Add chessboard visualisations

assets/packages.tex

@@ -74,6 +74,11 @@
 \usepackage{graphicx}
 \usepackage{subcaption}
 
+% Drawing chessboards (background)
+\usepackage{chessboard}
+\setchessboard{showmover=false}
+
+
 % Algorithms
 % \usepackage[ruled,vlined]{algorithm2e}
 

chapters/2_background.tex

@@ -86,8 +86,8 @@ problem format these techniques expect. \Cref{sec:back-other-languages}
 introduces alternative \cmls\ and compares their functionality to \minizinc{}.
 Then,\cref{sec:back-term} survey the closely related technologies in the field
 of \glspl{trs}. Finally, \cref{sec:back-mzn-interpreter} explores the process
-that the current \minizinc\ interpreter uses to translate a \minizinc{} instance
+that the current \minizinc\ interpreter uses to translate a \minizinc{}
-into a solver-level constraint model.
+instance into a solver-level constraint model.
 
 \section{\glsentrytext{minizinc}}%
 \label{sec:back-minizinc}
@@ -99,8 +99,8 @@ library of constraints allow users to easily model complex problems.
 
 \begin{listing}
   \mznfile{assets/mzn/back_knapsack.mzn}
-  \caption{\label{lst:back-mzn-knapsack} A \minizinc\ model describing a 0-1 knapsack
+	\caption{\label{lst:back-mzn-knapsack} A \minizinc\ model describing a 0-1
-    problem}
+	knapsack problem}
 \end{listing}
 
 \begin{example}%
@@ -684,7 +684,7 @@ track of \gls{domain} changes using a \gls{trail}. For every \gls{domain} change
 that is made during propagation (after the first search decision), the reverse
 change is stored on in a list. Whenever a new search decision is made, the
 current position of the list is tagged. If the \solver{} now needs to undo a
-search decision (\ie\ \gls{backtrack}), it can apply all change until it reaches
+search decision (\ie\ \gls{backtrack}), it can apply all changes until it reaches
 the change that is tagged with the search decision. Because all changes before
 the tagged point on the \gls{trail} were made before the search decision was
 made, it is guaranteed that these \gls{domain} changes do not depend on the
@@ -730,19 +730,82 @@ constraints otherwise.
 
   When solving the problem, initially no values can be eliminated from the
   \glspl{domain} of the \variables{}. The first propagation will happen when the
-  first queen is placed on the board. The search space will then look like this:
+	first queen is placed on the board, the first search decision.
 
-  \jip{Insert image of n-queens board with one queen assigned.}
+	\Cref{fig:back-nqueens} visualises the domain propagation of placing a queen
-
+	on the d3 square of an eight by eight chess board. When the queen it placed
-  Now that one queen is placed on the board the propagators for the
+	on the board in \cref{sfig:back-nqueens-1}, it fixes the value of row 4 to
-  \mzninline{all_different} constraints can start propagating. They can eliminate
+	the value 3. This implicitly eliminates any possibility of placing another
-  all values that are on the same row or diagonal as the queen placed on the
+	queen in the row. Now that one queen is placed on the board the propagators
-  board.
+	for the \mzninline{all_different} constraints can start propagating. As show
-
+	in \cref{sfig:back-nqueens-2}, the first \mzninline{all_different} constraint
-  \jip{Insert previous image with all impossible moves marked red.}
+	can now stop other queens to be placed in the same column. It eliminates the
+	value 3 from the domains of the queens in the remaining rows. Similarly, the
+	other \mzninline{all_different} constraints remove all values that correspond
+	to positions on the same diagonal as the placed queen, shown in
+	\cref{sfig:back-nqueens-3,sfig:back-nqueens-4}.
 
+	The propagation of the first placed queen severely limits the positions where
+	a second queen can be placed.
 \end{example}
 
+\begin{figure}
+  \centering
+  \begin{subfigure}[b]{.48\columnwidth}
+    \centering
+		\chessboard[
+			setwhite={Qd3},
+			pgfstyle=cross,
+			color=red,
+			markarea={d1-d2,d4-d8},
+		]
+    \caption{\label{sfig:back-nqueens-1} Assign a queen to d3}
+  \end{subfigure}%
+  \hspace{0.04\columnwidth}%
+  \begin{subfigure}[b]{.48\columnwidth}
+    \centering
+		\chessboard[
+			setwhite={Qd3},
+			pgfstyle=cross,
+			color=red!35!white,
+			markareas={d1-d2,d4-d8},
+			pgfstyle=cross,
+			color=red,
+			markareas={a3-c3,e3-h3},
+		]
+    \caption{\label{sfig:back-nqueens-2} Propagate rows}
+  \end{subfigure}
+  \begin{subfigure}[b]{.48\columnwidth}
+    \centering
+		\chessboard[
+			setwhite={Qd3},
+			pgfstyle=cross,
+			color=red!35!white,
+			markareas={d1-d2,d4-d8,a3-c3,e3-h3},
+			pgfstyle=cross,
+			color=red,
+			markareas={b1-b1,c2-c2,e4-e4,f5-f5,g6-g6,h7-h7},
+		]
+    \caption{\label{sfig:back-nqueens-3} Propagate upwards diagonal}
+  \end{subfigure}%
+  \hspace{0.04\columnwidth}%
+  \begin{subfigure}[b]{.48\columnwidth}
+    \centering
+		\chessboard[
+			setwhite={Qd3},
+			pgfstyle=cross,
+			color=red!35!white,
+			markareas={d1-d2,d4-d8,a3-c3,e3-h3,b1-b1,c2-c2,e4-e4,f5-f5,g6-g6,h7-h7},
+			pgfstyle=cross,
+			color=red,
+			markareas={a6-a6,b5-b5,c4-c4,e2-e2,f1-f1},
+			]
+    \caption{\label{sfig:back-nqueens-4} Propagate downward diagonal}
+  \end{subfigure}
+	\caption{\label{fig:back-nqueens} An example of domain propagation when a
+	queen gets assigned in the N-Queens problem.}
+\end{figure}
+
 In \gls{cp} solving there is a trade-off between the amount of time spend
 propagating a constraint and the amount of search that is otherwise required.
 The golden standard for a \gls{propagator} is to be \gls{domain-con}, meaning