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1 // Part 1 about finding Knight's tours |
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2 //===================================== |
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3 |
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4 type Pos = (Int, Int) // a position on a chessboard |
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5 type Path = List[Pos] // a path...a list of positions |
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6 |
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7 // for measuring time |
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8 def time_needed[T](code: => T) : T = { |
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9 val start = System.nanoTime() |
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10 val result = code |
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11 val end = System.nanoTime() |
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12 println(f"Time needed: ${(end - start) / 1.0e9}%3.3f secs.") |
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13 result |
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14 } |
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15 |
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16 // for printing a board |
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17 def print_board(dim: Int, path: Path): Unit = { |
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18 println |
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19 for (i <- 0 until dim) { |
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20 for (j <- 0 until dim) { |
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21 print(f"${path.reverse.indexOf((j, dim - i - 1))}%3.0f ") |
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22 } |
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23 println |
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24 } |
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25 } |
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26 |
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27 |
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28 // 1 mark |
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29 |
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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 |
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33 assert(is_legal(8, Nil, (3, 4)) == true) |
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34 assert(is_legal(8, List((4, 1), (1, 0)), (4, 1)) == false) |
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35 assert(is_legal(2, Nil, (0, 0)) == true) |
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36 |
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37 |
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38 def add_pair(x: Pos, y: Pos): Pos = |
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39 (x._1 + y._1, x._2 + y._2) |
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40 |
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41 def moves(x: Pos): List[Pos] = |
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42 List(( 1, 2),( 2, 1),( 2, -1),( 1, -2), |
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43 (-1, -2),(-2, -1),(-2, 1),(-1, 2)).map(add_pair(x, _)) |
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44 |
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45 // 1 mark |
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46 |
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47 def legal_moves(dim: Int, path: Path, x: Pos): List[Pos] = |
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48 moves(x).filter(is_legal(dim, path, _)) |
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49 |
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50 assert(legal_moves(8, Nil, (2,2)) == |
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51 List((3,4), (4,3), (4,1), (3,0), (1,0), (0,1), (0,3), (1,4))) |
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52 assert(legal_moves(8, Nil, (7,7)) == List((6,5), (5,6))) |
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53 assert(legal_moves(8, List((4,1), (1,0)), (2,2)) == |
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54 List((3,4), (4,3), (3,0), (0,1), (0,3), (1,4))) |
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55 assert(legal_moves(8, List((6,6)), (7,7)) == List((6,5), (5,6))) |
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56 assert(legal_moves(1, Nil, (0,0)) == List()) |
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57 assert(legal_moves(2, Nil, (0,0)) == List()) |
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58 assert(legal_moves(3, Nil, (0,0)) == List((1,2), (2,1))) |
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59 |
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60 // 2 marks |
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61 |
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62 def count_tours(dim: Int, path: Path): Int = { |
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63 if (path.length == dim * dim) 1 |
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64 else |
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65 (for (x <- legal_moves(dim, path, path.head)) yield count_tours(dim, x::path)).sum |
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66 } |
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67 |
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68 def enum_tours(dim: Int, path: Path): List[Path] = { |
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69 if (path.length == dim * dim) List(path) |
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70 else |
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71 (for (x <- legal_moves(dim, path, path.head)) yield enum_tours(dim, x::path)).flatten |
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72 } |
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73 |
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74 // as far as tasks go |
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75 // test cases for CW |
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76 |
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77 |
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78 |
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79 def count_all_tours(dim: Int) = { |
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80 for (i <- (0 until dim).toList; |
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81 j <- (0 until dim).toList) yield count_tours(dim, List((i, j))) |
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82 } |
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83 |
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84 def enum_all_tours(dim: Int): List[Path] = { |
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85 (for (i <- (0 until dim).toList; |
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86 j <- (0 until dim).toList) yield enum_tours(dim, List((i, j)))).flatten |
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87 } |
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88 |
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89 |
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90 println("Number of tours starting from (0, 0)") |
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91 |
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92 for (dim <- 1 to 5) { |
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93 println(s"${dim} x ${dim} " + time_needed(0, count_tours(dim, List((0, 0))))) |
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94 } |
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95 |
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96 println("Number of tours starting from all fields") |
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97 |
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98 for (dim <- 1 to 5) { |
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99 println(s"${dim} x ${dim} " + time_needed(0, count_all_tours(dim))) |
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100 } |
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101 |
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102 for (dim <- 1 to 5) { |
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103 val ts = enum_tours(dim, List((0, 0))) |
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104 println(s"${dim} x ${dim} ") |
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105 if (ts != Nil) { |
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106 print_board(dim, ts.head) |
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107 println(ts.head) |
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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 // 1 mark |
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113 |
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114 def first(xs: List[Pos], f: Pos => Option[Path]): Option[Path] = xs match { |
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115 case Nil => None |
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116 case x::xs => { |
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117 val result = f(x) |
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118 if (result.isDefined) result else first(xs, f) |
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119 } |
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120 } |
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121 |
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122 // test cases |
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123 def foo(x: (Int, Int)) = if (x._1 > 3) Some(List(x)) else None |
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124 |
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125 first(List((1, 0),(2, 0),(3, 0),(4, 0)), foo) |
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126 first(List((1, 0),(2, 0),(3, 0)), foo) |
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127 |
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128 |
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129 // 1 mark |
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130 |
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131 def first_tour(dim: Int, path: Path): Option[Path] = { |
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132 if (path.length == dim * dim) Some(path) |
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133 else |
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134 first(legal_moves(dim, path, path.head), (x: Pos) => first_tour(dim, x::path)) |
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135 } |
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136 |
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137 |
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138 |
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139 /* |
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140 for (dim <- 1 to 8) { |
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141 val t = first_tour(dim, List((0, 0))) |
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142 println(s"${dim} x ${dim} " + (if (t == None) "" else { print_board(dim, t.get) ; "" })) |
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143 } |
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144 */ |
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145 |
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146 // 15 secs for 8 x 8 |
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147 val ts1 = time_needed(0,first_tour(8, List((0, 0))).get) |
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148 |
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149 // no result for 4 x 4 |
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150 val ts2 = time_needed(0, first_tour(4, List((0, 0)))) |
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151 |
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152 // 0.3 secs for 6 x 6 |
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153 val ts3 = time_needed(0, first_tour(6, List((0, 0)))) |
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154 |
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155 // 15 secs for 8 x 8 |
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156 time_needed(0, print_board(8, first_tour(8, List((0, 0))).get)) |
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157 |