1 // Part 1 about finding Knight's tours |
1 // Part 1 about finding and counting Knight's tours |
2 //===================================== |
2 //================================================== |
3 |
3 |
4 // If you need any auxiliary function, feel free to |
4 //object CW8a { // for preparing the jar |
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 |
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9 |
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10 |
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11 type Pos = (Int, Int) // a position on a chessboard |
6 type Pos = (Int, Int) // a position on a chessboard |
12 type Path = List[Pos] // a path...a list of positions |
7 type Path = List[Pos] // a path...a list of positions |
13 |
8 |
14 //(1) Complete the function that tests whether the position x |
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15 // is inside the board and not yet element in the path. |
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16 |
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17 def is_legal(dim: Int, path: Path, x: Pos) : Boolean = { |
10 // for measuring time in the JAR |
18 if ((x._1 < dim && x._2 < dim) && !(path.contains(x))) |
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19 true |
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20 else |
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21 false |
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22 } |
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23 |
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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] = { |
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32 val legalMovesList = List((x._1 + 1, x._2 + 2), (x._1 + 2, x._2 + 1), (x._1 + 2, x._2 - 1), (x._1 + 1, x._2 - 2), (x._1 - 1, x._2 - 2), (x._1 - 2, x._2 - 1), (x._1 - 2, x._2 + 1), (x._1 - 1, x._2 + 2)) |
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33 |
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34 for (i <- legalMovesList |
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35 if (is_legal(dim, path, i))) |
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36 yield i |
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37 |
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38 } |
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39 |
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40 |
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41 //some test cases |
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42 // |
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43 //assert(legal_moves(8, Nil, (2,2)) == |
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44 // List((3,4), (4,3), (4,1), (3,0), (1,0), (0,1), (0,3), (1,4))) |
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45 //assert(legal_moves(8, Nil, (7,7)) == List((6,5), (5,6))) |
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46 //assert(legal_moves(8, List((4,1), (1,0)), (2,2)) == |
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47 // List((3,4), (4,3), (3,0), (0,1), (0,3), (1,4))) |
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48 //assert(legal_moves(8, List((6,6)), (7,7)) == List((6,5), (5,6))) |
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49 |
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50 |
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51 //(3) Complete the two recursive functions below. |
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52 // They exhaustively search for knight's tours starting from the |
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53 // given path. The first function counts all possible tours, |
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54 // and the second collects all tours in a list of paths. |
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55 |
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56 def count_tours(dim: Int, path: Path) : Int = { |
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57 if (path.size == (dim ^ 2)){ |
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58 List(path).size |
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59 } else { |
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60 val totalTours = legal_moves(dim, path, path.head) |
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61 totalTours.map(element => count_tours(dim, element :: path)).sum |
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62 } |
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63 } |
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64 |
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65 def enum_tours(dim: Int, path: Path) : List[Path] = { |
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66 if (path.size == (dim ^ 2)){ |
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67 List(path) |
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68 } else { |
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69 val totalEnums = legal_moves(dim, path, path.head) |
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70 totalEnums.map(element => enum_tours(dim, element :: path)).flatten |
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71 } |
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72 } |
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73 |
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74 |
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75 //(5) Implement a first-function that finds the first |
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76 // element, say x, in the list xs where f is not None. |
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77 // In that case Return f(x), otherwise None. If possible, |
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78 // calculate f(x) only once. |
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79 |
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80 def first(xs: List[Pos], f: Pos => Option[Path]) : Option[Path] = { |
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81 if (xs eq Nil) { |
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82 None |
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83 } else { |
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84 if (f(xs.head) != None) { |
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85 f(xs.head) |
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86 } else { |
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87 first(xs.tail, f) |
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88 } |
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89 } |
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90 |
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91 } |
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92 |
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93 |
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94 // test cases |
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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 |
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102 |
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103 //(6) Implement a function that uses the first-function from (5) for |
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104 // trying out onward moves, and searches recursively for a |
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105 // knight tour on a dim * dim-board. |
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106 |
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107 |
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108 //def first_tour(dim: Int, path: Path) : Option[Path] = ... |
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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 |
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114 |
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115 /* Helper functions |
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116 |
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117 |
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118 // for measuring time |
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119 def time_needed[T](code: => T) : T = { |
11 def time_needed[T](code: => T) : T = { |
120 val start = System.nanoTime() |
12 val start = System.nanoTime() |
121 val result = code |
13 val result = code |
122 val end = System.nanoTime() |
14 val end = System.nanoTime() |
123 println(f"Time needed: ${(end - start) / 1.0e9}%3.3f secs.") |
15 println(f"Time needed: ${(end - start) / 1.0e9}%3.3f secs.") |
124 result |
16 result |
125 } |
17 } |
126 |
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127 // can be called for example with |
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128 // time_needed(count_tours(dim, List((0, 0)))) |
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129 // in order to print out the time that is needed for |
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130 // running count_tours |
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131 |
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132 // for printing a board |
19 // for printing a board |
133 def print_board(dim: Int, path: Path): Unit = { |
20 def print_board(dim: Int, path: Path): Unit = { |
134 println |
21 println |
135 for (i <- 0 until dim) { |
22 for (i <- 0 until dim) { |
138 } |
25 } |
139 println |
26 println |
140 } |
27 } |
141 } |
28 } |
142 |
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) |
143 |
32 |
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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 } |
144 */ |
117 */ |
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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 |