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\( \newcommand {\multicolumn }[3]{#3}\)
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\(\let \Check \check \)
\(\let \Tilde \tilde \)
\(\let \Acute \acute \)
\(\let \Grave \grave \)
\(\let \Dot \dot \)
\(\let \Ddot \ddot \)
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The g-Myopic Process
The \(\GMyopic \) process is an adversarial version of \(\TwoChoice \), where whenever the load difference of the two bins is at most \(g\), it allocates randomly to one of the two samples. More formally, it can be described as follows:
The \(\GMyopic \) process:
Parameter: A threshold \(g \in \N \).
Iteration: At each step \(t = 1, 2, \ldots \),
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• Sample two bins \(i_1\) and \(i_2\) uniformly at random.
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• If \(|y_{i_1}^t - y_{i_2}^t| \leq g\), then allocate randomly to \(j \in \{i_1, i_2\}\).
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• Otherwise, allocate to \(j \in \{i_1, i_2\}\) satisfying \(y_j^t = \min \{ y_{i_1}^t, y_{i_2}^t \}\).
This process is like a mild version of the \(\GBounded \) process, where the adversary allocates to the heavier bin, when given a chance. However, it was shown that the two processes have the same asymptotic gap of \(\Theta (\frac {g}{\log g} \cdot \log \log n)\).
.