1 // Part 1 about finding and counting Knight's tours |
1 // Part 1 about finding Knight's tours |
2 //================================================== |
2 //===================================== |
3 |
3 |
4 //object CW8a { // for preparing the jar |
4 // If you need any auxiliary function, feel free to |
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5 // implement it, but do not make any changes to the |
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6 // templates below. Also have a look whether the functions |
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7 // at the end are of any help. |
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8 |
5 |
9 |
6 type Pos = (Int, Int) // a position on a chessboard |
10 type Pos = (Int, Int) // a position on a chessboard |
7 type Path = List[Pos] // a path...a list of positions |
11 type Path = List[Pos] // a path...a list of positions |
8 |
12 |
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13 //(1) Complete the function that tests whether the position x |
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14 // is inside the board and not yet element in the path. |
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15 |
10 // for measuring time in the JAR |
16 def is_legal(dim: Int, path: Path, x: Pos) : Boolean = ((!(path.contains(x))) && (x._1 < dim) && (x._2 < dim)) |
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17 |
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18 |
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19 |
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20 //(2) Complete the function that calculates for a position x |
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21 // all legal onward moves that are not already in the path. |
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22 // The moves should be ordered in a "clockwise" manner. |
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23 |
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24 |
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25 def legal_moves(dim: Int, path: Path, x: Pos) : List[Pos] ={ |
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26 val y = List((x._1 + 1, x._2 + 2), |
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27 (x._1 + 2, x._2 + 1), |
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28 (x._1 + 2, x._2 - 1), |
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29 (x._1 + 1, x._2 - 2), |
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30 (x._1 - 1, x._2 - 2), |
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31 (x._1 - 2, x._2 - 1), |
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32 (x._1 - 2, x._2 + 1), |
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33 (x._1 - 1, x._2 + 2) |
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34 ) |
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35 y.filter(next => is_legal(dim, path, next)) |
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36 } |
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37 |
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38 //some test cases |
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39 // |
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40 //assert(legal_moves(8, Nil, (2,2)) == List((3,4), (4,3), (4,1), (3,0), (1,0), (0,1), (0,3), (1,4))) |
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41 //assert(legal_moves(8, Nil, (7,7)) == List((6,5), (5,6))) |
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42 //assert(legal_moves(8, List((4,1), (1,0)), (2,2)) == List((3,4), (4,3), (3,0), (0,1), (0,3), (1,4))) |
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43 //assert(legal_moves(8, List((6,6)), (7,7)) == List((6,5), (5,6))) |
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44 |
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45 |
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46 //(3) Complete the two recursive functions below. |
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47 // They exhaustively search for knight's tours starting from the |
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48 // given path. The first function counts all possible tours, |
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49 // and the second collects all tours in a list of paths. |
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50 |
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51 def count_tours(dim: Int, path: Path) : Int = { |
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52 if(path.length == dim*dim) 1 else |
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53 (for(i <- legal_moves(dim, path, path.head)) yield |
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54 count_tours(dim, i :: path) |
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55 ).sum |
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56 } |
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57 |
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58 def enum_tours(dim: Int, path: Path) : List[Path] ={ |
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59 if(path.length == dim*dim) List(path) else |
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60 (for(i <- legal_moves(dim, path, path.head)) yield |
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61 enum_tours(dim, i :: path) |
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62 ).flatten |
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63 } |
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64 |
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65 //(5) Implement a first-function that finds the first |
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66 // element, say x, in the list xs where f is not None. |
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67 // In that case Return f(x), otherwise None. If possible, |
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68 // calculate f(x) only once. |
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69 |
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70 def first(xs: List[Pos], f: Pos => Option[Path]) : Option[Path] = { |
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71 if(xs == Nil) None |
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72 else( |
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73 for(x <- xs) yield{ |
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74 val a = f(x) |
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75 if(a != None) a |
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76 else first(xs.drop(1), f) |
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77 } |
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78 ).head |
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79 } |
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80 |
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81 // test cases |
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82 //def foo(x: (Int, Int)) = if (x._1 > 3) Some(List(x)) else None |
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83 // |
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84 //first(List((1, 0),(2, 0),(3, 0),(4, 0)), foo) // Some(List((4,0))) |
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85 //first(List((1, 0),(2, 0),(3, 0)), foo) // None |
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86 |
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87 |
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88 |
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89 |
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90 //(6) Implement a function that uses the first-function from (5) for |
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91 // trying out onward moves, and searches recursively for a |
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92 // knight tour on a dim * dim-board. |
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93 |
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94 |
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95 // def first_tour(dim: Int, path: Path) : Option[Path] = { |
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96 // first(legal_moves(dim, path, path.head), (x : Pos => )) |
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97 // } |
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98 |
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99 /* Helper functions |
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100 |
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101 |
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102 // for measuring time |
11 def time_needed[T](code: => T) : T = { |
103 def time_needed[T](code: => T) : T = { |
12 val start = System.nanoTime() |
104 val start = System.nanoTime() |
13 val result = code |
105 val result = code |
14 val end = System.nanoTime() |
106 val end = System.nanoTime() |
15 println(f"Time needed: ${(end - start) / 1.0e9}%3.3f secs.") |
107 println(f"Time needed: ${(end - start) / 1.0e9}%3.3f secs.") |
16 result |
108 result |
17 } |
109 } |
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110 |
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111 // can be called for example with |
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112 // time_needed(count_tours(dim, List((0, 0)))) |
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113 // in order to print out the time that is needed for |
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114 // running count_tours |
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115 |
19 // for printing a board |
116 // for printing a board |
20 def print_board(dim: Int, path: Path): Unit = { |
117 def print_board(dim: Int, path: Path): Unit = { |
21 println |
118 println |
22 for (i <- 0 until dim) { |
119 for (i <- 0 until dim) { |
25 } |
122 } |
26 println |
123 println |
27 } |
124 } |
28 } |
125 } |
29 |
126 |
30 def is_legal(dim: Int, path: Path, x: Pos): Boolean = |
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31 0 <= x._1 && 0 <= x._2 && x._1 < dim && x._2 < dim && !path.contains(x) |
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32 |
127 |
33 // testcases |
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34 //assert(is_legal(8, Nil, (3, 4)) == true) |
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35 //assert(is_legal(8, List((4, 1), (1, 0)), (4, 1)) == false) |
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36 //assert(is_legal(2, Nil, (0, 0)) == true) |
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37 |
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38 |
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39 def add_pair(x: Pos, y: Pos): Pos = |
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40 (x._1 + y._1, x._2 + y._2) |
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41 |
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42 def moves(x: Pos): List[Pos] = |
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43 List(( 1, 2),( 2, 1),( 2, -1),( 1, -2), |
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44 (-1, -2),(-2, -1),(-2, 1),(-1, 2)).map(add_pair(x, _)) |
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45 |
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46 // 1 mark |
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47 |
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48 def legal_moves(dim: Int, path: Path, x: Pos): List[Pos] = |
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49 moves(x).filter(is_legal(dim, path, _)) |
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50 |
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51 // testcases |
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52 //assert(legal_moves(8, Nil, (2,2)) == |
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53 // List((3,4), (4,3), (4,1), (3,0), (1,0), (0,1), (0,3), (1,4))) |
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54 //assert(legal_moves(8, Nil, (7,7)) == List((6,5), (5,6))) |
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55 //assert(legal_moves(8, List((4,1), (1,0)), (2,2)) == |
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56 // List((3,4), (4,3), (3,0), (0,1), (0,3), (1,4))) |
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57 //assert(legal_moves(8, List((6,6)), (7,7)) == List((6,5), (5,6))) |
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58 //assert(legal_moves(1, Nil, (0,0)) == List()) |
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59 //assert(legal_moves(2, Nil, (0,0)) == List()) |
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60 //assert(legal_moves(3, Nil, (0,0)) == List((1,2), (2,1))) |
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61 |
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62 // 2 marks |
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63 |
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64 def tcount_tours(dim: Int, path: Path): Int = { |
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65 if (path.length == dim * dim) 1 |
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66 else |
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67 (for (x <- legal_moves(dim, path, path.head)) yield tcount_tours(dim, x::path)).sum |
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68 } |
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69 |
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70 def count_tours(dim: Int, path: Path) = |
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71 time_needed(tcount_tours(dim: Int, path: Path)) |
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72 |
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73 |
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74 def tenum_tours(dim: Int, path: Path): List[Path] = { |
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75 if (path.length == dim * dim) List(path) |
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76 else |
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77 (for (x <- legal_moves(dim, path, path.head)) yield tenum_tours(dim, x::path)).flatten |
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78 } |
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79 |
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80 def enum_tours(dim: Int, path: Path) = |
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81 time_needed(tenum_tours(dim: Int, path: Path)) |
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82 |
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83 // test cases |
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84 |
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85 /* |
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86 def count_all_tours(dim: Int) = { |
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87 for (i <- (0 until dim).toList; |
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88 j <- (0 until dim).toList) yield count_tours(dim, List((i, j))) |
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89 } |
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90 |
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91 def enum_all_tours(dim: Int): List[Path] = { |
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92 (for (i <- (0 until dim).toList; |
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93 j <- (0 until dim).toList) yield enum_tours(dim, List((i, j)))).flatten |
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94 } |
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95 |
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96 |
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97 println("Number of tours starting from (0, 0)") |
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98 |
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99 for (dim <- 1 to 5) { |
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100 println(s"${dim} x ${dim} " + time_needed(0, count_tours(dim, List((0, 0))))) |
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101 } |
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102 |
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103 println("Number of tours starting from all fields") |
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104 |
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105 for (dim <- 1 to 5) { |
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106 println(s"${dim} x ${dim} " + time_needed(0, count_all_tours(dim))) |
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107 } |
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108 |
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109 for (dim <- 1 to 5) { |
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110 val ts = enum_tours(dim, List((0, 0))) |
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111 println(s"${dim} x ${dim} ") |
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112 if (ts != Nil) { |
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113 print_board(dim, ts.head) |
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114 println(ts.head) |
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115 } |
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116 } |
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117 */ |
128 */ |
118 |
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119 // 1 mark |
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120 |
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121 def first(xs: List[Pos], f: Pos => Option[Path]): Option[Path] = xs match { |
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122 case Nil => None |
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123 case x::xs => { |
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124 val result = f(x) |
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125 if (result.isDefined) result else first(xs, f) |
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126 } |
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127 } |
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128 |
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129 // test cases |
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130 //def foo(x: (Int, Int)) = if (x._1 > 3) Some(List(x)) else None |
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131 // |
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132 //first(List((1, 0),(2, 0),(3, 0),(4, 0)), foo) |
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133 //first(List((1, 0),(2, 0),(3, 0)), foo) |
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134 |
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135 |
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136 // 1 mark |
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137 |
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138 def tfirst_tour(dim: Int, path: Path): Option[Path] = { |
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139 if (path.length == dim * dim) Some(path) |
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140 else |
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141 first(legal_moves(dim, path, path.head), (x:Pos) => tfirst_tour(dim, x::path)) |
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142 } |
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143 |
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144 def first_tour(dim: Int, path: Path) = |
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145 time_needed(tfirst_tour(dim: Int, path: Path)) |
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146 |
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147 |
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148 /* |
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149 for (dim <- 1 to 8) { |
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150 val t = first_tour(dim, List((0, 0))) |
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151 println(s"${dim} x ${dim} " + (if (t == None) "" else { print_board(dim, t.get) ; "" })) |
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152 } |
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153 */ |
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154 |
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155 // 15 secs for 8 x 8 |
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156 //val ts1 = time_needed(0,first_tour(8, List((0, 0))).get) |
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157 |
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158 // no result for 4 x 4 |
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159 //val ts2 = time_needed(0, first_tour(4, List((0, 0)))) |
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160 |
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161 // 0.3 secs for 6 x 6 |
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162 //val ts3 = time_needed(0, first_tour(6, List((0, 0)))) |
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163 |
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164 // 15 secs for 8 x 8 |
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165 //time_needed(0, print_board(8, first_tour(8, List((0, 0))).get)) |
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166 |
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167 |
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168 //} |
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169 |
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170 |
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