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// Scala Lecture 3
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//=================
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449
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// - Higher-Order functions
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448
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// - maps (behind for-comprehensions)
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// - Pattern-Matching
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def fib(n: Int) : Int = n match {
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case 0 => 1
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case 1 => 1
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case n => fib(n - 1) + fib(n - 2)
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}
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abstract class Rexp
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case object ZERO extends Rexp // matches nothing
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case object ONE extends Rexp // matches the empty string
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case class CHAR(c: Char) extends Rexp // matches a character c
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case class ALT(r1: Rexp, r2: Rexp) extends Rexp // alternative
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case class SEQ(r1: Rexp, r2: Rexp) extends Rexp // sequence
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case class STAR(r: Rexp) extends Rexp // star
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def depth(r: Rexp) : Int = r match {
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case ZERO => 1
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case ONE => 1
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case CHAR(_) => 1
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case ALT(r1, r2) => 1 + List(depth(r1), depth(r2)).max
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case SEQ(r1, r2) => 1 + List(depth(r1), depth(r2)).max
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case STAR(r1) => 1 + depth(r1)
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}
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// - String-Interpolations
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// String Interpolations
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//=======================
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def cube(n: Int) : Int = n * n * n
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val n = 3
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println("The cube of " + n + " is " + cube(n) + ".")
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println(s"The cube of $n is ${cube(n)}.")
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// or even
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println(s"The cube of $n is ${n * n * n}.")
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// helpful for debugging purposes
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//
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// "The most effective debugging tool is still careful
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// thought, coupled with judiciously placed print
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// statements."
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// — Brian W. Kernighan, in Unix for Beginners (1979)
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def gcd_db(a: Int, b: Int) : Int = {
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println(s"Function called with $a and $b.")
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if (b == 0) a else gcd_db(b, a % b)
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}
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gcd_db(48, 18)
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// naive quicksort with "On" function
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def sortOn(f: Int => Int, xs: List[Int]) : List[Int] = {
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if (xs.size < 2) xs
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else {
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val pivot = xs.head
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val (left, right) = xs.partition(f(_) < f(pivot))
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sortOn(f, left) ::: pivot :: sortOn(f, right.tail)
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}
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}
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sortOn(identity, List(99,99,99,98,10,-3,2))
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sortOn(n => - n, List(99,99,99,98,10,-3,2))
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// Recursion Again ;o)
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//====================
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// another well-known example: Towers of Hanoi
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//=============================================
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def move(from: Char, to: Char) =
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println(s"Move disc from $from to $to!")
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def hanoi(n: Int, from: Char, via: Char, to: Char) : Unit = {
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if (n == 0) ()
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else {
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hanoi(n - 1, from, to, via)
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move(from, to)
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hanoi(n - 1, via, from, to)
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}
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}
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hanoi(4, 'A', 'B', 'C')
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// User-defined Datatypes
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//========================
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abstract class Tree
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case class Leaf(x: Int) extends Tree
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case class Node(s: String, left: Tree, right: Tree) extends Tree
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val lf = Leaf(20)
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val tr = Node("foo", Leaf(10), Leaf(23))
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val lst : List[Tree] = List(lf, tr)
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abstract class Colour
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case object Red extends Colour
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case object Green extends Colour
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case object Blue extends Colour
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case object Yellow extends Colour
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def fav_colour(c: Colour) : Boolean = c match {
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case Green => true
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case _ => false
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}
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fav_colour(Blue)
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// ... a tiny bit more useful: Roman Numerals
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sealed abstract class RomanDigit
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case object I extends RomanDigit
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case object V extends RomanDigit
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case object X extends RomanDigit
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case object L extends RomanDigit
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case object C extends RomanDigit
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case object D extends RomanDigit
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case object M extends RomanDigit
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type RomanNumeral = List[RomanDigit]
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List(X,I,M,A)
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/*
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I -> 1
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II -> 2
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III -> 3
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IV -> 4
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V -> 5
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VI -> 6
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VII -> 7
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VIII -> 8
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IX -> 9
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X -> 10
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*/
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def RomanNumeral2Int(rs: RomanNumeral): Int = rs match {
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case Nil => 0
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case M::r => 1000 + RomanNumeral2Int(r)
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case C::M::r => 900 + RomanNumeral2Int(r)
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case D::r => 500 + RomanNumeral2Int(r)
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case C::D::r => 400 + RomanNumeral2Int(r)
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case C::r => 100 + RomanNumeral2Int(r)
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case X::C::r => 90 + RomanNumeral2Int(r)
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case L::r => 50 + RomanNumeral2Int(r)
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case X::L::r => 40 + RomanNumeral2Int(r)
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case X::r => 10 + RomanNumeral2Int(r)
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case I::X::r => 9 + RomanNumeral2Int(r)
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case V::r => 5 + RomanNumeral2Int(r)
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case I::V::r => 4 + RomanNumeral2Int(r)
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case I::r => 1 + RomanNumeral2Int(r)
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}
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RomanNumeral2Int(List(I,V)) // 4
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RomanNumeral2Int(List(I,I,I,I)) // 4 (invalid Roman number)
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RomanNumeral2Int(List(V,I)) // 6
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RomanNumeral2Int(List(I,X)) // 9
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RomanNumeral2Int(List(M,C,M,L,X,X,I,X)) // 1979
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RomanNumeral2Int(List(M,M,X,V,I,I)) // 2017
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// expressions (essentially trees)
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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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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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val e = Plus(N(9), Times(N(3), N(4)))
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e.toString
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println(string(e))
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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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println(eval(e))
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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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// (....9 ....)
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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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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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// String interpolations as patterns
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val date = "2019-11-26"
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val s"$year-$month-$day" = date
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def parse_date(date: String) : Option[(Int, Int, Int)]= date match {
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case s"$year-$month-$day" => Some((day.toInt, month.toInt, year.toInt))
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case s"$day/$month/$year" => Some((day.toInt, month.toInt, year.toInt))
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case s"$day.$month.$year" => Some((day.toInt, month.toInt, year.toInt))
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case _ => None
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}
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parse_date("2019-11-26")
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parse_date("26/11/2019")
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parse_date("26.11.2019")
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// guards in pattern-matching
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def foo(xs: List[Int]) : String = xs match {
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case Nil => s"this list is empty"
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case x :: xs if x % 2 == 0
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=> s"the first elemnt is even"
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case x :: y :: rest if x == y
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=> s"this has two elemnts that are the same"
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case hd :: tl => s"this list is standard $hd::$tl"
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}
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foo(Nil)
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foo(List(1,2,3))
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foo(List(1,2))
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foo(List(1,1,2,3))
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foo(List(2,2,2,3))
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// Tail recursion
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//================
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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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fact(10) //ok
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fact(10000) // produces a stackoverflow
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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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factT(10, 1)
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println(factT(100000, 1))
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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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@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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// 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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def length(xs: List[Int]) : Int = xs match {
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case Nil => 0
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case _ :: tail => 1 + length(tail)
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}
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@tailrec
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def lengthT(xs: List[Int], acc : Int) : Int = xs match {
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case Nil => acc
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case _ :: tail => lengthT(tail, 1 + acc)
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}
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lengthT(List.fill(10000000)(1), 0)
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// Sudoku
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//========
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// uses Strings for games
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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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val allValues = "123456789".toList
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val indexes = (0 to 8).toList
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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) : Pos =
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(empty(game) % maxValue, empty(game) / maxValue)
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def get_row(game: String, y: Int) = indexes.map(col => game(y * maxValue + col))
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def get_col(game: String, x: Int) = indexes.map(row => game(x + row * maxValue))
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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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for (x <- (x0 until x0 + 3).toList;
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y <- (y0 until y0 + 3).toList) yield game(x + y * maxValue)
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}
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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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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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def candidates(game: String, pos: Pos): List[Char] =
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allValues.diff(toAvoid(game, pos))
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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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candidates(game, emptyPosition(game)).
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map(c => search(update(game, empty(game), c))).flatten
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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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def pretty(game: String): String =
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"\n" + (game.sliding(maxValue, maxValue).mkString(",\n"))
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// tail recursive version that searches
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// for all solutions
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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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searchT(List(game3), List()).map(pretty)
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// tail recursive version that searches
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// for a single solution
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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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search1T(List(game3)).map(pretty)
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414 |
time_needed(10, search1T(List(game3)))
|
|
415 |
|
158
|
416 |
|
155
|
417 |
// game with multiple solutions
|
|
418 |
val game3 = """.8...9743
|
|
419 |
|.5...8.1.
|
|
420 |
|.1.......
|
|
421 |
|8....5...
|
|
422 |
|...8.4...
|
|
423 |
|...3....6
|
|
424 |
|.......7.
|
|
425 |
|.3.5...8.
|
|
426 |
|9724...5.""".stripMargin.replaceAll("\\n", "")
|
|
427 |
|
158
|
428 |
searchT(List(game3), Nil).map(pretty)
|
155
|
429 |
search1T(List(game3)).map(pretty)
|
67
|
430 |
|
77
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
431 |
// Moral: Whenever a recursive function is resource-critical
|
158
|
432 |
// (i.e. works with large recursion depth), then you need to
|
77
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
433 |
// write it in tail-recursive fashion.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
434 |
//
|
155
|
435 |
// Unfortuantely, Scala because of current limitations in
|
|
436 |
// the JVM is not as clever as other functional languages. It can
|
77
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
437 |
// only optimise "self-tail calls". This excludes the cases of
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
438 |
// multiple functions making tail calls to each other. Well,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
439 |
// nothing is perfect.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
440 |
|