evaluation: lmdk-sel-sol
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\label{subsec:lmdk-sel-sol}
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The main idea of the privacy-preserving {\thething} selection component is to privately select extra {\thething} event timestamps, i.e.,~dummy {\thethings}, from the set of timestamps $T /\ L$ of the time series $S_T$ and add them to the original {\thething} set $L$.
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Selecting extra events, on top of the actual {\thethings}, as dummy {\thethings} can render actual ones indistinguishable.
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The goal is to select a list of sets with additional timestamps from a series of events at timestamps $T$ for a set of {\thethings} at $L \subseteq T$.
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Thus, we create a new set $L'$ such that $L \subset L' \subseteq T$.
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We generate a set of dummy {\thething} set options by adding regular event timestamps from $T /\ L$ to $L$ (Section~\ref{subsec:lmdk-set-opts}).
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Then (Section~\ref{subsec:lmdk-opt-sel}), we utilize the exponential mechanism, with a utility function that calculates an indicator for each of the options in the set based on how much it differs from the original {\thething} set $L$, and randomly select one ot the options that we created earlier.
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First, we generate a set of dummy {\thething} set options by adding regular event timestamps from $T /\ L$ to $L$ (Section~\ref{subsec:lmdk-set-opts}).
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Then, we utilize the exponential mechanism, with a utility function that calculates an indicator for each of the options in the set based on how much it differs from the original {\thething} set $L$, and randomly select one of the options (Section~\ref{subsec:lmdk-opt-sel}).
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This process provides an extra layer of privacy protection to {\thethings}, and thus allows the release, and thereafter processing, of {\thething} timestamps.
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% We utilize the exponential mechanism with a utility function that calculates an indicator for each of the options in the set that we selected in the previous step.
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% The utility depends on the positioning of the {\thething} timestamps of an option in the series, e.g.,~the distance from the previous/next {\thething}, the distance from the start/end of the series, etc.
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\subsubsection{{\Thething} set options}
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\subsubsection{{\Thething} set options generation}
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\label{subsec:lmdk-set-opts}
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This step aims to select a set of candidate {\thething} timestamps options either by randomizing the actual timestamps (Section~\ref{subsec:lmdk-rnd}), or by inserting dummy timestamps (Section~\ref{subsec:lmdk-dum-gen}) to the actual {\thething} timestamps.
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\paragraph{Dummy {\thething} generation}
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\label{subsec:lmdk-dum-gen}
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Selecting extra events, on top of the actual {\thethings}, as dummy {\thethings} can render actual ones indistinguishable.
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The goal is to select a list of sets with additional timestamps from a series of events at timestamps $\{t_n\}$ for a set of {\thethings} at $\{l_k\} \subseteq \{t_n\}$.
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Algorithms~\ref{algo:lmdk-sel-opt} and \ref{algo:lmdk-sel-heur} approach this problem with an optimal and heuristic methodology, respectively.
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Function \evalSeq evaluates the result of the union of $L$ and a timestamp combination from $T \setminus L$ by, e.g.,~estimating the standard deviation of all the distances from the previous/next {\thething}.
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\getOpts returns all the possible \emph{valid} sets of combinations \opt such that larger options contain all of the timestamps that are present in smaller ones.
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Each combination contains a set of timestamps with sizes $\left|L\right| + 1, \left|L\right| + 2, \dots, \left|T\right|$, where each one of them is a combination of $L$ with $x \in [1, \left|T\right| - \left|L\right|]$ timestamps from $T$.
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Function \calcMetric measures an indicator for the union of $\{l_k\}$ and a timestamp combination from $\{t_n\} \setminus \{l_k\}$.
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Function \evalSeq evaluates the result of \calcMetric by, e.g.,~estimating the standard deviation of all the distances from the previous/next {\thething}.
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Function \getOpts returns all possible \emph{valid} sets of combinations \opt such that $\{l_{k+i}\} \subset \{l_{k+j}\}, \forall i, j \in [k, n] \mid i < j$, i.e.,~larger options must contain all of the timestamps that are present in smaller ones.
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Each combination contains a set of timestamps with sizes $k + 1, k + 2, \dots, n$, where each one of them is a combination of $\{l_k\}$ with $x \in [1, n - k]$ timestamps from $\{t_n\}$.
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\paragraph{Optimal}
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Algorithm~\ref{algo:lmdk-sel-opt}, between Lines~{\ref{algo:lmdk-sel-opt-for-each}--\ref{algo:lmdk-sel-opt-end}} evaluates each option in \opts.
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It finds the option that is the most \emph{similar} to the original (Lines~{\ref{algo:lmdk-sel-opt-comparison}-\ref{algo:lmdk-sel-opt-end}}), i.e.,~the option that has an evaluation that differs the least from that of the sequence $T$ with {\thethings} $L$.
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\begin{algorithm}
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\caption{Optimal dummy {\thething} set options selection}
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\caption{Optimal dummy {\thething} set options generation}
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\label{algo:lmdk-sel-opt}
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\DontPrintSemicolon
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\KwData{$\{t_n\}, \{l_k\}$}
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\KwData{$T, L$}
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\SetKwInput{KwData}{Input}
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@ -43,11 +40,10 @@ Each combination contains a set of timestamps with sizes $k + 1, k + 2, \dots, n
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\BlankLine
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% Evaluate the original
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\metricOrig $\leftarrow$ \calcMetric{$\{t_n\}, \emptyset, \{l_k\}$}\;
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\evalOrig $\leftarrow$ \evalSeq{\metricOrig}\;
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\evalOrig $\leftarrow$ \evalSeq{$T, \emptyset, L$}\;
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% Get all possible option combinations
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\opts $\leftarrow$ \getOpts{$\{t_n\}, \{l_k\}$}\;
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\opts $\leftarrow$ \getOpts{$T, L$}\;
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% Track the minimum (best) evaluation
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\diffMin $\leftarrow$ $\infty$\;
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% Track the optimal sequence (the one with the best evaluation)
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\optim $\leftarrow$ $[]$\;
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\ForEach{\opt $\in$ \opts}{\label{algo:lmdk-sel-opt-for-each}
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\evalSum $\leftarrow 0$\;
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\ForEach{\opt $\in$ \opts}{ \label{algo:lmdk-sel-opt-for-each}
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\evalCur $\leftarrow 0$\;
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\ForEach{\opti $\in$ \opt}{
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\metricCur $\leftarrow$ \calcMetric{$\{t_n\}, \opti, \{l_k\}$}\;\label{algo:lmdk-sel-opt-comparison}
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\evalSum $\leftarrow$ \evalSum $+$ \evalSeq{\metricCur}\;
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\evalCur $\leftarrow$ \evalCur $+$ \evalSeq{$T, \opti, L$}/\#\opt\; \label{algo:lmdk-sel-opt-comparison}
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}
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% Compare with current optimal
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\diffCur $\leftarrow \left|\evalSum/\#\opt - \evalOrig\right|$\;
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\diffCur $\leftarrow \left|\evalCur - \evalOrig\right|$\;
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\If{\diffCur $<$ \diffMin}{
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\diffMin $\leftarrow$ \diffCur\;
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\optim $\leftarrow$ \opt\;
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}
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}
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}\label{algo:lmdk-sel-opt-end}
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} \label{algo:lmdk-sel-opt-end}
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\Return{\optim}
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\end{algorithm}
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Algorithm~\ref{algo:lmdk-sel-opt}, in particular, between Lines~{\ref{algo:lmdk-sel-opt-for-each}-\ref{algo:lmdk-sel-opt-end}} evaluates each option in \opts.
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It finds the option that is the most \emph{similar} to the original (Lines~{\ref{algo:lmdk-sel-opt-comparison}-\ref{algo:lmdk-sel-opt-end}}), i.e.,~the option that has an evaluation that differs the least from that of the sequence $\{t_n\}$ with {\thethings} $\{l_k\}$.
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Algorithm~\ref{algo:lmdk-sel-opt} guarantees to return the optimal set of dummy {\thethings} with regard to the original set $L$.
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However, it is rather costly in terms of complexity: given $n$ regular events and a combination of size $r$, it requires $\mathcal{O}(C(n, r) + 2^C(n, r))$ time and $\mathcal{O}(r*C(n, r))$ space.
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Next, we present a heuristic solution with improved time and space requirements.
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\paragraph{Heuristic}
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Algorithm~\ref{algo:lmdk-sel-heur}, follows an incremental methodology.
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At each step it selects a new timestamp that corresponds to a regular ({non-\thething}) event from $T \setminus L$.
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\begin{algorithm}
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\caption{Heuristic dummy {\thething} set options selection}
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@ -81,30 +81,28 @@ It finds the option that is the most \emph{similar} to the original (Lines~{\ref
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\DontPrintSemicolon
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\KwData{$\{t_n\}, \{l_k\}$}
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\KwData{$T, L$}
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\KwResult{\optim}
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\BlankLine
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% Evaluate the original
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\metricOrig $\leftarrow$ \calcMetric{$\{t_n\}, \emptyset, \{l_k\}$}\;
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\evalOrig $\leftarrow$ \evalSeq{\metricOrig}\;
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\evalOrig $\leftarrow$ \evalSeq{$T, \emptyset, L$}\;
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% Get all possible option combinations
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\optim $\leftarrow$ $[]$\;
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$\{l_{k'}\} \leftarrow \{l_k\}$\;
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$L' \leftarrow L$\;
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\While{$\{l_{k'}\} \neq \{t_n\}$}{\label{algo:lmdk-sel-heur-while}
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\While{$L' \neq T$}{\label{algo:lmdk-sel-heur-while}
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% Track the minimum (best) evaluation
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\diffMin $\leftarrow$ $\infty$\;
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\optimi $\leftarrow$ $0$\;
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\optimi $\leftarrow$ Null\;
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% Find the combinations for one more point
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\ForEach{\reg $\in \{t_n\} \setminus \{l_{k'}\}$}{
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\ForEach{\reg $\in T \setminus L'$}{
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% Evaluate current
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\metricCur $\leftarrow$ \calcMetric{$\{t_n\}, \reg, \{l_{k'}\}$}\;\label{algo:lmdk-sel-heur-comparison}
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\evalCur $\leftarrow$ \evalSeq{\metricCur}\;
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\evalCur $\leftarrow$ \evalSeq{$T, \reg, L'$}\; \label{algo:lmdk-sel-heur-comparison}
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% Compare evaluations
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\diffCur $\leftarrow$ $\left|\evalCur - \evalOrig\right|$\;
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@ -116,27 +114,31 @@ It finds the option that is the most \emph{similar} to the original (Lines~{\ref
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}
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% Save new point to landmarks
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$k' \leftarrow k' + 1$\;
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$l_{k'} \leftarrow \optimi$\;
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$L'.add(\optimi)$\;
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% Add new option
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\optim.add($\{l_{k'}\} \setminus \{l_k\}$)\;
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\optim.append($L' \setminus L$)\;
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}\label{algo:lmdk-sel-heur-end}
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\Return{\optim}
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\end{algorithm}
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Algorithm~\ref{algo:lmdk-sel-heur}, follows an incremental methodology.
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At each step it selects a new timestamp that corresponds to a regular ({non-\thething}) event from $\{t_n\} \setminus \{l_k\}$.
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Similar to Algorithm~\ref{algo:lmdk-sel-opt}, the selection is done based on a predefined metric (Lines~{\ref{algo:lmdk-sel-heur-comparison}-\ref{algo:lmdk-sel-heur-comparison-end}}).
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This process (Lines~{\ref{algo:lmdk-sel-heur-while}-\ref{algo:lmdk-sel-heur-end}}) goes on until we select a set that is equal to the size of the series of events, i.e.,~$\{l_{k'}\} = \{t_n\}$.
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This process (Lines~{\ref{algo:lmdk-sel-heur-while}-\ref{algo:lmdk-sel-heur-end}}) goes on until we select a set that is equal to the size of the series of events, i.e.,~$L' = T$.
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Note that the reverse heuristic approach, i.e.,~starting with $\{t_n\}$ {\thethings} and removing until $\{l_k\}$, performs worse than and occasionally the same with Algorithm~\ref{algo:lmdk-sel-heur}.
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In terms of complexity: given $n$ regular events it requires $\mathcal{O}(n^2)$ time and space.
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Note that the reverse heuristic approach, i.e.,~starting with $T$ {\thethings} and removing until $L$, performs similarly with Algorithm~\ref{algo:lmdk-sel-heur}.
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\mk{WIP: Histograms}
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\subsubsection{Privacy-preserving option selection}
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\label{subsec:lmdk-opt-sel}
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\mk{WIP}
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% Nearby events
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Events that occur at recent timestamps are more likely to reveal sensitive information regarding the users involved~\cite{kellaris2014differentially}.
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Thus, taking into account more recent events with respect to {\thethings} can result in less privacy loss and better privacy protection overall.
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