\documentclass{article}
\usepackage{chessboard}
\usepackage[LSBC4,T1]{fontenc}
\usepackage{../style}

\begin{document}

\setchessboard{smallboard,
               zero,
               showmover=false,
               boardfontencoding=LSBC4,
               hlabelformat=\arabic{ranklabel},
               vlabelformat=\arabic{filelabel}}

\mbox{}\\[-18mm]\mbox{}

\section*{Coursework 7 (Scala, Knight's Tour)}

This coursework is worth 10\%. It is about searching and
backtracking. The first part is due on 23 November at 11pm; the
second, more advanced part, is due on 30 November at 11pm. You are
asked to implement Scala programs that solve various versions of the
\textit{Knight's Tour Problem} on a chessboard. Note the second part
might include material you have not yet seen in the first two
lectures.  Make sure the files you submit can be processed by just
calling \texttt{scala <<filename.scala>>}.\bigskip

\noindent
\textbf{Important:} Do not use any mutable data structures in your
submissions! They are not needed. This means you cannot use
\texttt{ListBuffer}s, for example. Do not use \texttt{return} in your
code! It has a different meaning in Scala, than in Java.
Do not use \texttt{var}! This declares a mutable variable. Feel free to
copy any code you need from files \texttt{knight1.scala},
\texttt{knight2.scala} and \texttt{knight3.scala}. Make sure the
functions you submit are defined on the ``top-level'' of Scala, not
inside a class or object. Also note that the running time of
each part will be restricted to a maximum of 360 seconds on my
laptop.

 
\subsection*{Disclaimer}

It should be understood that the work you submit represents
your own effort. You have not copied from anyone else. An
exception is the Scala code I showed during the lectures or
uploaded to KEATS, which you can freely use.\medskip

\subsection*{Background}

The \textit{Knight's Tour Problem} is about finding a tour such that
the knight visits every field on an $n\times n$ chessboard once. For
example on a $5\times 5$ chessboard, a knight's tour is:


\chessboard[maxfield=d4, 
            pgfstyle= {[base,at={\pgfpoint{0pt}{-0.5ex}}]text},
            text = \small 24, markfield=Z4,
            text = \small 11, markfield=a4,
            text = \small  6, markfield=b4,
            text = \small 17, markfield=c4,
            text = \small  0, markfield=d4,
            text = \small 19, markfield=Z3,
            text = \small 16, markfield=a3,
            text = \small 23, markfield=b3,
            text = \small 12, markfield=c3,
            text = \small  7, markfield=d3,
            text = \small 10, markfield=Z2,
            text = \small  5, markfield=a2,
            text = \small 18, markfield=b2,
            text = \small  1, markfield=c2,
            text = \small 22, markfield=d2,
            text = \small 15, markfield=Z1,
            text = \small 20, markfield=a1,
            text = \small  3, markfield=b1,
            text = \small  8, markfield=c1,
            text = \small 13, markfield=d1,
            text = \small  4, markfield=Z0,
            text = \small  9, markfield=a0,
            text = \small 14, markfield=b0,
            text = \small 21, markfield=c0,
            text = \small  2, markfield=d0
           ]

\noindent
The tour starts in the right-upper corner, then moves to field
$(3,2)$, then $(4,0)$ and so on. There are no knight's tours on
$2\times 2$, $3\times 3$ and $4\times 4$ chessboards, but for every
bigger board there is. 

A knight's tour is called \emph{closed}, if the last step in the tour
is within a knight's move to the beginning of the tour. So the above
knight's tour is \underline{not} closed because the last
step on field $(0, 4)$ is not within the reach of the first step on
$(4, 4)$. It turns out there is no closed knight's tour on a $5\times
5$ board. But there are on a $6\times 6$ board and on bigger ones, for
example

\chessboard[maxfield=e5, 
            pgfstyle= {[base,at={\pgfpoint{0pt}{-0.5ex}}]text},
            text = \small 10, markfield=Z5,
            text = \small  5, markfield=a5,
            text = \small 18, markfield=b5,
            text = \small 25, markfield=c5,
            text = \small 16, markfield=d5,
            text = \small  7, markfield=e5,
            text = \small 31, markfield=Z4,
            text = \small 26, markfield=a4,
            text = \small  9, markfield=b4,
            text = \small  6, markfield=c4,
            text = \small 19, markfield=d4,
            text = \small 24, markfield=e4,
            % 4  11  30  17   8  15 
            text = \small  4, markfield=Z3,
            text = \small 11, markfield=a3,
            text = \small 30, markfield=b3,
            text = \small 17, markfield=c3,
            text = \small  8, markfield=d3,
            text = \small 15, markfield=e3,
            %29  32  27   0  23  20 
            text = \small 29, markfield=Z2,
            text = \small 32, markfield=a2,
            text = \small 27, markfield=b2,
            text = \small  0, markfield=c2,
            text = \small 23, markfield=d2,
            text = \small 20, markfield=e2,
            %12   3  34  21  14   1 
            text = \small 12, markfield=Z1,
            text = \small  3, markfield=a1,
            text = \small 34, markfield=b1,
            text = \small 21, markfield=c1,
            text = \small 14, markfield=d1,
            text = \small  1, markfield=e1,
            %33  28  13   2  35  22 
            text = \small 33, markfield=Z0,
            text = \small 28, markfield=a0,
            text = \small 13, markfield=b0,
            text = \small  2, markfield=c0,
            text = \small 35, markfield=d0,
            text = \small 22, markfield=e0,
            vlabel=false,
            hlabel=false
           ]


\noindent
where the 35th move can join up again with the 0th move.

If you cannot remember how a knight moves in chess, or never played
chess, below are all potential moves indicated for two knights, one on
field $(2, 2)$ (blue moves) and another on $(7, 7)$ (red moves):


\chessboard[maxfield=g7,
            color=blue!50,
            linewidth=0.2em,
            shortenstart=0.5ex,
            shortenend=0.5ex,
            markstyle=cross,
            markfields={a4, c4, Z3, d3, Z1, d1, a0, c0},
            color=red!50,
            markfields={f5, e6},
            setpieces={Ng7, Nb2}]

\subsection*{Part 1 (7 Marks)}

You are asked to implement the knight's tour problem such that the
dimension of the board can be changed.  Therefore most functions will
take the dimension of the board as an argument.  The fun with this
problem is that even for small chessboard dimensions it has already an
incredibly large search space---finding a tour is like finding a
needle in a haystack. In the first task we want to see how far we get
with exhaustively exploring the complete search space for small
chessboards.\medskip

\noindent
Let us first fix the basic datastructures for the implementation.  The
board dimension is an integer (we will never go beyond board sizes of
$50 \times 50$).  A \emph{position} (or field) on the chessboard is
a pair of integers, like $(0, 0)$. A \emph{path} is a list of
positions. The first (or 0th move) in a path is the last element in
this list; and the last move in the path is the first element. For
example the path for the $5\times 5$ chessboard above is represented
by

\[
\texttt{List($\underbrace{\texttt{(0, 4)}}_{24}$,
  $\underbrace{\texttt{(2, 3)}}_{23}$, ...,
  $\underbrace{\texttt{(3, 2)}}_1$, $\underbrace{\texttt{(4, 4)}}_0$)}
\]

\noindent
Suppose the dimension of a chessboard is $n$, then a path is a
\emph{tour} if the length of the path is $n \times n$, each element
occurs only once in the path, and each move follows the rules of how a
knight moves (see above for the rules).


\subsubsection*{Tasks (file knight1.scala)}

\begin{itemize}
\item[(1a)] Implement an \texttt{is-legal-move} function that takes a
  dimension, a path and a position as argument and tests whether the
  position is inside the board and not yet element in the
  path. \hfill[1 Mark]

\item[(1b)] Implement a \texttt{legal-moves} function that calculates for a
  position all legal onward moves. If the onward moves are
  placed on a circle, you should produce them starting from
  ``12-oclock'' following in clockwise order.  For example on an
  $8\times 8$ board for a knight on position $(2, 2)$ and otherwise
  empty board, the legal-moves function should produce the onward
  positions in this order:

  \begin{center}
  \texttt{List((3,4), (4,3), (4,1), (3,0), (1,0), (0,1), (0,3), (1,4))}
  \end{center}

  If the board is not empty, then maybe some of the moves need to be
  filtered out from this list.  For a knight on field $(7, 7)$ and an
  empty board, the legal moves are

  \begin{center}
  \texttt{List((6,5), (5,6))}
  \end{center}
  \mbox{}\hfill[1 Mark]

\item[(1c)] Implement two recursive functions (count-tours and
  enum-tours). They each take a dimension and a path as
  arguments. They exhaustively search for tours starting
  from the given path. The first function counts all possible 
  tours (there can be none for certain board sizes) and the second
  collects all tours in a list of paths.\hfill[2 Marks]
\end{itemize}

\noindent \textbf{Test data:} For the marking, the functions in (1c)
will be called with board sizes up to $5 \times 5$. If you search
for tours on a $5 \times 5$ board starting only from field $(0, 0)$,
there are 304 of tours. If you try out every field of a $5 \times
5$-board as a starting field and add up all tours, you obtain
1728. A $6\times 6$ board is already too large to be searched
exhaustively.\footnote{For your interest, the number of tours on
  $6\times 6$, $7\times 7$ and $8\times 8$ are 6637920, 165575218320,
  19591828170979904, respectively.}

\subsubsection*{Tasks (file knight2.scala)}

\begin{itemize}
\item[(2a)] Implement a first-function. This function takes a list of
  positions and a function $f$ as arguments. The function $f$ takes a
  position as argument and produces an optional path. So $f$'s type is
  \texttt{Pos => Option[Path]}. The idea behind the first-function is
  as follows:

  \[
  \begin{array}{lcl}
  \textit{first}(\texttt{Nil}, f) & \dn & \texttt{None}\\  
  \textit{first}(x\!::\!xs, f) & \dn & \begin{cases}
    f(x) & \textit{if}\;f(x) \not=\texttt{None}\\
    \textit{first}(xs, f) & \textit{otherwise}\\
                              \end{cases}
  \end{array}
  \]

  \noindent That is, we want to find the first position where the
  result of $f$ is not \texttt{None}, if there is one. Note that you
  do not (need to) know anything about the function $f$ except its
  type, namely \texttt{Pos => Option[Path]}. There is one additional
  point however you should take into account when implementing
  \textit{first}: you will need to calculate what the result of $f(x)$
  is; your code should do this only \textbf{once}!\\\mbox{}\hfill[1 Mark]
  
\item[(2b)] Implement a first-tour function that uses the
  first-function from (2a), and searches recursively for a tour.
  As there might not be such a tour at all, the first-tour function
  needs to return an \texttt{Option[Path]}.\\\mbox{}\hfill[2 Marks]
\end{itemize}

\noindent
\textbf{Testing} The first tour function will be called with board
sizes of up to $8 \times 8$. 


\subsection*{Part 2 (3 Marks)}

As you should have seen in Part 1, a naive search for tours
beyond $8 \times 8$ boards and also searching for closed tours
takes too much time. There is a heuristic, called Warnsdorf's rule
that can speed up finding a tour. This heuristic states that a knight
is moved so that it always proceeds to the field from which the
knight will have the \underline{fewest} onward moves.  For example for
a knight on field $(1, 3)$, the field $(0, 1)$ has the fewest possible
onward moves, namely 2.

\chessboard[maxfield=g7,
            pgfstyle= {[base,at={\pgfpoint{0pt}{-0.5ex}}]text},
            text = \small 3, markfield=Z5,
            text = \small 7, markfield=b5,
            text = \small 7, markfield=c4,
            text = \small 7, markfield=c2,
            text = \small 5, markfield=b1,
            text = \small 2, markfield=Z1,
            setpieces={Na3}]

\noindent
Warnsdorf's rule states that the moves on the board above should be
tried in the order

\[
(0, 1), (0, 5), (2, 1), (2, 5), (3, 4), (3, 2)
\]

\noindent
Whenever there are ties, the corresponding onward moves can be in any
order.  When calculating the number of onward moves for each field, we
do not count moves that revisit any field already visited.

\subsubsection*{Tasks (file knight3.scala)}

\begin{itemize}
\item[(3a)] Write a function ordered-moves that calculates a list of
  onward moves like in (1b) but orders them according to the
  Warnsdorf’s rule. That means moves with the fewest legal onward moves
  should come first (in order to be tried out first). \hfill[1 Mark]
  
\item[(3b)] Implement a first-closed-tour-heuristic function that searches for a
  \textbf{closed} tour on a $6\times 6$ board. It should use the
  first-function from (2a) and tries out onward moves according to
  the ordered-moves function from (3a). It is more likely to find
  a solution when started in the middle of the board (that is
  position $(dimension / 2, dimension / 2)$). \hfill[1 Mark]

\item[(3c)] Implement a first-tour-heuristic function for boards up to $50\times 50$.
  It is the same function  as in (3b) but searches for tours. You have
  to be careful to write a tail-recursive version of the first-tour-heuristic
  function otherwise you will get problems with stack-overflows. \hfill[1 Mark]
\end{itemize}  

\end{document}

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