1 // Preliminary Part 1 finding and counting Knight's tours |
1 // Preliminary Part about finding Knight's tours |
2 //======================================================== |
2 //=============================================== |
3 |
3 |
4 object CW8a { |
4 |
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5 object CW8a { |
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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 |
5 |
13 |
6 type Pos = (Int, Int) // a position on a chessboard |
14 type Pos = (Int, Int) // a position on a chessboard |
7 type Path = List[Pos] // a path...a list of positions |
15 type Path = List[Pos] // a path...a list of positions |
8 |
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. |
9 |
19 |
10 // for measuring time in the JAR |
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 |
11 def time_needed[T](code: => T) : T = { |
117 def time_needed[T](code: => T) : T = { |
12 val start = System.nanoTime() |
118 val start = System.nanoTime() |
13 val result = code |
119 val result = code |
14 val end = System.nanoTime() |
120 val end = System.nanoTime() |
15 println(f"Time needed: ${(end - start) / 1.0e9}%3.3f secs.") |
121 println(f"Time needed: ${(end - start) / 1.0e9}%3.3f secs.") |
16 result |
122 result |
17 } |
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 |
19 // for printing a board |
133 // for printing a board |
20 def print_board(dim: Int, path: Path): Unit = { |
134 def print_board(dim: Int, path: Path): Unit = { |
21 println |
135 println |
22 for (i <- 0 until dim) { |
136 for (i <- 0 until dim) { |
25 } |
139 } |
26 println |
140 println |
27 } |
141 } |
28 } |
142 } |
29 |
143 |
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 |
144 |
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 */ |
145 */ |
118 |
146 |
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 } |
147 } |
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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