1 // Preliminary Part about finding Knight's tours |
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2 //=============================================== |
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3 |
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4 |
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5 object CW9a { |
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6 |
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7 // If you need any auxiliary function, feel free to |
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8 // implement it, but do not make any changes to the |
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9 // templates below. Also have a look whether the functions |
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10 // at the end are of any help. |
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11 |
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12 |
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13 |
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14 type Pos = (Int, Int) // a position on a chessboard |
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15 type Path = List[Pos] // a path...a list of positions |
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16 |
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17 //(1) Complete the function that tests whether the position x |
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18 // is inside the board and not yet element in the path. |
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19 |
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20 def is_legal(dim: Int, path: Path, x: Pos) : Boolean = { |
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21 if ((!(path.contains(x))) && (x._1 >= 0) && (x._2 >= 0) && (x._1 < dim) && (x._2 < dim)) |
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22 true |
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23 else false |
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24 } |
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25 |
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26 //(2) Complete the function that calculates for a position x |
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27 // all legal onward moves that are not already in the path. |
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28 // The moves should be ordered in a "clockwise" manner. |
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29 |
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30 |
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31 def legal_moves(dim: Int, path: Path, x: Pos) : List[Pos] = {//List[Pos] |
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32 val changes = List((1,2),(2,1),(2,-1),(1,-2),(-1,-2),(-2,-1),(-2,1),(-1,2)) |
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33 val returnList = (for ((y,z) <- changes) yield( |
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34 //println(y,z)-2,-1 |
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35 if ((is_legal(dim,path,((x._1 + y) , (x._2 + z)))) == true) |
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36 Some(x._1 + y , x._2 + z) |
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37 else |
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38 None |
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39 )) |
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40 returnList.flatten |
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41 } |
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42 |
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43 |
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44 //some testcases |
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45 // |
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46 //assert(legal_moves(8, Nil, (2,2)) == |
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47 //List((3,4), (4,3), (4,1), (3,0), (1,0), (0,1), (0,3), (1,4))) |
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48 //assert(legal_moves(8, Nil, (7,7)) == List((6,5), (5,6))) |
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49 //assert(legal_moves(8, List((4,1), (1,0)), (2,2)) == |
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50 // List((3,4), (4,3), (3,0), (0,1), (0,3), (1,4))) |
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51 //assert(legal_moves(8, List((6,6)), (7,7)) == List((6,5), (5,6))) |
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52 |
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53 |
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54 //(3) Complete the two recursive functions below. |
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55 // They exhaustively search for knight's tours starting from the |
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56 // given path. The first function counts all possible tours, |
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57 // and the second collects all tours in a list of paths. |
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58 |
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59 def count_tours(dim: Int, path: Path) : Int = (dim,path) match {//Int |
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60 case (_, Nil) => 0 |
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61 case (0, path) => 0 |
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62 case (dim, path) => { if (legal_moves(dim,path, path.head).size == 0) |
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63 if(path.size < dim*dim) |
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64 0 |
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65 else |
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66 1 |
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67 else (for (j <- legal_moves(dim,path, path.head)) yield count_tours(dim,j::path)).sum |
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68 } |
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69 } |
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70 |
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71 def enum_tours(dim: Int, path: Path) : List[Path] = (dim,path) match { |
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72 case (_, Nil) => Nil |
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73 case (0, path) => Nil |
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74 case (dim, path) => { if (legal_moves(dim,path, path.head).size == 0) |
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75 if(path.size < dim*dim) |
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76 Nil |
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77 else |
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78 List(path) |
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79 else (for (j <- legal_moves(dim,path, path.head)) yield enum_tours(dim,j::path)).flatten |
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80 } |
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81 |
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82 } |
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83 |
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84 |
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85 //(4) Implement a first-function that finds the first |
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86 // element, say x, in the list xs where f is not None. |
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87 // In that case Return f(x), otherwise None. If possible, |
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88 // calculate f(x) only once. |
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89 |
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90 //def first(xs: List[Pos], f: Pos => Option[Path]) : Option[Path] = ... |
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91 |
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92 |
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93 // testcases |
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94 // |
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95 //def foo(x: (Int, Int)) = if (x._1 > 3) Some(List(x)) else None |
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96 // |
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97 //first(List((1, 0),(2, 0),(3, 0),(4, 0)), foo) // Some(List((4,0))) |
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98 //first(List((1, 0),(2, 0),(3, 0)), foo) // None |
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99 |
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100 |
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101 //(5) Implement a function that uses the first-function from (5) for |
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102 // trying out onward moves, and searches recursively for a |
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103 // knight tour on a dim * dim-board. |
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104 |
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105 |
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106 //def first_tour(dim: Int, path: Path) : Option[Path] = ... |
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107 |
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108 |
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109 |
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110 |
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111 |
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112 |
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113 /* Helper functions |
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114 |
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115 |
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116 // for measuring time |
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117 def time_needed[T](code: => T) : T = { |
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118 val start = System.nanoTime() |
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119 val result = code |
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120 val end = System.nanoTime() |
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121 println(f"Time needed: ${(end - start) / 1.0e9}%3.3f secs.") |
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122 result |
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123 } |
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124 |
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125 // can be called for example with |
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126 // time_needed(count_tours(dim, List((0, 0)))) |
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127 // in order to print out the time that is needed for |
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128 // running count_tours |
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129 |
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130 |
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131 |
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132 |
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133 // for printing a board |
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134 def print_board(dim: Int, path: Path): Unit = { |
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135 println |
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136 for (i <- 0 until dim) { |
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137 for (j <- 0 until dim) { |
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138 print(f"${path.reverse.indexOf((j, dim - i - 1))}%3.0f ") |
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139 } |
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140 println |
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141 } |
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142 } |
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143 |
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144 |
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145 */ |
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146 |
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147 } |
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