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\documentclass{article}

\usepackage{chessboard}

\usepackage[LSBC4,T1]{fontenc}

\let\clipbox\relax

\usepackage{../styles/style}

\usepackage{../styles/langs}

\usepackage{disclaimer}

\usepackage{ulem}

\begin{document}

\setchessboard{smallboard,

               zero,

               showmover=false,

               boardfontencoding=LSBC4,

               hlabelformat=\arabic{ranklabel},

               vlabelformat=\arabic{filelabel}}

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

\section*{Main Part 4 (Scala, 11 Marks)}

\mbox{}\hfill\textit{``The problem with object-oriented languages is they’ve got all this implicit,}\\

\mbox{}\hfill\textit{environment that they carry around with them. You wanted a banana but}\\

\mbox{}\hfill\textit{what you got was a gorilla holding the banana and the entire jungle.''}\smallskip\\

\mbox{}\hfill\textit{ --- Joe Armstrong (creator of the Erlang programming language)}\medskip\bigskip

\noindent

This part is about searching and backtracking. You are asked to

implement Scala programs that solve various versions of the

\textit{Knight's Tour Problem} on a chessboard.

\medskip 

% Note the core, more advanced, part might include material you have not

%yet seen in the first three lectures. \bigskip

\IMPORTANTNONE{}

\noindent

Also note that the running time of each part will be restricted to a

maximum of 30 seconds on my laptop: If you calculate a result once,

try to avoid to calculate the result again. Feel free to copy any code

you need from files \texttt{knight1.scala}, \texttt{knight2.scala} and

\texttt{knight3.scala}.

\DISCLAIMER{}

\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

This 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},

            boardfontsize=12pt,labelfontsize=9pt]}

\subsection*{Reference Implementation}

%\mbox{}\alert{}\textcolor{red}{You need to download \texttt{knight1.jar} from K%EATS. The one

%supplied with github does not contain the correct code. See Scala coursework

%section on KEATS.}\medskip

\noindent

This Scala part comes with three reference implementations in form of

\texttt{jar}-files. This allows you to run any test cases on your own

computer. For example you can call Scala on the command line with the

option \texttt{-cp knight1.jar} and then query any function from the

\texttt{knight1.scala} template file. As usual you have to

prefix the calls with \texttt{M4a}, \texttt{M4b}, \texttt{M4c} and \texttt{M4d}.

Since some of the calls are time sensitive, I included some timing

information. For example

\begin{lstlisting}[language={},numbers=none,basicstyle=\ttfamily\small]

$ scala -cp knight1.jar

scala> M4a.enum_tours(5, List((0, 0))).length

Time needed: 1.722 secs.

res0: Int = 304

scala> M4a.print_board(8, M4a.first_tour(8, List((0, 0))).get)

Time needed: 15.411 secs.

 51  46  55  44  53   4  21  12 

 56  43  52   3  22  13  24   5 

 47  50  45  54  25  20  11  14 

 42  57   2  49  40  23   6  19 

 35  48  41  26  61  10  15  28 

 58   1  36  39  32  27  18   7 

 37  34  31  60   9  62  29  16 

  0  59  38  33  30  17   8  63 

\end{lstlisting}%$

\subsection*{Hints}

\noindent

Useful list functions: \texttt{.contains(..)} checks

whether an element is in a list, \texttt{.flatten} turns a list of

lists into just a list, \texttt{\_::\_} puts an element on the head of

the list, \texttt{.head} gives you the first element of a list (make

sure the list is not \texttt{Nil}); a useful option function:

\texttt{.isDefined} returns true, if an option is \texttt{Some(..)};

anonymous functions can be constructed using \texttt{(x:Int) => ...},

this function takes an \texttt{Int} as an argument; 

a useful list function: \texttt{.sortBy} sorts a list

according to a component given by the function; a function can be

tested to be tail-recursive by annotation \texttt{@tailrec}, which is

made available by importing \texttt{scala.annotation.tailrec}.\medskip

%%\newpage

\subsection*{Tasks}

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.

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*{Task 1 (file knight1.scala)}

\begin{itemize}

\item[(1)] Implement an \texttt{is\_legal} function that takes a

  dimension, a path and a position as arguments and tests whether the

  position is inside the board and not yet element in the

  path. \hfill[1 Mark]

\item[(2)] 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-o'clock'' following in clockwise order.  For example on an

  $8\times 8$ board for a knight at 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[(3)] Implement two recursive functions (\texttt{count\_tours} and

  \texttt{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. These functions will be

  called with a path containing a single position---the starting field.

  They are expected to extend this path so as to find all tours starting

  from the given position.\\

  \mbox{}\hfill[1 Mark]

\end{itemize}

\noindent \textbf{Test data:} For the marking, the functions in (3)

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.}\smallskip

\begin{itemize}

\item[(4)] Implement a \texttt{first}-function. This function takes a list of

  positions and a function $f$ as arguments; $f$ is the name we give to

  this argument). 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 \texttt{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

  `inside' \texttt{first}, you do not (need to) know anything about

  the argument $f$ except its type, namely \texttt{Pos =>

    Option[Path]}. If you want to find out what the result of $f$ is

  on a particular argument, say $x$, you can just write $f(x)$. 

  There is one additional point however you should

  take into account when implementing \texttt{first}: you will need to

  calculate what the result of $f(x)$ is; your code should do this

  only \textbf{once} and for as \textbf{few} elements in the list as

  possible! Do not calculate $f(x)$ for all elements and then see which 

  is the first \texttt{Some}.\\\mbox{}\hfill[1 Mark]

\item[(5)] Implement a \texttt{first\_tour} function that uses the

  \texttt{first}-function from (4), and searches recursively for single tour.

  As there might not be such a tour at all, the \texttt{first\_tour} function

  needs to return a value of type

  \texttt{Option[Path]}.\\\mbox{}\hfill[1 Mark]  

\end{itemize}

\noindent

\textbf{Testing:} The \texttt{first\_tour} function will be called with board

sizes of up to $8 \times 8$.

\bigskip

%%\newpage

\subsubsection*{Task 2 (file knight2.scala)}

\noindent

As you should have seen in the earlier parts, a naive search for tours beyond

$8 \times 8$ boards and also searching for closed tours even on small

boards takes too much time. There is a heuristics, called \emph{Warnsdorf's

Rule} that can speed up finding a tour. This heuristics 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.

\begin{itemize}

\item[(6)] Write a function \texttt{ordered\_moves} that calculates a list of

  onward moves like in (2) but orders them according to 

  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[(7)] Implement a \texttt{first\_closed\_tour\_heuristics}

  function that searches for a single

  \textbf{closed} tour on a $6\times 6$ board. It should try out

  onward moves according to

  the \texttt{ordered\_moves} function from (6). 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[(8)] Implement a \texttt{first\_tour\_heuristics} function

  for boards up to

  $30\times 30$.  It is the same function as in (7) but searches for

  tours (not just closed tours). It might be called with any field on the

  board as starting field.\\

  %You have to be careful to write a

  %tail-recursive function of the \texttt{first\_tour\_heuristics} function

  %otherwise you will get problems with stack-overflows.\\

  \mbox{}\hfill[1 Mark]

\end{itemize}    

\subsubsection*{Task 3 (file knight3.scala)}

\begin{itemize}

\item[(9)] Implement a function \texttt{tour\_on\_mega\_board} which is

  the same function as in (8), \textbf{but} should be able to

  deal with boards up to

  $70\times 70$ \textbf{within 30 seconds} (on my laptop). This will be tested

  by starting from field $(0, 0)$. You have to be careful to

  write a tail-recursive function otherwise you will get problems

  with stack-overflows. Please observe the requirements about

  the submissions: no tricks involving \textbf{.par}.\medskip

  The timelimit of 30 seconds is with respect to the laptop on which the

  marking will happen. You can roughly estimate how well your

  implementation performs by running \texttt{knight3.jar} on your

  computer. For example the reference implementation shows

  on my laptop:

  \begin{lstlisting}[language={},numbers=none,basicstyle=\ttfamily\small]

$ scala -cp knight3.jar

scala> M4c.tour_on_mega_board(70, List((0, 0)))

Time needed: 9.484 secs.

...<<long_list>>...

\end{lstlisting}%$

  \mbox{}\hfill[1 Mark]

\end{itemize}

\subsubsection*{Task 4 (file knight4.scala)}

\begin{itemize}

\item[(10)] In this task we want to solve the problem of finding a single

  tour (if there exists one) on what is sometimes called ``mutilated''

  chessboards, for example rectangular chessbords or chessboards where

  fields are missing. For this implement a search function

  \begin{center}

    \begin{tabular}{@{}l@{}}

    \texttt{def one\_tour\_pred(dim: Int, path: Path,}\\

      \texttt{\phantom{def one\_tour\_pred(}n: Int, f: Pos => Boolean): Option[Path]}

      \end{tabular}

  \end{center}

  which has, like before, the dimension and current path as arguments,

  and in addtion it takes an integer, which specifies the length of

  the longest path (or length of the path after which the search

  should backtrack), and a function from positions to Booleans. This

  function acts as a predicate in order to restrict which onward legal

  moves should be considered in the search. The function

  \texttt{one\_tour\_pred} should return a single tour

  (\texttt{Some}-case), if one or more tours exist, and \texttt{None}, if no

  tour exists. For example when called with 

  \begin{center}

  \texttt{one\_tour\_pred(7, List((0, 0)), 35, x => x.\_1 < 5)}

  \end{center}  

  we are looking for a tour starting from position \texttt{(0,0)} on a

  7 $\times$ 5 board, where the maximum length of the path is 35. The

  predicate \texttt{x => x.\_1 < 5} ensures that no position with an

  x-coordinate bigger than 4 is considered. One possible solution is

  \begin{lstlisting}[language={},numbers=none,basicstyle=\ttfamily\small]

   0   13   22   33   28   11   20 

  23   32    1   12   21   34   27 

  14    7   16   29    2   19   10 

  31   24    5    8   17   26    3 

   6   15   30   25    4    9   18 

  -1   -1   -1   -1   -1   -1   -1 

  -1   -1   -1   -1   -1   -1   -1

\end{lstlisting}%$

where \texttt{print\_board} prints a \texttt{-1} for all fields that

have not been visited.

  \mbox{}\hfill[2 Marks]

\end{itemize}

\end{document}

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