\(\newcommand{\footnotename}{footnote}\)
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\(\newcommand {\mathnormal }[1]{{#1}}\)
\(\newcommand \ensuremath [1]{#1}\)
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\(\newcommand {\setlength }[2]{}\)
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\(\newcommand {\cline }[1]{}\)
\(\newcommand {\directlua }[1]{\text {(directlua)}}\)
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\(\def \LWRabsorbquotenumber "#1 {}\)
\(\newcommand {\LWRabsorboption }[1][]{}\)
\(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\)
\(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\)
\(\def \mathcode #1={\mathchar }\)
\(\let \delcode \mathcode \)
\(\let \delimiter \mathchar \)
\(\def \oe {\unicode {x0153}}\)
\(\def \OE {\unicode {x0152}}\)
\(\def \ae {\unicode {x00E6}}\)
\(\def \AE {\unicode {x00C6}}\)
\(\def \aa {\unicode {x00E5}}\)
\(\def \AA {\unicode {x00C5}}\)
\(\def \o {\unicode {x00F8}}\)
\(\def \O {\unicode {x00D8}}\)
\(\def \l {\unicode {x0142}}\)
\(\def \L {\unicode {x0141}}\)
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\(\def \SS {\unicode {x1E9E}}\)
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\(\def \P {\unicode {x00B6}}\)
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\( \newcommand {\multicolumn }[3]{#3}\)
\(\require {textcomp}\)
\(\newcommand {\intertext }[1]{\text {#1}\notag \\}\)
\(\let \Hat \hat \)
\(\let \Check \check \)
\(\let \Tilde \tilde \)
\(\let \Acute \acute \)
\(\let \Grave \grave \)
\(\let \Dot \dot \)
\(\let \Ddot \ddot \)
\(\let \Breve \breve \)
\(\let \Bar \bar \)
\(\let \Vec \vec \)
\(\newcommand {\tcbset }[1]{}\)
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\(\newcommand {\Batched }{b\text {-}{\rm B{\small ATCHED}}}\)
\(\newcommand {\OneChoice }{{\rm O{\small NE}}\text {-}{\rm C{\small HOICE}}}\)
\(\newcommand {\TwoChoice }{{\rm T{\small WO}}\text {-}{\rm C{\small HOICE}}}\)
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\(\newcommand {\TwoSample }{{\rm T{\small WO}}\text {-}{\rm S{\small AMPLE}}}\)
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\(\newcommand {\MeanThinning }{{\rm M{\small EAN}\text {-}T{\small HINNING}}}\)
\(\newcommand {\TwoThinning }{{\rm T{\small WO}\text {-}T{\small HINNING}}}\)
\(\newcommand {\DThinning }{d\text {-}{\rm T{\small HINNING}}}\)
\(\newcommand {\Twinning }{{\rm T{\small WINNING}}}\)
\(\newcommand {\Threshold }{{\rm T{\small HRESHOLD}}}\)
\(\newcommand {\Thinning }{{\rm T{\small HINNING}}}\)
\(\newcommand {\KThreshold }{k\text {-}{\rm T{\small HRESHOLD}}}\)
\(\newcommand {\OnePlusBeta }{(1+\beta )}\)
\(\newcommand {\Packing }{{\rm P{\small ACKING}}}\)
\(\newcommand {\Quantile }{{\rm Q{\small UANTILE}}}\)
\(\newcommand {\Memory }{{\rm M{\small EMORY}}}\)
\(\newcommand {\N }{\mathbb {N}}\)
\(\newcommand {\R }{\mathbb {R}}\)
\(\newcommand {\Oh }{\mathcal {O}}\)
\(\newcommand {\AdvTauDelay }{\tau \text {-}{\rm A{\small DV}}\text {-}{\rm D{\small ELAY}}}\)
\(\newcommand {\RandomTauDelay }{\tau \text {-}{\rm R{\small AND}}\text {-}{\rm D{\small ELAY}}}\)
\(\newcommand {\TauDelay }{\tau \text {-}{\rm D{\small ELAY}}}\)
\(\newcommand {\SigmaNoisyLoad }{\sigma \text {-}{\rm N{\small OISY}}\text {-}{\rm L{\small OAD}}}\)
\(\newcommand {\Heterogeneous }{{\rm H{\small ETEROGENEOUS}}}\)
\(\newcommand {\Gap }{\mathrm {Gap}}\)
\(\newcommand {\eps }{\epsilon }\)
The τ -Delay Setting
The \(\TauDelay \) setting is an outdated information setting where an adversary can report any of the loads from the last \(\tau \) steps.
The \(\TauDelay \) process:
Parameter: A delay \(\tau \in \N \).
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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• An adversary chooses two load estimates \(\tilde {x}_{i_1}^t \in [x_{i_1}^{t-\tau }, x_{i_1}^{t-1}]\) and \(\tilde {x}_{i_2}^t \in [x_{i_1}^{t-\tau }, x_{i_2}^{t-1}]\).
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• Allocate to bin \(i \in \{ i_1, i_2 \}\) satisfying \(\tilde {x}_i^t = \min \{ \tilde {x}_{i_1}^t, \tilde {x}_{i_2}^t \}\).
The \(\Batched \) setting is an instance of the \(\TauDelay \) setting for \(\tau = b\). Another concrete instance of the \(\TauDelay \) setting is the \(\AdvTauDelay \) process where the adversary always tries to flip the decision of \(\TwoChoice \). Further, in the \(\RandomTauDelay \) process the adversary reports the load estimates from a random step in the last \(\tau \) steps.
.