1 // Main Part 4 about finding Knight's tours |
1 // Part 1 about finding and counting Knight's tours |
2 //========================================== |
2 //================================================== |
3 import scala.annotation.tailrec |
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4 |
3 |
5 |
4 object M4a { // for preparing the jar |
6 object M4a { |
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7 |
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8 // If you need any auxiliary functions, feel free to |
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9 // implement them, but do not make any changes to the |
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10 // templates below. Also have a look whether the functions |
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11 // at the end of the file are of any help. |
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12 |
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13 |
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14 |
5 |
15 type Pos = (Int, Int) // a position on a chessboard |
6 type Pos = (Int, Int) // a position on a chessboard |
16 type Path = List[Pos] // a path...a list of positions |
7 type Path = List[Pos] // a path...a list of positions |
17 |
8 |
18 //(1) Complete the function that tests whether the position x |
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19 // is inside the board and not yet element in the path. |
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20 |
9 |
21 def is_legal(dim: Int, path: Path, x: Pos) : Boolean = { |
10 // for measuring time in the JAR |
22 (x._1 < dim) && (x._2 < dim) && (!path.contains(x)) |
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23 } |
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24 |
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25 |
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26 |
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27 //(2) Complete the function that calculates for a position x |
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28 // all legal onward moves that are not already in the path. |
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29 // The moves should be ordered in a "clockwise" manner. |
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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 movesets = List( |
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33 (x._1 + 1, x._2 + 2), |
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34 (x._1 + 2, x._2 + 1), |
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35 (x._1 + 2, x._2 - 1), |
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36 (x._1 + 1, x._2 - 2), |
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37 (x._1 - 1, x._2 - 2), |
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38 (x._1 - 2, x._2 - 1), |
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39 (x._1 - 2, x._2 + 1), |
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40 (x._1 - 1, x._2 + 2) |
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41 ) |
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42 movesets.filter(is_legal(dim, path, _)) |
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43 } |
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44 |
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45 |
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46 //some testcases |
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47 // |
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48 //assert(legal_moves(8, Nil, (2,2)) == |
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49 // List((3,4), (4,3), (4,1), (3,0), (1,0), (0,1), (0,3), (1,4))) |
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50 //assert(legal_moves(8, Nil, (7,7)) == List((6,5), (5,6))) |
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51 //assert(legal_moves(8, List((4,1), (1,0)), (2,2)) == |
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52 // List((3,4), (4,3), (3,0), (0,1), (0,3), (1,4))) |
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53 //assert(legal_moves(8, List((6,6)), (7,7)) == List((6,5), (5,6))) |
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54 |
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55 |
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56 //(3) Complete the two recursive functions below. |
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57 // They exhaustively search for knight's tours starting from the |
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58 // given path. The first function counts all possible tours, |
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59 // and the second collects all tours in a list of paths. |
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60 |
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61 def count_tours(dim: Int, path: Path) : Int = { |
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62 if (dim <= 4) 0 else { |
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63 if (path.length >= (dim * dim)) 1 else { |
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64 val movesets = legal_moves(dim, path, path.head) |
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65 (for (move <- movesets) yield count_tours(dim, move :: path)).sum |
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66 } |
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67 } |
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68 } |
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69 |
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70 def enum_tours(dim: Int, path: Path) : List[Path] = { |
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71 if (dim <= 4) Nil else { |
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72 if (path.length >= (dim * dim)) List(path) else { |
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73 val movesets = legal_moves(dim, path, path.head) |
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74 (for (move <- movesets) yield enum_tours(dim, move :: path)).flatten |
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75 } |
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76 } |
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77 } |
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78 |
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79 |
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80 //(4) Implement a first-function that finds the first |
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81 // element, say x, in the list xs where f is not None. |
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82 // In that case Return f(x), otherwise None. If possible, |
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83 // calculate f(x) only once. |
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84 |
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85 @tailrec |
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86 def first(xs: List[Pos], f: Pos => Option[Path]) : Option[Path] = { |
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87 xs match { |
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88 case Nil => None |
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89 case head :: rest => { |
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90 val result = f(head) |
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91 if (result.isEmpty) first(rest, f) else result |
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92 } |
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93 } |
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94 } |
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95 |
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96 |
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97 // testcases |
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98 // |
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99 //def foo(x: (Int, Int)) = if (x._1 > 3) Some(List(x)) else None |
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100 // |
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101 //first(List((1, 0),(2, 0),(3, 0),(4, 0)), foo) // Some(List((4,0))) |
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102 //first(List((1, 0),(2, 0),(3, 0)), foo) // None |
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103 |
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104 |
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105 //(5) Implement a function that uses the first-function from (4) for |
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106 // trying out onward moves, and searches recursively for a |
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107 // knight tour on a dim * dim-board. |
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108 |
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109 def first_tour(dim: Int, path: Path) : Option[Path] = ??? |
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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 = { |
11 def time_needed[T](code: => T) : T = { |
118 val start = System.nanoTime() |
12 val start = System.nanoTime() |
119 val result = code |
13 val result = code |
120 val end = System.nanoTime() |
14 val end = System.nanoTime() |
121 println(f"Time needed: ${(end - start) / 1.0e9}%3.3f secs.") |
15 println(f"Time needed: ${(end - start) / 1.0e9}%3.3f secs.") |
122 result |
16 result |
123 } |
17 } |
124 |
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125 // can be called for example with |
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126 // |
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127 // time_needed(count_tours(dim, List((0, 0)))) |
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128 // |
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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 |
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133 // for printing a board |
19 // for printing a board |
134 def print_board(dim: Int, path: Path): Unit = { |
20 def print_board(dim: Int, path: Path): Unit = { |
135 println() |
21 println() |
136 for (i <- 0 until dim) { |
22 for (i <- 0 until dim) { |
139 } |
25 } |
140 println() |
26 println() |
141 } |
27 } |
142 } |
28 } |
143 |
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) |
144 |
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 def legal_moves(dim: Int, path: Path, x: Pos): List[Pos] = |
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47 moves(x).filter(is_legal(dim, path, _)) |
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48 |
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49 |
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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(8, Nil, (0,1)) == List((1,3), (2,2), (2,0))) |
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59 //assert(legal_moves(1, Nil, (0,0)) == List()) |
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60 //assert(legal_moves(2, Nil, (0,0)) == List()) |
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61 //assert(legal_moves(3, Nil, (0,0)) == List((1,2), (2,1))) |
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62 |
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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 } |
145 */ |
117 */ |
146 |
118 |
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119 |
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120 def first(xs: List[Pos], f: Pos => Option[Path]): Option[Path] = xs match { |
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121 case Nil => None |
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122 case x::xs => { |
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123 val result = f(x) |
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124 if (result.isDefined) result else first(xs, f) |
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125 } |
147 } |
126 } |
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127 |
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128 // test cases |
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129 //def foo(x: (Int, Int)) = if (x._1 > 3) Some(List(x)) else None |
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130 // |
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131 //first(List((1, 0),(2, 0),(3, 0),(4, 0)), foo) |
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132 //first(List((1, 0),(2, 0),(3, 0)), foo) |
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133 |
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134 |
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135 def tfirst_tour(dim: Int, path: Path): Option[Path] = { |
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136 if (path.length == dim * dim) Some(path) |
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137 else |
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138 first(legal_moves(dim, path, path.head), (x:Pos) => tfirst_tour(dim, x::path)) |
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139 } |
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140 |
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141 def first_tour(dim: Int, path: Path) = |
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142 time_needed(tfirst_tour(dim: Int, path: Path)) |
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143 |
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144 |
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145 /* |
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146 for (dim <- 1 to 8) { |
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147 val t = first_tour(dim, List((0, 0))) |
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148 println(s"${dim} x ${dim} " + (if (t == None) "" else { print_board(dim, t.get) ; "" })) |
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149 } |
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150 */ |
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151 |
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152 // 15 secs for 8 x 8 |
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153 //val ts1 = time_needed(0,first_tour(8, List((0, 0))).get) |
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154 //??val ts1 = time_needed(0,first_tour(8, List((1, 1))).get) |
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155 |
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156 // no result for 4 x 4 |
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157 //val ts2 = time_needed(0, first_tour(4, List((0, 0)))) |
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158 |
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159 // 0.3 secs for 6 x 6 |
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160 //val ts3 = time_needed(0, first_tour(6, List((0, 0)))) |
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161 |
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162 // 15 secs for 8 x 8 |
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163 //time_needed(0, print_board(8, first_tour(8, List((0, 0))).get)) |
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164 |
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165 |
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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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171 |