abstract class Rexp 
case object NULL extends Rexp
case object EMPTY extends Rexp
case class CHAR(c: Char) extends Rexp
case class ALT(r1: Rexp, r2: Rexp) extends Rexp
case class SEQ(r1: Rexp, r2: Rexp) extends Rexp 
case class STAR(r: Rexp) extends Rexp 
case class NTIMES(r: Rexp, n: Int) extends Rexp 

def simp(r: Rexp): Rexp = r match {
  case ALT(r1, r2) => {
    (simp(r1), simp(r2)) match {
      case (NULL, r2s) => r2s
      case (r1s, NULL) => r1s
      case (r1s, r2s) => if (r1s == r2s) r1s else ALT(r1s, r2s)
    }
  }
  case SEQ(r1, r2) => {
    (simp(r1), simp(r2)) match {
      case (NULL, _) => NULL
      case (_, NULL) => NULL
      case (EMPTY, r2s) => r2s
      case (r1s, EMPTY) => r1s
      case (r1s, r2s) => SEQ(r1s, r2s)
    }
  }
  case NTIMES(r, n) => NTIMES(simp(r), n)    
  case r => r
}

def nullable (r: Rexp) : Boolean = r match {
  case NULL => false
  case EMPTY => true
  case CHAR(_) => false
  case ALT(r1, r2) => nullable(r1) || nullable(r2)
  case SEQ(r1, r2) => nullable(r1) && nullable(r2)
  case STAR(_) => true
  case NTIMES(r, i) => if (i == 0) true else nullable(r)
}

def der (c: Char, r: Rexp) : Rexp = r match {
  case NULL => NULL
  case EMPTY => NULL
  case CHAR(d) => if (c == d) EMPTY else NULL
  case ALT(r1, r2) => ALT(der(c, r1), der(c, r2))
  case SEQ(r1, r2) => 
    if (nullable(r1)) ALT(SEQ(der(c, r1), r2), der(c, r2))
    else SEQ(der(c, r1), r2)
  case STAR(r) => SEQ(der(c, r), STAR(r))
  case NTIMES(r, i) => 
    if (i == 0) NULL else SEQ(der(c, r), NTIMES(r, i - 1))
}

def ders (s: List[Char], r: Rexp) : Rexp = s match {
  case Nil => r
  case c::s => ders(s, simp(der(c, r)))
}

def matches(r: Rexp, s: String) : Boolean = nullable(ders(s.toList, r))


//one or zero
def OPT(r: Rexp) = ALT(r, EMPTY)

def EVIL(n: Int) = SEQ(NTIMES(OPT(CHAR('a')), n), NTIMES(CHAR('a'), n))

def time_needed[T](i: Int, code: => T) = {
  val start = System.nanoTime()
  for (j <- 1 to i) code
  val end = System.nanoTime()
  (end - start)/(i * 1.0e9)
}


for (i <- 1 to 12001 by 500) {
  println(i + " " + "%.5f".format(time_needed(1, matches(EVIL(i), "a" * i))))
}