// Preliminary Part about finding Knight's tours+ −
//===============================================+ −
+ −
+ −
object CW9a {+ −
+ −
// If you need any auxiliary function, feel free to + −
// implement it, but do not make any changes to the+ −
// templates below. Also have a look whether the functions+ −
// at the end are of any help.+ −
+ −
+ −
+ −
type Pos = (Int, Int) // a position on a chessboard + −
type Path = List[Pos] // a path...a list of positions+ −
+ −
//(1) Complete the function that tests whether the position x+ −
// is inside the board and not yet element in the path.+ −
+ −
def is_legal(dim: Int, path: Path, x: Pos) : Boolean = { + −
if ((!(path.contains(x))) && (x._1 >= 0) && (x._2 >= 0) && (x._1 < dim) && (x._2 < dim))+ −
true+ −
else false+ −
}+ −
+ −
//(2) Complete the function that calculates for a position x+ −
// all legal onward moves that are not already in the path. + −
// The moves should be ordered in a "clockwise" manner.+ −
+ −
+ −
def legal_moves(dim: Int, path: Path, x: Pos) : List[Pos] = {//List[Pos]+ −
val changes = List((1,2),(2,1),(2,-1),(1,-2),(-1,-2),(-2,-1),(-2,1),(-1,2))+ −
val returnList = (for ((y,z) <- changes) yield(+ −
//println(y,z)-2,-1+ −
if ((is_legal(dim,path,((x._1 + y) , (x._2 + z)))) == true)+ −
Some(x._1 + y , x._2 + z)+ −
else+ −
None+ −
))+ −
returnList.flatten+ −
}+ −
+ −
+ −
//some testcases+ −
//+ −
//assert(legal_moves(8, Nil, (2,2)) == + −
//List((3,4), (4,3), (4,1), (3,0), (1,0), (0,1), (0,3), (1,4)))+ −
//assert(legal_moves(8, Nil, (7,7)) == List((6,5), (5,6)))+ −
//assert(legal_moves(8, List((4,1), (1,0)), (2,2)) == + −
// List((3,4), (4,3), (3,0), (0,1), (0,3), (1,4)))+ −
//assert(legal_moves(8, List((6,6)), (7,7)) == List((6,5), (5,6)))+ −
+ −
+ −
//(3) Complete the two recursive functions below. + −
// They exhaustively search for knight's tours starting from the + −
// given path. The first function counts all possible tours, + −
// and the second collects all tours in a list of paths.+ −
+ −
def count_tours(dim: Int, path: Path) : Int = (dim,path) match {//Int+ −
case (_, Nil) => 0+ −
case (0, path) => 0+ −
case (dim, path) => { if (legal_moves(dim,path, path.head).size == 0) + −
if(path.size < dim*dim) + −
0 + −
else + −
1+ −
else (for (j <- legal_moves(dim,path, path.head)) yield count_tours(dim,j::path)).sum+ −
}+ −
}+ −
+ −
def enum_tours(dim: Int, path: Path) : List[Path] = (dim,path) match {+ −
case (_, Nil) => Nil+ −
case (0, path) => Nil+ −
case (dim, path) => { if (legal_moves(dim,path, path.head).size == 0) + −
if(path.size < dim*dim) + −
Nil+ −
else + −
List(path)+ −
else (for (j <- legal_moves(dim,path, path.head)) yield enum_tours(dim,j::path)).flatten+ −
}+ −
+ −
}+ −
+ −
+ −
//(4) Implement a first-function that finds the first + −
// element, say x, in the list xs where f is not None. + −
// In that case Return f(x), otherwise None. If possible,+ −
// calculate f(x) only once.+ −
+ −
//def first(xs: List[Pos], f: Pos => Option[Path]) : Option[Path] = ...+ −
+ −
+ −
// testcases+ −
//+ −
//def foo(x: (Int, Int)) = if (x._1 > 3) Some(List(x)) else None+ −
//+ −
//first(List((1, 0),(2, 0),(3, 0),(4, 0)), foo) // Some(List((4,0)))+ −
//first(List((1, 0),(2, 0),(3, 0)), foo) // None+ −
+ −
+ −
//(5) Implement a function that uses the first-function from (5) for+ −
// trying out onward moves, and searches recursively for a+ −
// knight tour on a dim * dim-board.+ −
+ −
+ −
//def first_tour(dim: Int, path: Path) : Option[Path] = ...+ −
+ −
+ −
+ −
+ −
+ −
+ −
/* Helper functions+ −
+ −
+ −
// for measuring time+ −
def time_needed[T](code: => T) : T = {+ −
val start = System.nanoTime()+ −
val result = code+ −
val end = System.nanoTime()+ −
println(f"Time needed: ${(end - start) / 1.0e9}%3.3f secs.")+ −
result+ −
}+ −
+ −
// can be called for example with+ −
// time_needed(count_tours(dim, List((0, 0))))+ −
// in order to print out the time that is needed for + −
// running count_tours+ −
+ −
+ −
+ −
+ −
// for printing a board+ −
def print_board(dim: Int, path: Path): Unit = {+ −
println+ −
for (i <- 0 until dim) {+ −
for (j <- 0 until dim) {+ −
print(f"${path.reverse.indexOf((j, dim - i - 1))}%3.0f ")+ −
}+ −
println+ −
} + −
}+ −
+ −
+ −
*/+ −
+ −
}+ −