1 // Part 1 about finding and counting Knight's tours |
1 // Part 1 about finding Knight's tours |
2 //================================================== |
2 //===================================== |
3 |
3 |
4 type Pos = (Int, Int) // a position on a chessboard |
4 type Pos = (Int, Int) // a position on a chessboard |
5 type Path = List[Pos] // a path...a list of positions |
5 type Path = List[Pos] // a path...a list of positions |
6 |
6 |
|
7 // for measuring time |
|
8 def time_needed[T](code: => T) : T = { |
|
9 val start = System.nanoTime() |
|
10 val result = code |
|
11 val end = System.nanoTime() |
|
12 println(f"Time needed: ${(end - start) / 1.0e9}%3.3f secs.") |
|
13 result |
|
14 } |
|
15 |
|
16 // for printing a board |
7 def print_board(dim: Int, path: Path): Unit = { |
17 def print_board(dim: Int, path: Path): Unit = { |
8 println |
18 println |
9 for (i <- 0 until dim) { |
19 for (i <- 0 until dim) { |
10 for (j <- 0 until dim) { |
20 for (j <- 0 until dim) { |
11 print(f"${path.reverse.indexOf((j, dim - i - 1))}%3.0f ") |
21 print(f"${path.reverse.indexOf((j, dim - i - 1))}%3.0f ") |
18 // 1 mark |
28 // 1 mark |
19 |
29 |
20 def is_legal(dim: Int, path: Path, x: Pos): Boolean = |
30 def is_legal(dim: Int, path: Path, x: Pos): Boolean = |
21 0 <= x._1 && 0 <= x._2 && x._1 < dim && x._2 < dim && !path.contains(x) |
31 0 <= x._1 && 0 <= x._2 && x._1 < dim && x._2 < dim && !path.contains(x) |
22 |
32 |
23 assert(is_legal(8, Nil)((3,4)) == true) |
33 assert(is_legal(8, Nil, (3, 4)) == true) |
24 assert(is_legal(8, List((4,1), (1,0)))((4,1)) == false) |
34 assert(is_legal(8, List((4, 1), (1, 0)), (4, 1)) == false) |
25 assert(is_legal(2, Nil)((0,0)) == true) |
35 assert(is_legal(2, Nil, (0, 0)) == true) |
26 |
36 |
27 |
37 |
28 def add_pair(x: Pos)(y: Pos): Pos = |
38 def add_pair(x: Pos, y: Pos): Pos = |
29 (x._1 + y._1, x._2 + y._2) |
39 (x._1 + y._1, x._2 + y._2) |
30 |
40 |
31 def moves(x: Pos): List[Pos] = |
41 def moves(x: Pos): List[Pos] = |
32 List(( 1, 2),( 2, 1),( 2, -1),( 1, -2), |
42 List(( 1, 2),( 2, 1),( 2, -1),( 1, -2), |
33 (-1, -2),(-2, -1),(-2, 1),(-1, 2)).map(add_pair(x)) |
43 (-1, -2),(-2, -1),(-2, 1),(-1, 2)).map(add_pair(x, _)) |
34 |
44 |
35 // 1 mark |
45 // 1 mark |
36 |
46 |
37 def legal_moves(dim: Int, path: Path, x: Pos): List[Pos] = |
47 def legal_moves(dim: Int, path: Path, x: Pos): List[Pos] = |
38 moves(x).filter(is_legal(dim, path)) |
48 moves(x).filter(is_legal(dim, path, _)) |
39 |
49 |
40 assert(legal_moves(8, Nil, (2,2)) == |
50 assert(legal_moves(8, Nil, (2,2)) == |
41 List((3,4), (4,3), (4,1), (3,0), (1,0), (0,1), (0,3), (1,4))) |
51 List((3,4), (4,3), (4,1), (3,0), (1,0), (0,1), (0,3), (1,4))) |
42 assert(legal_moves(8, Nil, (7,7)) == List((6,5), (5,6))) |
52 assert(legal_moves(8, Nil, (7,7)) == List((6,5), (5,6))) |
43 assert(legal_moves(8, List((4,1), (1,0)), (2,2)) == |
53 assert(legal_moves(8, List((4,1), (1,0)), (2,2)) == |
94 println(ts.head) |
107 println(ts.head) |
95 } |
108 } |
96 } |
109 } |
97 |
110 |
98 |
111 |
|
112 // 1 mark |
|
113 |
|
114 def first(xs: List[Pos], f: Pos => Option[Path]): Option[Path] = xs match { |
|
115 case Nil => None |
|
116 case x::xs => { |
|
117 val result = f(x) |
|
118 if (result.isDefined) result else first(xs, f) |
|
119 } |
|
120 } |
|
121 |
|
122 // test cases |
|
123 def foo(x: (Int, Int)) = if (x._1 > 3) Some(List(x)) else None |
|
124 |
|
125 first(List((1, 0),(2, 0),(3, 0),(4, 0)), foo) |
|
126 first(List((1, 0),(2, 0),(3, 0)), foo) |
|
127 |
|
128 |
|
129 // 1 mark |
|
130 |
|
131 def first_tour(dim: Int, path: Path): Option[Path] = { |
|
132 if (path.length == dim * dim) Some(path) |
|
133 else |
|
134 first(legal_moves(dim, path, path.head), (x: Pos) => first_tour(dim, x::path)) |
|
135 } |
|
136 |
|
137 |
|
138 |
|
139 /* |
|
140 for (dim <- 1 to 8) { |
|
141 val t = first_tour(dim, List((0, 0))) |
|
142 println(s"${dim} x ${dim} " + (if (t == None) "" else { print_board(dim, t.get) ; "" })) |
|
143 } |
|
144 */ |
|
145 |
|
146 // 15 secs for 8 x 8 |
|
147 val ts1 = time_needed(0,first_tour(8, List((0, 0))).get) |
|
148 |
|
149 // no result for 4 x 4 |
|
150 val ts2 = time_needed(0, first_tour(4, List((0, 0)))) |
|
151 |
|
152 // 0.3 secs for 6 x 6 |
|
153 val ts3 = time_needed(0, first_tour(6, List((0, 0)))) |
|
154 |
|
155 // 15 secs for 8 x 8 |
|
156 time_needed(0, print_board(8, first_tour(8, List((0, 0))).get)) |
|
157 |