// 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.
// Cool Stuff in Scala
//=====================
// Implicits or How to Pimp your Library
//======================================
//
// For example adding your own methods to Strings:
// Imagine you want to increment strings, like
//
// "HAL".increment
//
// you can avoid ugly fudges, like a MyString, by
// using implicit conversions.
implicit class MyString(s: String) {
def increment = s.map(c => (c + 1).toChar)
}
"HAL".increment
// Abstract idea:
// In that version implicit conversions were used to solve the
// late extension problem; namely, given a class C and a class T,
// how to have C extend T without touching or recompiling C.
// Conversions add a wrapper when a member of T is requested
// from an instance of C.
//Another example (TimeUnit in 2.13?)
import scala.concurrent.duration.{TimeUnit,SECONDS,MINUTES}
case class Duration(time: Long, unit: TimeUnit) {
def +(o: Duration) =
Duration(time + unit.convert(o.time, o.unit), unit)
}
implicit class Int2Duration(that: Int) {
def seconds = new Duration(that, SECONDS)
def minutes = new Duration(that, MINUTES)
}
5.seconds + 2.minutes //Duration(125L, SECONDS )
2.minutes + 60.seconds
// Regular expressions - the power of DSLs in Scala
//==================================================
abstract class Rexp
case object ZERO extends Rexp // nothing
case object ONE extends Rexp // the empty string
case class CHAR(c: Char) extends Rexp // a character c
case class ALT(r1: Rexp, r2: Rexp) extends Rexp // alternative r1 + r2
case class SEQ(r1: Rexp, r2: Rexp) extends Rexp // sequence r1 . r2
case class STAR(r: Rexp) extends Rexp // star r*
// writing (ab)* in the format above is
// tedious
val r0 = STAR(SEQ(CHAR('a'), CHAR('b')))
// some convenience for typing in regular expressions
import scala.language.implicitConversions
import scala.language.reflectiveCalls
def charlist2rexp(s: List[Char]): Rexp = s match {
case Nil => ONE
case c::Nil => CHAR(c)
case c::s => SEQ(CHAR(c), charlist2rexp(s))
}
implicit def string2rexp(s: String): Rexp =
charlist2rexp(s.toList)
"(a|b)"
val r1 = STAR("ab")
val r2 = (STAR("ab")) | (STAR("ba"))
val r3 = STAR(SEQ("ab", ALT("a", "b")))
implicit def RexpOps (r: Rexp) = new {
def | (s: Rexp) = ALT(r, s)
def % = STAR(r)
def ~ (s: Rexp) = SEQ(r, s)
}
implicit def stringOps (s: String) = new {
def | (r: Rexp) = ALT(s, r)
def | (r: String) = ALT(s, r)
def % = STAR(s)
def ~ (r: Rexp) = SEQ(s, r)
def ~ (r: String) = SEQ(s, r)
}
//example regular expressions
val digit = ("0" | "1" | "2" | "3" | "4" |
"5" | "6" | "7" | "8" | "9")
val sign = "+" | "-" | ""
val number = sign ~ digit ~ digit.%
// Mind-Blowing Regular Expressions
// same examples using the internal regexes
val evil = "(a*)*b"
println("a" * 100)
("a" * 10000).matches(evil)
("a" * 10).matches(evil)
("a" * 10000).matches(evil)
("a" * 20000).matches(evil)
("a" * 50000).matches(evil)
time_needed(1, ("a" * 50000).matches(evil))