// Scala Lecture 4+
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//=================+
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+
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+
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// expressions (essentially trees)+
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+
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abstract class Exp+
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case class N(n: Int) extends Exp                  // for numbers+
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case class Plus(e1: Exp, e2: Exp) extends Exp+
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case class Times(e1: Exp, e2: Exp) extends Exp+
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+
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def string(e: Exp) : String = e match {+
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  case N(n) => s"$n"+
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  case Plus(e1, e2) => s"(${string(e1)} + ${string(e2)})" +
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  case Times(e1, e2) => s"(${string(e1)} * ${string(e2)})"+
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}+
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+
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val e = Plus(N(9), Times(N(3), N(4)))+
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println(string(e))+
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+
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def eval(e: Exp) : Int = e match {+
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  case N(n) => n+
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  case Plus(e1, e2) => eval(e1) + eval(e2) +
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  case Times(e1, e2) => eval(e1) * eval(e2) +
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}+
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+
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println(eval(e))+
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+
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// simplification rules:+
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// e + 0, 0 + e => e +
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// e * 0, 0 * e => 0+
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// e * 1, 1 * e => e+
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//+
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// (....0  ....)+
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+
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def simp(e: Exp) : Exp = e match {+
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  case N(n) => N(n)+
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  case Plus(e1, e2) => (simp(e1), simp(e2)) match {+
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    case (N(0), e2s) => e2s+
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    case (e1s, N(0)) => e1s+
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    case (e1s, e2s) => Plus(e1s, e2s)+
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  }  +
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  case Times(e1, e2) => (simp(e1), simp(e2)) match {+
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    case (N(0), _) => N(0)+
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    case (_, N(0)) => N(0)+
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    case (N(1), e2s) => e2s+
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    case (e1s, N(1)) => e1s+
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    case (e1s, e2s) => Times(e1s, e2s)+
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  }  +
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}+
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+
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+
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val e2 = Times(Plus(N(0), N(1)), Plus(N(0), N(9)))+
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println(string(e2))+
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println(string(simp(e2)))+
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+
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+
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// Tokens and Reverse Polish Notation+
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abstract class Token+
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case class T(n: Int) extends Token+
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case object PL extends Token+
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case object TI extends Token+
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+
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// transfroming an Exp into a list of tokens+
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def rp(e: Exp) : List[Token] = e match {+
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  case N(n) => List(T(n))+
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  case Plus(e1, e2) => rp(e1) ::: rp(e2) ::: List(PL) +
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  case Times(e1, e2) => rp(e1) ::: rp(e2) ::: List(TI) +
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}+
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println(string(e2))+
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println(rp(e2))+
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+
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def comp(ls: List[Token], st: List[Int] = Nil) : Int = (ls, st) match {+
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  case (Nil, st) => st.head +
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  case (T(n)::rest, st) => comp(rest, n::st)+
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  case (PL::rest, n1::n2::st) => comp(rest, n1 + n2::st)+
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  case (TI::rest, n1::n2::st) => comp(rest, n1 * n2::st)+
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}+
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+
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comp(rp(e))+
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+
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def proc(s: String) : Token = s match {+
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  case  "+" => PL+
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  case  "*" => TI+
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  case  _ => T(s.toInt) +
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}+
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+
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comp("1 2 + 4 * 5 + 3 +".split(" ").toList.map(proc), Nil)+
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+
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+
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+
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+
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// Sudoku +
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//========+
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+
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// THE POINT OF THIS CODE IS NOT TO BE SUPER+
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// EFFICIENT AND FAST, just explaining exhaustive+
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// depth-first search+
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+
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+
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val game0 = """.14.6.3..+
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              |62...4..9+
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              |.8..5.6..+
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              |.6.2....3+
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              |.7..1..5.+
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              |5....9.6.+
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              |..6.2..3.+
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              |1..5...92+
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              |..7.9.41.""".stripMargin.replaceAll("\\n", "")+
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+
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candidates(game0, (0, 0))+
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+
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type Pos = (Int, Int)+
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val EmptyValue = '.'+
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val MaxValue = 9+
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+
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val allValues = "123456789".toList+
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val indexes = (0 to 8).toList+
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+
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+
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def empty(game: String) = game.indexOf(EmptyValue)+
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def isDone(game: String) = empty(game) == -1 +
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def emptyPosition(game: String) = +
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  (empty(game) % MaxValue, empty(game) / MaxValue)+
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+
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+
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def get_row(game: String, y: Int) = +
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  indexes.map(col => game(y * MaxValue + col))+
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def get_col(game: String, x: Int) = +
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  indexes.map(row => game(x + row * MaxValue))+
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+
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get_row(game0, 0)+
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+
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def get_box(game: String, pos: Pos): List[Char] = {+
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    def base(p: Int): Int = (p / 3) * 3+
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    val x0 = base(pos._1)+
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    val y0 = base(pos._2)+
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    val ys = (y0 until y0 + 3).toList+
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    (x0 until x0 + 3).toList.flatMap(x => ys.map(y => game(x + y * MaxValue)))+
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}+
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+
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//get_row(game0, 0)+
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//get_row(game0, 1)+
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//get_col(game0, 0)+
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//get_box(game0, (3, 1))+
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+
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+
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// this is not mutable!!+
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def update(game: String, pos: Int, value: Char): String = +
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  game.updated(pos, value)+
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+
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def toAvoid(game: String, pos: Pos): List[Char] = +
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  (get_col(game, pos._1) ++ get_row(game, pos._2) ++ get_box(game, pos))+
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+
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def candidates(game: String, pos: Pos): List[Char] = +
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  allValues.diff(toAvoid(game, pos))+
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+
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//candidates(game0, (0,0))+
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+
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def pretty(game: String): String = +
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  "\n" + (game.sliding(MaxValue, MaxValue).mkString("\n"))+
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+
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def search(game: String): List[String] = {+
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  if (isDone(game)) List(game)+
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  else {+
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    val cs = candidates(game, emptyPosition(game))+
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    cs.map(c => search(update(game, empty(game), c))).toList.flatten+
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  }+
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}+
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+
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List(List("sol1"), List("sol2", "sol3")).flatten+
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+
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search(game0).map(pretty)+
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+
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val game1 = """23.915...+
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              |...2..54.+
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              |6.7......+
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              |..1.....9+
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              |89.5.3.17+
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              |5.....6..+
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              |......9.5+
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              |.16..7...+
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              |...329..1""".stripMargin.replaceAll("\\n", "")+
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+
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search(game1).map(pretty)+
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+
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// a game that is in the hard category+
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val game2 = """8........+
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              |..36.....+
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              |.7..9.2..+
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              |.5...7...+
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              |....457..+
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              |...1...3.+
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              |..1....68+
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              |..85...1.+
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              |.9....4..""".stripMargin.replaceAll("\\n", "")+
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+
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search(game2).map(pretty)+
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+
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// game with multiple solutions+
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val game3 = """.8...9743+
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              |.5...8.1.+
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              |.1.......+
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              |8....5...+
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              |...8.4...+
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              |...3....6+
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              |.......7.+
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              |.3.5...8.+
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              |9724...5.""".stripMargin.replaceAll("\\n", "")+
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+
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search(game3).map(pretty).foreach(println)+
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+
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// for measuring time+
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def time_needed[T](i: Int, code: => T) = {+
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  val start = System.nanoTime()+
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  for (j <- 1 to i) code+
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  val end = System.nanoTime()+
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  s"${(end - start) / 1.0e9} secs"+
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}+
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+
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time_needed(1, search(game2))+
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+
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+
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+
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// Tail recursion+
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//================+
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+
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@tailrec+
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def fact(n: BigInt): BigInt = +
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  if (n == 0) 1 else n * fact(n - 1)+
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+
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+
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fact(10)          +
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fact(1000)        +
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fact(100000)       +
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+
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def factB(n: BigInt): BigInt = +
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  if (n == 0) 1 else n * factB(n - 1)+
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+
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def factT(n: BigInt, acc: BigInt): BigInt =+
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  if (n == 0) acc else factT(n - 1, n * acc)+
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+
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+
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factB(1000)+
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+
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+
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+
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+
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factT(10, 1)+
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println(factT(500000, 1))+
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+
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+
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+
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+
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+
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// there is a flag for ensuring a function is tail recursive+
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import scala.annotation.tailrec+
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+
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@tailrec+
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def factT(n: BigInt, acc: BigInt): BigInt =+
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  if (n == 0) acc else factT(n - 1, n * acc)+
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+
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factT(100000, 1)+
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+
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// for tail-recursive functions the Scala compiler+
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// generates loop-like code, which does not need+
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// to allocate stack-space in each recursive+
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// call; Scala can do this only for tail-recursive+
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// functions+
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+
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// tail recursive version that searches +
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// for all Sudoku solutions+
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+
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@tailrec+
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def searchT(games: List[String], sols: List[String]): List[String] = games match {+
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  case Nil => sols+
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  case game::rest => {+
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    if (isDone(game)) searchT(rest, game::sols)+
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    else {+
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      val cs = candidates(game, emptyPosition(game))+
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      searchT(cs.map(c => update(game, empty(game), c)) ::: rest, sols)+
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    }+
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  }+
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}+
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+
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searchT(List(game3), List()).map(pretty)+
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+
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+
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// tail recursive version that searches +
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// for a single solution+
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+
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def search1T(games: List[String]): Option[String] = games match {+
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  case Nil => None+
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  case game::rest => {+
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    if (isDone(game)) Some(game)+
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    else {+
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      val cs = candidates(game, emptyPosition(game))+
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      search1T(cs.map(c => update(game, empty(game), c)) ::: rest)+
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    }+
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  }+
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}+
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+
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search1T(List(game3)).map(pretty)+
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time_needed(1, search1T(List(game3)))+
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time_needed(1, search1T(List(game2)))+
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+
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// game with multiple solutions+
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val game3 = """.8...9743+
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              |.5...8.1.+
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              |.1.......+
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              |8....5...+
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              |...8.4...+
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              |...3....6+
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              |.......7.+
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              |.3.5...8.+
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              |9724...5.""".stripMargin.replaceAll("\\n", "")+
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+
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searchT(List(game3), Nil).map(pretty)+
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search1T(List(game3)).map(pretty)+
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+
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// Moral: Whenever a recursive function is resource-critical+
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// (i.e. works with a large recursion depth), then you need to+
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// write it in tail-recursive fashion.+
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// +
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// Unfortuantely, Scala because of current limitations in +
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// the JVM is not as clever as other functional languages. It can +
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// only optimise "self-tail calls". This excludes the cases of +
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// multiple functions making tail calls to each other. Well,+
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// nothing is perfect. +
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+
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+
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+
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+
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+
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+
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+
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// Cool Stuff in Scala+
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//=====================+
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+
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+
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// Implicits or How to Pimp your Library+
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//======================================+
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//+
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// For example adding your own methods to Strings:+
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// Imagine you want to increment strings, like+
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//+
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//     "HAL".increment+
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//+
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// you can avoid ugly fudges, like a MyString, by+
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// using implicit conversions.+
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+
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+
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implicit class MyString(s: String) {+
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  def increment = s.map(c => (c + 1).toChar) +
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}+
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+
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"HAL".increment+
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+
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+
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// Abstract idea:+
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// In that version implicit conversions were used to solve the +
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// late extension problem; namely, given a class C and a class T, +
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// how to have C extend T without touching or recompiling C. +
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// Conversions add a wrapper when a member of T is requested +
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// from an instance of C.+
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+
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//Another example (TimeUnit in 2.13?)+
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+
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import scala.concurrent.duration.{TimeUnit,SECONDS,MINUTES}+
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+
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case class Duration(time: Long, unit: TimeUnit) {+
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  def +(o: Duration) = +
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    Duration(time + unit.convert(o.time, o.unit), unit)+
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}+
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+
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implicit class Int2Duration(that: Int) {+
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  def seconds = new Duration(that, SECONDS)+
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  def minutes = new Duration(that, MINUTES)+
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}+
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+
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5.seconds + 2.minutes   //Duration(125L, SECONDS )+
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2.minutes + 60.seconds+
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+
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+
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+
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+
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// Regular expressions - the power of DSLs in Scala+
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//==================================================+
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+
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abstract class Rexp+
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case object ZERO extends Rexp                     // nothing+
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case object ONE extends Rexp                      // the empty string+
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case class CHAR(c: Char) extends Rexp             // a character c+
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case class ALT(r1: Rexp, r2: Rexp) extends Rexp   // alternative  r1 + r2+
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case class SEQ(r1: Rexp, r2: Rexp) extends Rexp   // sequence     r1 . r2  +
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case class STAR(r: Rexp) extends Rexp             // star         r*+
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+
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+
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+
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// writing (ab)* in the format above is +
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// tedious+
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val r0 = STAR(SEQ(CHAR('a'), CHAR('b')))+
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+
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+
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// some convenience for typing in regular expressions+
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import scala.language.implicitConversions    +
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import scala.language.reflectiveCalls +
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+
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def charlist2rexp(s: List[Char]): Rexp = s match {+
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  case Nil => ONE+
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  case c::Nil => CHAR(c)+
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  case c::s => SEQ(CHAR(c), charlist2rexp(s))+
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}+
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+
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implicit def string2rexp(s: String): Rexp = +
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  charlist2rexp(s.toList)+
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+
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"(a|b)"+
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+
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val r1 = STAR("ab")+
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val r2 = (STAR("ab")) | (STAR("ba"))+
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val r3 = STAR(SEQ("ab", ALT("a", "b")))+
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+
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implicit def RexpOps (r: Rexp) = new {+
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  def | (s: Rexp) = ALT(r, s)+
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  def % = STAR(r)+
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  def ~ (s: Rexp) = SEQ(r, s)+
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}+
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+
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implicit def stringOps (s: String) = new {+
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  def | (r: Rexp) = ALT(s, r)+
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  def | (r: String) = ALT(s, r)+
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  def % = STAR(s)+
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  def ~ (r: Rexp) = SEQ(s, r)+
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  def ~ (r: String) = SEQ(s, r)+
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}+
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+
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//example regular expressions+
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val digit = ("0" | "1" | "2" | "3" | "4" | +
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              "5" | "6" | "7" | "8" | "9")+
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val sign = "+" | "-" | ""+
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val number = sign ~ digit ~ digit.% +
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+
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+
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+
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// Mind-Blowing Regular Expressions+
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+
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// same examples using the internal regexes+
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val evil = "(a*)*b"+
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+
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+
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println("a" * 100)+
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+
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("a" * 10000).matches(evil)+
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("a" * 10).matches(evil)+
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("a" * 10000).matches(evil)+
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("a" * 20000).matches(evil)+
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("a" * 50000).matches(evil)+
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+
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time_needed(1, ("a" * 50000).matches(evil))+
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