diff -r 19b75e899d37 -r 9c03b5e89a2a cws/cw03.tex --- a/cws/cw03.tex Fri Apr 26 17:29:30 2024 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,445 +0,0 @@ -% !TEX program = xelatex -\documentclass{article} -\usepackage{chessboard} -\usepackage[LSBC4,T1]{fontenc} -\let\clipbox\relax -\usepackage{../style} -\usepackage{../langs} -\usepackage{disclaimer} - -\begin{document} - -\setchessboard{smallboard, - zero, - showmover=false, - boardfontencoding=LSBC4, - hlabelformat=\arabic{ranklabel}, - vlabelformat=\arabic{filelabel}} - -\mbox{}\\[-18mm]\mbox{} - -\section*{Part 8 (Scala)} - -\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. The preliminary part (4\%) is -due on \cwEIGHT{} at 4pm; the core part is due on \cwEIGHTa{} at 4pm. -Note the core, more advanced, part might include material you have not -yet seen in the first three lectures. \bigskip - -\IMPORTANT{} -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} - -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{CW8a}, \texttt{CW8b} and \texttt{CW8c}. -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> CW8a.enum_tours(5, List((0, 0))).length -Time needed: 1.722 secs. -res0: Int = 304 - -scala> CW8a.print_board(8, CW8a.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 -\textbf{Preliminary Part} 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.\medskip - - -\noindent -\textbf{Core Part} 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 - - - - -\subsection*{Preliminary Part (4 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. -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[(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[2 Marks] -\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 - - -\subsection*{Core Part (6 Marks)} - - -\subsubsection*{Tasks (file knight1.scala cont.)} - -\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 - -\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. - -\subsubsection*{Tasks (file knight2.scala)} - -\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 (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> CW8c.tour_on_mega_board(70, List((0, 0))) -Time needed: 9.484 secs. -...<>... -\end{lstlisting}%$ - - \mbox{}\hfill[1 Mark] -\end{itemize} -\bigskip - - - - -\end{document} - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: t -%%% End: