--- a/testing3/knight1.scala Wed Nov 28 23:26:47 2018 +0000
+++ b/testing3/knight1.scala Thu Nov 29 17:15:11 2018 +0000
@@ -1,13 +1,105 @@
-// Part 1 about finding and counting Knight's tours
-//==================================================
+// Part 1 about finding Knight's tours
+//=====================================
-//object CW8a { // for preparing the jar
+// 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.
-// for measuring time in the JAR
+def is_legal(dim: Int, path: Path, x: Pos) : Boolean = ((!(path.contains(x))) && (x._1 < dim) && (x._2 < dim))
+
+
+
+//(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] ={
+ val y = 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)
+ )
+ y.filter(next => is_legal(dim, path, next))
+}
+
+//some test cases
+//
+//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 = {
+ if(path.length == dim*dim) 1 else
+ (for(i <- legal_moves(dim, path, path.head)) yield
+ count_tours(dim, i :: path)
+ ).sum
+}
+
+def enum_tours(dim: Int, path: Path) : List[Path] ={
+ if(path.length == dim*dim) List(path) else
+ (for(i <- legal_moves(dim, path, path.head)) yield
+ enum_tours(dim, i :: path)
+ ).flatten
+}
+
+//(5) 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] = {
+ if(xs == Nil) None
+ else(
+ for(x <- xs) yield{
+ val a = f(x)
+ if(a != None) a
+ else first(xs.drop(1), f)
+ }
+ ).head
+}
+
+// test cases
+//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
+
+
+
+
+//(6) 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] = {
+// first(legal_moves(dim, path, path.head), (x : Pos => ))
+// }
+
+/* Helper functions
+
+
+// for measuring time
def time_needed[T](code: => T) : T = {
val start = System.nanoTime()
val result = code
@@ -16,6 +108,11 @@
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
@@ -27,144 +124,5 @@
}
}
-def is_legal(dim: Int, path: Path, x: Pos): Boolean =
- 0 <= x._1 && 0 <= x._2 && x._1 < dim && x._2 < dim && !path.contains(x)
-// testcases
-//assert(is_legal(8, Nil, (3, 4)) == true)
-//assert(is_legal(8, List((4, 1), (1, 0)), (4, 1)) == false)
-//assert(is_legal(2, Nil, (0, 0)) == true)
-
-
-def add_pair(x: Pos, y: Pos): Pos =
- (x._1 + y._1, x._2 + y._2)
-
-def moves(x: Pos): List[Pos] =
- List(( 1, 2),( 2, 1),( 2, -1),( 1, -2),
- (-1, -2),(-2, -1),(-2, 1),(-1, 2)).map(add_pair(x, _))
-
-// 1 mark
-
-def legal_moves(dim: Int, path: Path, x: Pos): List[Pos] =
- moves(x).filter(is_legal(dim, path, _))
-
-// 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)))
-//assert(legal_moves(1, Nil, (0,0)) == List())
-//assert(legal_moves(2, Nil, (0,0)) == List())
-//assert(legal_moves(3, Nil, (0,0)) == List((1,2), (2,1)))
-
-// 2 marks
-
-def tcount_tours(dim: Int, path: Path): Int = {
- if (path.length == dim * dim) 1
- else
- (for (x <- legal_moves(dim, path, path.head)) yield tcount_tours(dim, x::path)).sum
-}
-
-def count_tours(dim: Int, path: Path) =
- time_needed(tcount_tours(dim: Int, path: Path))
-
-
-def tenum_tours(dim: Int, path: Path): List[Path] = {
- if (path.length == dim * dim) List(path)
- else
- (for (x <- legal_moves(dim, path, path.head)) yield tenum_tours(dim, x::path)).flatten
-}
-
-def enum_tours(dim: Int, path: Path) =
- time_needed(tenum_tours(dim: Int, path: Path))
-
-// test cases
-
-/*
-def count_all_tours(dim: Int) = {
- for (i <- (0 until dim).toList;
- j <- (0 until dim).toList) yield count_tours(dim, List((i, j)))
-}
-
-def enum_all_tours(dim: Int): List[Path] = {
- (for (i <- (0 until dim).toList;
- j <- (0 until dim).toList) yield enum_tours(dim, List((i, j)))).flatten
-}
-
-
-println("Number of tours starting from (0, 0)")
-
-for (dim <- 1 to 5) {
- println(s"${dim} x ${dim} " + time_needed(0, count_tours(dim, List((0, 0)))))
-}
-
-println("Number of tours starting from all fields")
-
-for (dim <- 1 to 5) {
- println(s"${dim} x ${dim} " + time_needed(0, count_all_tours(dim)))
-}
-
-for (dim <- 1 to 5) {
- val ts = enum_tours(dim, List((0, 0)))
- println(s"${dim} x ${dim} ")
- if (ts != Nil) {
- print_board(dim, ts.head)
- println(ts.head)
- }
-}
*/
-
-// 1 mark
-
-def first(xs: List[Pos], f: Pos => Option[Path]): Option[Path] = xs match {
- case Nil => None
- case x::xs => {
- val result = f(x)
- if (result.isDefined) result else first(xs, f)
- }
-}
-
-// test cases
-//def foo(x: (Int, Int)) = if (x._1 > 3) Some(List(x)) else None
-//
-//first(List((1, 0),(2, 0),(3, 0),(4, 0)), foo)
-//first(List((1, 0),(2, 0),(3, 0)), foo)
-
-
-// 1 mark
-
-def tfirst_tour(dim: Int, path: Path): Option[Path] = {
- if (path.length == dim * dim) Some(path)
- else
- first(legal_moves(dim, path, path.head), (x:Pos) => tfirst_tour(dim, x::path))
-}
-
-def first_tour(dim: Int, path: Path) =
- time_needed(tfirst_tour(dim: Int, path: Path))
-
-
-/*
-for (dim <- 1 to 8) {
- val t = first_tour(dim, List((0, 0)))
- println(s"${dim} x ${dim} " + (if (t == None) "" else { print_board(dim, t.get) ; "" }))
-}
-*/
-
-// 15 secs for 8 x 8
-//val ts1 = time_needed(0,first_tour(8, List((0, 0))).get)
-
-// no result for 4 x 4
-//val ts2 = time_needed(0, first_tour(4, List((0, 0))))
-
-// 0.3 secs for 6 x 6
-//val ts3 = time_needed(0, first_tour(6, List((0, 0))))
-
-// 15 secs for 8 x 8
-//time_needed(0, print_board(8, first_tour(8, List((0, 0))).get))
-
-
-//}
-
-