^t \leq \frac {t}{n}\), then allocate to \(i_1\).
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• Otherwise, allocate to \(i_2\).
It was shown that this process achieves w.h.p. an \(\Oh (\log n)\) gap, which is tight for any \(m = \Omega (n \log n)\). This process makes in expectation \(2 - \epsilon \) samples per allocation.
.
Relative-Threshold processes
A generalisation of this process is the \(\RelativeThreshold (f)\) processes where \(f = f(n)\) is the fixed offset, so \(\MeanThinning \) has \(f(n) = 0\).
The \(\RelativeThreshold (f)\) process:
Iteration: At each step \(t = 1, 2, \ldots \),
-
• Sample two bins \(i_1 = i_1^t\) and \(i_2 = i_2^t\) uniformly at random.
-
• If \(x_{i_1}^t \leq \frac {t}{n} + f(n)\), then allocate to \(i_1\).
-
• Otherwise, allocate to \(i_2\).
.