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The \((1+\beta )\)-process is a process mixing the \(\TwoChoice \) process (with probability \(\beta \in (0, 1]\)) and the \(\OneChoice \) process (with probability \(1 - \beta \)). More formally, it can be defined as follows.
The \((1+\beta )\)-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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• With probability \(\beta \) allocate to bin \(j \in \{ i_1, i_2 \}\) such that \(x_{j}^t = \min \{ x_{i_1}^t, x_{i_2}^t \}\).
-
• Otherwise, allocate to bin \(i_1\).
The process was introduced by Mitzenmacher (1999) as a model of \(\TwoChoice \) with erroneous comparisons. Peres, Talwar and Wieder (2015) proved that this process has w.h.p. an \(\Oh (\frac {\log n}{\beta })\) gap
for \(\beta \in (0, 1]\). This process is also known to work well in the \(\Batched \) setting.
.
References
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[1] M. Mitzenmacher. “On the analysis of randomized load balancing schemes”. In: Theory Comput. Syst. 32.3 (1999), pp. 361–386. issn: 1432-4350. doi: 10.1007/s002240000122.
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[2] Yuval Peres, Kunal Talwar, and Udi Wieder. “Graphical balanced allocations and the (1 + β)-choice process”. In: Random Structures & Algorithms 47.4 (2015), pp. 760–775. issn:
1042-9832. doi: 10.1002/rsa.20558.