// Scala Lecture 4

//=================

// 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)))

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

//

// (....0  ....)

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)))

// Tokens and Reverse Polish Notation

abstract class Token

case class T(n: Int) extends Token

case object PL extends Token

case object TI extends Token

// transfroming an Exp into a list of tokens

def rp(e: Exp) : List[Token] = e match {

  case N(n) => List(T(n))

  case Plus(e1, e2) => rp(e1) ::: rp(e2) ::: List(PL) 

  case Times(e1, e2) => rp(e1) ::: rp(e2) ::: List(TI) 

}

println(string(e2))

println(rp(e2))

def comp(ls: List[Token], st: List[Int] = Nil) : Int = (ls, st) match {

  case (Nil, st) => st.head 

  case (T(n)::rest, st) => comp(rest, n::st)

  case (PL::rest, n1::n2::st) => comp(rest, n1 + n2::st)

  case (TI::rest, n1::n2::st) => comp(rest, n1 * n2::st)

}

comp(rp(e))

def proc(s: String) : Token = s match {

  case  "+" => PL

  case  "*" => TI

  case  _ => T(s.toInt) 

}

comp("1 2 + 4 * 5 + 3 +".split(" ").toList.map(proc), Nil)

// Sudoku 

//========

// THE POINT OF THIS CODE IS NOT TO BE SUPER

// EFFICIENT AND FAST, just explaining exhaustive

// depth-first search

val game0 = """.14.6.3..

              |62...4..9

              |.8..5.6..

              |.6.2....3

              |.7..1..5.

              |5....9.6.

              |..6.2..3.

              |1..5...92

              |..7.9.41.""".stripMargin.replaceAll("\\n", "")

candidates(game0, (0, 0))

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) = 

  (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))

get_row(game0, 0)

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)

    val ys = (y0 until y0 + 3).toList

    (x0 until x0 + 3).toList.flatMap(x => ys.map(y => game(x + y * MaxValue)))

}

//get_row(game0, 0)

//get_row(game0, 1)

//get_col(game0, 0)

//get_box(game0, (3, 1))

// this is not mutable!!

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))

//candidates(game0, (0,0))

def pretty(game: String): String = 

  "\n" + (game.sliding(MaxValue, MaxValue).mkString("\n"))

def search(game: String): List[String] = {

  if (isDone(game)) List(game)

  else {

    val cs = candidates(game, emptyPosition(game))

    cs.map(c => search(update(game, empty(game), c))).toList.flatten

  }

}

List(List("sol1"), List("sol2", "sol3")).flatten

search(game0).map(pretty)

val game1 = """23.915...

              |...2..54.

              |6.7......

              |..1.....9

              |89.5.3.17

              |5.....6..

              |......9.5

              |.16..7...

              |...329..1""".stripMargin.replaceAll("\\n", "")

search(game1).map(pretty)

// a game that is in the hard category

val game2 = """8........

              |..36.....

              |.7..9.2..

              |.5...7...

              |....457..

              |...1...3.

              |..1....68

              |..85...1.

              |.9....4..""".stripMargin.replaceAll("\\n", "")

search(game2).map(pretty)

// 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", "")

search(game3).map(pretty).foreach(println)

// for measuring time

def time_needed[T](i: Int, code: => T) = {

  val start = System.nanoTime()

  for (j <- 1 to i) code

  val end = System.nanoTime()

  s"${(end - start) / 1.0e9} secs"

}

time_needed(1, search(game2))

// Tail recursion

//================

@tailrec

def fact(n: BigInt): BigInt = 

  if (n == 0) 1 else n * fact(n - 1)

fact(10)          

fact(1000)        

fact(100000)       

def factB(n: BigInt): BigInt = 

  if (n == 0) 1 else n * factB(n - 1)

def factT(n: BigInt, acc: BigInt): BigInt =

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

factB(1000)

factT(10, 1)

println(factT(500000, 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)

factT(100000, 1)

// 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

// tail recursive version that searches 

// for all Sudoku solutions

@tailrec

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(1, search1T(List(game3)))

time_needed(1, search1T(List(game2)))

// 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 a 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.