Given a set of {\thethings} at respective timestamps $\{l_k\}$ in a series of events at $\{t_n\}$, such that $\{l_k\}\subseteq\{t_n\}$, a data publisher might release this information by:
\begin{enumerate}
\item Selecting a set of options (Section~\ref{subsec:lmdk-set-opts}) consisting of different possible versions of $\{l_k\}$.
\mk{`option' or `candidate'?}
This could be:
\begin{itemize}
\item either a random set of $k$ other timestamps similar to the actual {\thething} timestamps (Section~\ref{subsec:lmdk-rnd}),
\item or a set including $\{l_k\}$ and $x \in[1, n - k]$ additional dummy timestamps (Section~\ref{subsec:lmdk-dum-gen}).
\end{itemize}
\item Releasing a privacy-preserving version of the {\thething} timestamps (Section~\ref{subsec:priv-opt-sel}).
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.
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.
\end{enumerate}
Following this process allows the release, and thereafter processing, of {\thething} timestamps.
Thus, we provide an extra layer of privacy protection when we separate {\thethings} from regular events.
\subsubsection{{\Thething} set options}
\label{subsec:lmdk-set-opts}
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.
\paragraph{{\Thething} randomization}
\label{subsec:lmdk-rnd}
A simple way to select a set of timestamps without disclosing the actual {\thethings} is by \emph{randomly} selecting an equally sized set of timestamps.
The randomization of the process, as we will discuss in more detail in Section~\ref{subsec:priv-opt-sel}, will depend on the positioning of the {\thethings} in the series of events.
In more detail, given a set of {\thething} timestamps $\{l_k\}\subseteq\{t_n\}$, where $\{t_n\}$ is an event sequence, we need to select all possible sets of size $k$ from $\{t_n\}$.
However, the introduction of randomization could impact arbitrarily the effectiveness of non-uniform privacy-protection methods.
This applies mainly in cases where we try to achieve optimal privacy-protection of {\thething} events while maximizing the utility of the data that corresponds to the rest of the series of events.
As a consequence, it is possible to end up providing lower levels of protection to {\thething} data than the one necessary, i.e.,~worse than the users' privacy-protection expectations.
The methodology that we present next (Section~\ref{subsec:lmdk-dum-gen}) attempts to tackle the aforementioned shortcoming.
\paragraph{Dummy {\thething} generation}
\label{subsec:lmdk-dum-gen}
Selecting extra events, on top of the actual {\thethings}, as dummy {\thethings} can render actual ones indistinguishable.
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 $\{l_k\}\subseteq\{t_n\}$.
Algorithms~\ref{algo:lmdk-sel-opt} and \ref{algo:lmdk-sel-heur} approach this problem with an optimal and heuristic methodology, respectively.
Function \calcMetric measures an indicator for the union of $\{l_k\}$ and a timestamp combination from $\{t_n\}\setminus\{l_k\}$.
Function \evalSeq evaluates the result of \calcMetric by, e.g.,~estimating the standard deviation of all the distances from the previous/next {\thething}.
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.
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 $\{t_n\}$.
\begin{algorithm}
\caption{Optimal dummy {\thething} set options selection}
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.
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}$\{l_k\}$.
\begin{algorithm}
\caption{Heuristic dummy {\thething} set options selection}
Algorithm~\ref{algo:lmdk-sel-heur}, follows an incremental methodology.
At each step it selects a new timestamp that corresponds to a regular ({non-\thething}) event from $\{t_n\}\setminus\{l_k\}$.
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}}).
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.,~$\{l_{k'}\}=\{t_n\}$.
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}.
Events that occur at recent timestamps are more likely to reveal sensitive information regarding the users involved~\cite{kellaris2014differentially}.
Thus, taking into account more recent events with respect to {\thethings} can result in less privacy loss and better privacy protection overall.
This leads to worse data utility.
% Depending on the {\thething} discovery technique
The values of events near a {\thething} are usually similar to that of the latter.
Therefore, privacy-preserving mechanisms are likely to approximate their values based on the nearest {\thething} instead of investing extra privacy budget to perturb their actual values; thus, spending less privacy budget.
Saving privacy budget for releasing perturbed versions of actual event values can bring about better data utility.
% Distant events
However, indicating the existence of randomized/dummy {\thethings} nearby actual {\thethings} can increase the adversarial confidence regarding the location of the latter within a series of events.
Hence, choosing randomized/dummy {\thethings} far from the actual {\thethings} (and thus less relevant) can limit the final privacy loss.
