problem: Reviewed lmdk-set-opts

text/problem/theotherthing/main.tex

@@ -39,10 +39,16 @@ In Example~\ref{ex:lmdk-risk}, we demonstrate the extreme case of the applicatio
 \SetKwData{evalCur}{evalCur}
 \SetKwData{evalOrig}{evalOrig}
 \SetKwData{evalSum}{evalSum}
+\SetKwData{h}{h}
+\SetKwData{hi}{h$_i$}
+\SetKwData{hist}{hist}
+\SetKwData{histCur}{histCur}
+\SetKwData{histTmp}{histTmp}
 \SetKwData{metricCur}{metricCur}
 \SetKwData{metricOrig}{metricOrig}
 \SetKwData{opt}{opt}
 \SetKwData{opti}{opt$_i$}
+\SetKwData{opts}{opts}
 \SetKwData{optim}{optim}
 \SetKwData{optimi}{optim$_i$}
 \SetKwData{opts}{opts}
@@ -51,7 +57,10 @@ In Example~\ref{ex:lmdk-risk}, we demonstrate the extreme case of the applicatio
 \SetKwFunction{calcMetric}{calcMetric}
 \SetKwFunction{evalSeq}{evalSeq}
 \SetKwFunction{getCombs}{getCombs}
+\SetKwFunction{getDiff}{getDiff}
+\SetKwFunction{getHist}{getHist}
 \SetKwFunction{getOpts}{getOpts}
+\SetKwFunction{getNorm}{getNorm}
 
 \input{problem/theotherthing/contribution}
 \input{problem/theotherthing/problem}

text/problem/theotherthing/solution.tex

@@ -73,7 +73,7 @@ Next, we present a heuristic solution with improved time and space requirements.
 
 \paragraph{Heuristic}
 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 $T \setminus L$.
+At each step it selects a new timestamp, that corresponds to a regular ({non-\thething}) event from $T \setminus L$, to create an option.
 
 \begin{algorithm}
   \caption{Heuristic dummy {\thething} set options selection}
@@ -89,7 +89,7 @@ At each step it selects a new timestamp that corresponds to a regular ({non-\the
   \evalOrig $\leftarrow$ \evalSeq{$T, \emptyset, L$}\;
 
   % Get all possible option combinations
-  \optim $\leftarrow$ $[]$\;
+  \opts $\leftarrow$ $[]$\;
 
   $L' \leftarrow L$\;
 
@@ -110,28 +110,87 @@ At each step it selects a new timestamp that corresponds to a regular ({non-\the
       \If{\diffCur $<$ \diffMin}{
         \diffMin $\leftarrow$ \diffCur\;
         \optimi $\leftarrow$ \reg\;
-      }\label{algo:lmdk-sel-heur-comparison-end}
+      }\label{algo:lmdk-sel-heur-cmp-end}
     }
 
     % Save new point to landmarks
     $L'$.add(\optimi)\;
 
     % Add new option
-    \optim.append($L' \setminus L$)\;
+    \opts.append($L' \setminus L$)\;
   }\label{algo:lmdk-sel-heur-end}
 
-  \Return{\optim}
+  \Return{\opts}
 \end{algorithm}
 
-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}}).
+Similar to Algorithm~\ref{algo:lmdk-sel-opt}, it selects new options based on a predefined metric (Lines~{\ref{algo:lmdk-sel-heur-comparison}-\ref{algo:lmdk-sel-heur-cmp-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.,~$L' = T$.
 
-In terms of complexity: given $n$ regular events it requires $\mathcal{O}(n^2)$ time and space.
+In terms of complexity, given $n$ regular events it requires $\mathcal{O}(n^2)$ time and space.
 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}.
 
 
+\paragraph{Partitioned}
+We improve the complexity of Algorithm~\ref{algo:lmdk-sel-opt} by partitioning the {\thething} timestamp sequence $L$.
+Algorithm~\ref{algo:lmdk-sel-hist}, \getHist generates a histogram from $L$ with bins of size \h.
+We find \h by using the Freedman–Diaconis rule which is resilient to outliers and takes into account the data variability and data size~\cite{meshgi2015expanding}.
+For every possible histogram version, the \getDiff function finds the difference between two histograms; for this operation we utilize the Euclidean distance~(see Section~\ref{subsec:sel-utl} for more details).
 
-\mk{WIP: Histograms}
+\begin{algorithm}
+  \caption{Partitioned dummy {\thething} set options selection}
+  \label{algo:lmdk-sel-hist}
+
+  \DontPrintSemicolon
+
+  \KwData{$T, L$}
+  \KwResult{\opts}
+  \BlankLine
+
+  \hist, \h $\leftarrow$ \getHist{$T, L$}\;
+
+  \histCur $\leftarrow$ hist\;
+
+  \opts $\leftarrow$ $[]$\;
+
+  \While{sum($L'$) $\neq$ len($T$)}{ \label{algo:lmdk-sel-hist-while}
+    % Track the minimum (best) evaluation
+    \diffMin $\leftarrow$ $\infty$\;
+
+    % The candidate option
+    \opt $\leftarrow$ \histCur\;
+
+    % Check every possibility
+    \ForEach{\hi \reg $L'$}{ \label{algo:lmdk-sel-hist-cmp-start}
+
+      % Can we add one more point?
+      \If{\hi $+$ $1$ $\leq$ \h}{
+        \histTmp $\leftarrow$ \histCur\;
+        \histTmp$[i]$ $\leftarrow$ \histTmp$[i]$ $+$ $1$\;
+        % Find difference from original
+        \diffCur $\leftarrow$ \getDiff{\hist, \histTmp}\;
+
+        % Remember if it is the best that you've seen
+        \If{\diffCur $<$ \diffMin}{ \label{algo:lmdk-sel-hist-cmp}
+          \diffMin $\leftarrow$ \diffCur\;
+          \opt $\leftarrow$ \histTmp\;
+        }
+
+      }
+
+    } \label{algo:lmdk-sel-hist-cmp-end}
+
+    % Update current histogram
+    \histCur $\leftarrow$ \opt\;
+    % Add current best to options
+    \opts $\leftarrow$ \opt\;
+
+  } \label{algo:lmdk-sel-hist-end}
+
+  \Return{\opts}
+\end{algorithm}
+
+Between Lines~{\ref{algo:lmdk-sel-hist-cmp-start}-\ref{algo:lmdk-sel-hist-cmp-end}} we check every possible histogram version by incrementing each bin by $1$ and comparing it to the original (Line~\ref{algo:lmdk-sel-hist-cmp}).
+In the end of the process, we return \opts which contains all the versions of \hist that are closest to \hist for all possible sizes of \hist.
 
 
 \subsubsection{Privacy-preserving option selection}