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\(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\)
\(\def \mathcode #1={\mathchar }\)
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\( \newcommand {\multicolumn }[3]{#3}\)
\(\require {textcomp}\)
\(\newcommand {\intertext }[1]{\text {#1}\notag \\}\)
\(\let \Hat \hat \)
\(\let \Check \check \)
\(\let \Tilde \tilde \)
\(\let \Acute \acute \)
\(\let \Grave \grave \)
\(\let \Dot \dot \)
\(\let \Ddot \ddot \)
\(\let \Breve \breve \)
\(\let \Bar \bar \)
\(\let \Vec \vec \)
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\(\newcommand {\Batched }{b\text {-}{\rm B{\small ATCHED}}}\)
\(\newcommand {\OneChoice }{{\rm O{\small NE}}\text {-}{\rm C{\small HOICE}}}\)
\(\newcommand {\TwoChoice }{{\rm T{\small WO}}\text {-}{\rm C{\small HOICE}}}\)
\(\newcommand {\DSample }{d\text {-}{\rm S{\small AMPLE}}}\)
\(\newcommand {\TwoSample }{{\rm T{\small WO}}\text {-}{\rm S{\small AMPLE}}}\)
\(\newcommand {\Graphical }{{\rm G{\small RAPHICAL}}}\)
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\(\newcommand {\Twinning }{{\rm T{\small WINNING}}}\)
\(\newcommand {\Threshold }{{\rm T{\small HRESHOLD}}}\)
\(\newcommand {\Thinning }{{\rm T{\small HINNING}}}\)
\(\newcommand {\KThreshold }{k\text {-}{\rm T{\small HRESHOLD}}}\)
\(\newcommand {\OnePlusBeta }{(1+\beta )}\)
\(\newcommand {\Packing }{{\rm P{\small ACKING}}}\)
\(\newcommand {\Quantile }{{\rm Q{\small UANTILE}}}\)
\(\newcommand {\Memory }{{\rm M{\small EMORY}}}\)
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\(\newcommand {\R }{\mathbb {R}}\)
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\(\newcommand {\TauDelay }{\tau \text {-}{\rm D{\small ELAY}}}\)
\(\newcommand {\SigmaNoisyLoad }{\sigma \text {-}{\rm N{\small OISY}}\text {-}{\rm L{\small OAD}}}\)
\(\newcommand {\Heterogeneous }{{\rm H{\small ETEROGENEOUS}}}\)
\(\newcommand {\Gap }{\mathrm {Gap}}\)
\(\newcommand {\eps }{\epsilon }\)
Batched Setting
Berenbrink, Czumaj, Englert, Friedetzky and Nagel (2012) introduced \(\Batched \) setting where balls are allocated in batches of \(b\) balls. More formally, for any balanced allocation process with probability allocation vector \(p\), the setting is defined as follows.
The \(\Batched \) setting:
Parameter: The probability allocation vector \(p^t\) for the process.
Iteration: At each step \(t = 0 \cdot b, 1 \cdot b, 2 \cdot b\ldots \),
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• Sample \(b\) bins \(i_1, \ldots , i_b\) from \([n]\) according to \(p^t\).
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• For each \(i \in [n]\), update
\[ x_i^{t+b} = x_i^t + \sum _{j = 1}^b \mathbf {1}_{i_j = i}. \]
For the \(\TwoChoice \) process, it was shown that this process achieves an \(\Oh (\frac {\log n}{\log \log n})\) for when \(b = n\) and an \(\Oh (\frac {b}{n})\) gap for \(b \geq n \log n\).
.
References
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[1] Petra Berenbrink et al. “Multiple-Choice Balanced Allocation in (Almost) Parallel”. In: Proceedings of 16th International Workshop on Approximation, Randomization, and Combinatorial Optimization (RANDOM’12). Berlin Heidelberg: Springer-Verlag, 2012, pp. 411–422. doi: 10.1007/978-3-642-32512-0_35.