Cartesian product of two lists

Question

Given a map where a digit is associated to several characters

scala> val conversion = Map("0" -> List("A", "B"), "1" -> List("C", "D"))
conversion: scala.collection.immutable.Map[java.lang.String,List[java.lang.String]] =
  Map(0 -> List(A, B), 1 -> List(C, D))

I want to generate all possible character sequences based on a sequence of digits. Examples:

"00" -> List("AA", "AB", "BA", "BB")
"01" -> List("AC", "AD", "BC", "BD")

I can do this with for comprehensions

scala> val number = "011"
number: java.lang.String = 011

Create a sequence of possible characters per index

scala> val values = number map { case c => conversion(c.toString) }
values: scala.collection.immutable.IndexedSeq[List[java.lang.String]] =
  Vector(List(A, B), List(C, D), List(C, D))

Generate all the possible character sequences

scala> for {
     | a <- values(0)
     | b <- values(1)
     | c <- values(2)
     | } yield a+b+c
res13: List[java.lang.String] = List(ACC, ACD, ADC, ADD, BCC, BCD, BDC, BDD)

Here things get ugly and it will only work for sequences of three digits. Is there any way to achieve the same result for any sequence length?

Solution 1

The following suggestion is not using a for-comprehension. But I don't think it's a good idea after all, because as you noticed you'd be tied to a certain length of your cartesian product.

scala> def cartesianProduct[T](xss: List[List[T]]): List[List[T]] = xss match {
     |   case Nil => List(Nil)
     |   case h :: t => for(xh <- h; xt <- cartesianProduct(t)) yield xh :: xt
     | }
cartesianProduct: [T](xss: List[List[T]])List[List[T]]

scala> val conversion = Map('0' -> List("A", "B"), '1' -> List("C", "D"))
conversion: scala.collection.immutable.Map[Char,List[java.lang.String]] = Map(0 -> List(A, B), 1 -> List(C, D))

scala> cartesianProduct("01".map(conversion).toList)
res9: List[List[java.lang.String]] = List(List(A, C), List(A, D), List(B, C), List(B, D))

Why not tail-recursive?

Note that above recursive function is not tail-recursive. This isn't a problem, as xss will be short unless you have a lot of singleton lists in xss. This is the case, because the size of the result grows exponentially with the number of non-singleton elements of xss.

Solution 2

I could come up with this:

val conversion = Map('0' -> Seq("A", "B"), '1' -> Seq("C", "D"))

def permut(str: Seq[Char]): Seq[String] = str match {
  case Seq()  => Seq.empty
  case Seq(c) => conversion(c)
  case Seq(head, tail @ _*) =>
    val t = permut(tail)
    conversion(head).flatMap(pre => t.map(pre + _))
}

permut("011")

Comments

[Vlad Patryshev]

Strictly speaking, what you are trying to get is not a cartesian product. If you have List[T], you do not guarantee the size, and also the list should be homogeneous.

[Mark Jayxcela]

Thanks Sciss, I accepted ziggystar's answer since he posted it before and both answers are pretty good. I especially like your idea of decomposing the for in a chain of flatMaps.

[ziggystar]

@Mark Sciss' answer differs only syntactically from mine (at least for the part building the combinations). He uses map and flatMap and I use the for-comprehension syntactic sugar, which gets translated to Sciss' code by the compiler. Only other difference is that I have extracted a method for the cartesian product.

[Mark Jayxcela]

@ziggystar Great, now I understand better what is happening behind the scenes.