// Scala Lecture 3
//=================

// - Higher-Order functions
// - maps (behind for-comprehensions)

// - Pattern-Matching

def fib(n: Int) : Int = n match {
  case 0 => 1
  case 1 =>  1
  case n => fib(n - 1) + fib(n - 2)
}


abstract class Rexp
case object ZERO extends Rexp                      // matches nothing
case object ONE extends Rexp                       // matches the empty string
case class CHAR(c: Char) extends Rexp              // matches a character c
case class ALT(r1: Rexp, r2: Rexp) extends Rexp    // alternative
case class SEQ(r1: Rexp, r2: Rexp) extends Rexp    // sequence
case class STAR(r: Rexp) extends Rexp              // star

def depth(r: Rexp) : Int = r match {
  case ZERO => 1
  case ONE => 1
  case CHAR(_) => 1
  case ALT(r1, r2) => 1 + List(depth(r1), depth(r2)).max
  case SEQ(r1, r2) => 1 + List(depth(r1), depth(r2)).max
  case STAR(r1) => 1 + depth(r1)
}


// - String-Interpolations

// String Interpolations
//=======================

def cube(n: Int) : Int = n * n * n

val n = 3
println("The cube of " + n + " is " + cube(n) + ".")

println(s"The cube of $n is ${cube(n)}.")

// or even

println(s"The cube of $n is ${n * n * n}.")

// helpful for debugging purposes
//
//     "The most effective debugging tool is still careful 
//          thought, coupled with judiciously placed print 
//                                             statements."
//       — Brian W. Kernighan, in Unix for Beginners (1979)


def gcd_db(a: Int, b: Int) : Int = {
  println(s"Function called with $a and $b.")
  if (b == 0) a else gcd_db(b, a % b)
}

gcd_db(48, 18)


// naive quicksort with "On" function

def sortOn(f: Int => Int, xs: List[Int]) : List[Int] = {
  if (xs.size < 2) xs
  else {
   val pivot = xs.head
   val (left, right) = xs.partition(f(_) < f(pivot))
   sortOn(f, left) ::: pivot :: sortOn(f, right.tail)
  }
} 

sortOn(identity, List(99,99,99,98,10,-3,2)) 
sortOn(n => - n, List(99,99,99,98,10,-3,2))


// Recursion Again ;o)
//====================


// another well-known example: Towers of Hanoi
//=============================================

def move(from: Char, to: Char) =
  println(s"Move disc from $from to $to!")

def hanoi(n: Int, from: Char, via: Char, to: Char) : Unit = {
  if (n == 0) ()
  else {
    hanoi(n - 1, from, to, via)
    move(from, to)
    hanoi(n - 1, via, from, to)
  }
} 

hanoi(4, 'A', 'B', 'C')



// User-defined Datatypes
//========================

abstract class Tree
case class Leaf(x: Int) extends Tree
case class Node(s: String, left: Tree, right: Tree) extends Tree 

val lf = Leaf(20)
val tr = Node("foo", Leaf(10), Leaf(23))

val lst : List[Tree] = List(lf, tr)


abstract class Colour
case object Red extends Colour 
case object Green extends Colour 
case object Blue extends Colour
case object Yellow extends Colour


def fav_colour(c: Colour) : Boolean = c match {
  case Green => true
  case _  => false 
}

fav_colour(Blue)


// ... a tiny bit more useful: Roman Numerals

sealed abstract class RomanDigit 
case object I extends RomanDigit 
case object V extends RomanDigit 
case object X extends RomanDigit 
case object L extends RomanDigit 
case object C extends RomanDigit 
case object D extends RomanDigit 
case object M extends RomanDigit 

type RomanNumeral = List[RomanDigit] 

List(X,I,M,A)

/*
I    -> 1
II   -> 2
III  -> 3
IV   -> 4
V    -> 5
VI   -> 6
VII  -> 7
VIII -> 8
IX   -> 9
X    -> 10
*/

def RomanNumeral2Int(rs: RomanNumeral): Int = rs match { 
  case Nil => 0
  case M::r    => 1000 + RomanNumeral2Int(r)  
  case C::M::r => 900 + RomanNumeral2Int(r)
  case D::r    => 500 + RomanNumeral2Int(r)
  case C::D::r => 400 + RomanNumeral2Int(r)
  case C::r    => 100 + RomanNumeral2Int(r)
  case X::C::r => 90 + RomanNumeral2Int(r)
  case L::r    => 50 + RomanNumeral2Int(r)
  case X::L::r => 40 + RomanNumeral2Int(r)
  case X::r    => 10 + RomanNumeral2Int(r)
  case I::X::r => 9 + RomanNumeral2Int(r)
  case V::r    => 5 + RomanNumeral2Int(r)
  case I::V::r => 4 + RomanNumeral2Int(r)
  case I::r    => 1 + RomanNumeral2Int(r)
}

RomanNumeral2Int(List(I,V))             // 4
RomanNumeral2Int(List(I,I,I,I))         // 4 (invalid Roman number)
RomanNumeral2Int(List(V,I))             // 6
RomanNumeral2Int(List(I,X))             // 9
RomanNumeral2Int(List(M,C,M,L,X,X,I,X)) // 1979
RomanNumeral2Int(List(M,M,X,V,I,I))     // 2017


// expressions (essentially trees)

abstract class Exp
case class N(n: Int) extends Exp                  // for numbers
case class Plus(e1: Exp, e2: Exp) extends Exp
case class Times(e1: Exp, e2: Exp) extends Exp

def string(e: Exp) : String = e match {
  case N(n) => s"$n"
  case Plus(e1, e2) => s"(${string(e1)} + ${string(e2)})" 
  case Times(e1, e2) => s"(${string(e1)} * ${string(e2)})"
}

val e = Plus(N(9), Times(N(3), N(4)))
e.toString
println(string(e))

def eval(e: Exp) : Int = e match {
  case N(n) => n
  case Plus(e1, e2) => eval(e1) + eval(e2) 
  case Times(e1, e2) => eval(e1) * eval(e2) 
}

println(eval(e))

// simplification rules:
// e + 0, 0 + e => e 
// e * 0, 0 * e => 0
// e * 1, 1 * e => e
//
// (....9 ....)

def simp(e: Exp) : Exp = e match {
  case N(n) => N(n)
  case Plus(e1, e2) => (simp(e1), simp(e2)) match {
    case (N(0), e2s) => e2s
    case (e1s, N(0)) => e1s
    case (e1s, e2s) => Plus(e1s, e2s)
  }  
  case Times(e1, e2) => (simp(e1), simp(e2)) match {
    case (N(0), _) => N(0)
    case (_, N(0)) => N(0)
    case (N(1), e2s) => e2s
    case (e1s, N(1)) => e1s
    case (e1s, e2s) => Times(e1s, e2s)
  }  
}


val e2 = Times(Plus(N(0), N(1)), Plus(N(0), N(9)))
println(string(e2))
println(string(simp(e2)))



// String interpolations as patterns

val date = "2019-11-26"
val s"$year-$month-$day" = date

def parse_date(date: String) : Option[(Int, Int, Int)]= date match {
  case s"$year-$month-$day" => Some((day.toInt, month.toInt, year.toInt))
  case s"$day/$month/$year" => Some((day.toInt, month.toInt, year.toInt))
  case s"$day.$month.$year" => Some((day.toInt, month.toInt, year.toInt))
  case _ => None
} 

parse_date("2019-11-26")
parse_date("26/11/2019")
parse_date("26.11.2019")


// guards in pattern-matching

def foo(xs: List[Int]) : String = xs match {
  case Nil => s"this list is empty"
  case x :: xs if x % 2 == 0
     => s"the first elemnt is even"
  case x :: y :: rest if x == y
     => s"this has two elemnts that are the same"
  case hd :: tl => s"this list is standard $hd::$tl"
}

foo(Nil)
foo(List(1,2,3))
foo(List(1,2))
foo(List(1,1,2,3))
foo(List(2,2,2,3))

// Tail recursion
//================

def fact(n: BigInt): BigInt = 
  if (n == 0) 1 else n * fact(n - 1)

fact(10)              //ok
fact(10000)           // produces a stackoverflow


def factT(n: BigInt, acc: BigInt): BigInt =
  if (n == 0) acc else factT(n - 1, n * acc)

factT(10, 1)
println(factT(100000, 1))

// there is a flag for ensuring a function is tail recursive
import scala.annotation.tailrec

@tailrec
def factT(n: BigInt, acc: BigInt): BigInt =
  if (n == 0) acc else factT(n - 1, n * acc)



// for tail-recursive functions the Scala compiler
// generates loop-like code, which does not need
// to allocate stack-space in each recursive
// call; Scala can do this only for tail-recursive
// functions

def length(xs: List[Int]) : Int = xs match {
  case Nil => 0
  case _ :: tail => 1 + length(tail)
}

@tailrec
def lengthT(xs: List[Int], acc : Int) : Int = xs match {
  case Nil => acc
  case _ :: tail => lengthT(tail, 1 + acc)
}

lengthT(List.fill(10000000)(1), 0)


// Sudoku
//========

// uses Strings for games

type Pos = (Int, Int)
val emptyValue = '.'
val maxValue = 9

val allValues = "123456789".toList
val indexes = (0 to 8).toList


def empty(game: String) = game.indexOf(emptyValue)
def isDone(game: String) = empty(game) == -1 
def emptyPosition(game: String) : Pos = 
  (empty(game) % maxValue, empty(game) / maxValue)


def get_row(game: String, y: Int) = indexes.map(col => game(y * maxValue + col))
def get_col(game: String, x: Int) = indexes.map(row => game(x + row * maxValue))

def get_box(game: String, pos: Pos): List[Char] = {
    def base(p: Int): Int = (p / 3) * 3
    val x0 = base(pos._1)
    val y0 = base(pos._2)
    for (x <- (x0 until x0 + 3).toList;
         y <- (y0 until y0 + 3).toList) yield game(x + y * maxValue)
}         


def update(game: String, pos: Int, value: Char): String = 
  game.updated(pos, value)

def toAvoid(game: String, pos: Pos): List[Char] = 
  (get_col(game, pos._1) ++ get_row(game, pos._2) ++ get_box(game, pos))

def candidates(game: String, pos: Pos): List[Char] = 
  allValues.diff(toAvoid(game, pos))

def search(game: String): List[String] = {
  if (isDone(game)) List(game)
  else 
    candidates(game, emptyPosition(game)).
      map(c => search(update(game, empty(game), c))).flatten
}


def search1T(games: List[String]): Option[String] = games match {
  case Nil => None
  case game::rest => {
    if (isDone(game)) Some(game)
    else {
      val cs = candidates(game, emptyPosition(game))
      search1T(cs.map(c => update(game, empty(game), c)) ::: rest)
    }
  }
}

def pretty(game: String): String = 
  "\n" + (game.sliding(maxValue, maxValue).mkString(",\n"))


// tail recursive version that searches 
// for all solutions

def searchT(games: List[String], sols: List[String]): List[String] = games match {
  case Nil => sols
  case game::rest => {
    if (isDone(game)) searchT(rest, game::sols)
    else {
      val cs = candidates(game, emptyPosition(game))
      searchT(cs.map(c => update(game, empty(game), c)) ::: rest, sols)
    }
  }
}

searchT(List(game3), List()).map(pretty)


// tail recursive version that searches 
// for a single solution

def search1T(games: List[String]): Option[String] = games match {
  case Nil => None
  case game::rest => {
    if (isDone(game)) Some(game)
    else {
      val cs = candidates(game, emptyPosition(game))
      search1T(cs.map(c => update(game, empty(game), c)) ::: rest)
    }
  }
}

search1T(List(game3)).map(pretty)
time_needed(10, search1T(List(game3)))


// game with multiple solutions
val game3 = """.8...9743
              |.5...8.1.
              |.1.......
              |8....5...
              |...8.4...
              |...3....6
              |.......7.
              |.3.5...8.
              |9724...5.""".stripMargin.replaceAll("\\n", "")

searchT(List(game3), Nil).map(pretty)
search1T(List(game3)).map(pretty)

// Moral: Whenever a recursive function is resource-critical
// (i.e. works with large recursion depth), then you need to
// write it in tail-recursive fashion.
// 
// Unfortuantely, Scala because of current limitations in 
// the JVM is not as clever as other functional languages. It can 
// only optimise "self-tail calls". This excludes the cases of 
// multiple functions making tail calls to each other. Well,
// nothing is perfect.