// 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.