Algorithm goal

A binary search tree has a property that all the child values to the left of the node are less than its value, and all to the right are more (including recursively). Binary search trees are very powerful for lookups and range searches, they can be used bth as sets and as maps.

Test cases in Scala

assert(
  RedBlackTree.of[Int](1).toPlainTree.contains(PlainTree.leaf(value = 1))
)
assert(
  RedBlackTree
    .of(1, 2)
    .toPlainTree
    .contains(
      PlainTree(value = 1, left = None, right = Some(PlainTree.leaf(value = 2)))
    )
)
assert(
  RedBlackTree
    .of(1, 2, 3)
    .toPlainTree
    .contains(
      PlainTree(
        value = 2,
        left = Some(PlainTree.leaf(value = 1)),
        right = Some(PlainTree.leaf(value = 3))
      )
    )
)
assert(
  RedBlackTree
    .of(1, 2, 3, 4)
    .toPlainTree
    .contains(
      PlainTree(
        value = 2,
        left = Some(PlainTree.leaf(value = 1)),
        right = Some(
          PlainTree(
            value = 3,
            left = None,
            right = Some(PlainTree.leaf(value = 4))
          )
        )
      )
    )
)
assert(
  RedBlackTree
    .of(1, 3, 4, 6, 7, 8, 10, 14, 13, 15)
    .toPlainTree
    .contains(
      PlainTree(
        6,
        Some(
          PlainTree(
            3,
            Some(PlainTree(1, None, None)),
            Some(PlainTree(4, None, None))
          )
        ),
        Some(
          PlainTree(
            8,
            Some(PlainTree(7, None, None)),
            Some(
              PlainTree(
                13,
                Some(PlainTree(10, None, None)),
                Some(PlainTree(14, None, Some(PlainTree(15, None, None))))
              )
            )
          )
        )
      )
    ),
  "Wikipedia example works"
)

Algorithm in Scala

78 lines of Scala (compatible versions 2.13 & 3.0).

Explanation

As a general principle, a Red-Black tree uses a method by which it marks nodes of the tree as Red or Black, and when there is an B-R-R tree, this immediately has to be flattened/transformed into one that is flatter, almost like when you put two South poles of magnets together, they repel. We however recommend to read this explanation by the Cornell University (we normally do our own but the R-B topic goes well beyond Scala in this case).

Important to note: Scala does come with its own highly-optimized Red-Black tree implementations called TreeSet and TreeMap, which we highly recommend you check out and play with. (this is © from www.scala-algorithms.com)

Scala concepts & Hints

1. foldLeft and foldRight

   A 'fold' allows you to perform the equivalent of a for-loop, but with a lot less code.

   def foldMutable[I, O](initialState: O)(items: List[I])(f: (O, I) => O): O =
     items.foldLeft(initialState)(f)
   

   ▶ Run code in the IDE

2. Option Type

   The 'Option' type is used to describe a computation that either has a result or does not. In Scala, you can 'chain' Option processing, combine with lists and other data structures. For example, you can also turn a pattern-match into a function that return an Option, and vice-versa!

   assert(Option(1).flatMap(x => Option(x + 2)) == Option(3))
   
   assert(Option(1).flatMap(x => None) == None)
   

   ▶ Run code in the IDE

3. Ordering

   In Scala, the 'Ordering' type is a 'type class' that contains methods to determine an ordering of specific types.

   assert(List(3, 2, 1).sorted == List(1, 2, 3))
   
   assert(List(3, 2, 1).sorted(Ordering[Int].reverse) == List(3, 2, 1))
   
   assert(Ordering[Int].lt(1, 2))
   
   assert(!Ordering[Int].lt(2, 1))
   

   ▶ Run code in the IDE

4. Partial Function

   A Partial Function in Scala is similar to function type `A => Option[B]` (Option Type).

   def getNaming(num: Int): Option[String] =
     PartialFunction.condOpt(num) { case 1 => "One" }
   
   assert(getNaming(1) == Some("One"))
   
   assert(getNaming(2) == None)
   

   ▶ Run code in the IDE

5. Pattern Matching

   Pattern matching in Scala lets you quickly identify what you are looking for in a data, and also extract it.

   assert("Hello World".collect {
     case character if Character.isUpperCase(character) => character.toLower
   } == "hw")
   

   ▶ Run code in the IDE

6. Stack Safety

   Stack safety is present where a function cannot crash due to overflowing the limit of number of recursive calls.

   This function will work for n = 5, but will not work for n = 2000 (crash with java.lang.StackOverflowError) - however there is a way to fix it :-)

   In Scala Algorithms, we try to write the algorithms in a stack-safe way, where possible, so that when you use the algorithms, they will not crash on large inputs. However, stack-safe implementations are often more complex, and in some cases, overly complex, for the task at hand.

   def sum(from: Int, until: Int): Int =
     if (from == until) until else from + sum(from + 1, until)
   
   def thisWillSucceed: Int = sum(1, 5)
   
   def thisWillFail: Int = sum(1, 300)
   

   ▶ Run code in the IDE

7. State machine

   A state machine is the use of `sealed trait` to represent all the possible states (and transitions) of a 'machine' in a hierarchical form.

8. Type Class

   Type classes are one of Scala's most important super-powers: they enable you to add new behaviour to existing classes, without modifying those classes. In many languages, to add a behaviour to a class, you would typically extend it with an interface, and then implement methods against this interface.This, however, does not scale: especially when you have older libraries, you would be forced to make them depend on a new interface, and have to re-build everything.

   Type classes are used heavily in Apple's SwiftUI as "extensions" to enable powerful abstraction capabilities.

   Type classes enable you to do things like this:

   import Ordering.Implicits._
   
   type CommonType = (Int, String, Option[String])
   
   val a: CommonType = (1, "X", None)
   
   val b: CommonType = (2, "A", Some("B"))
   
   assert(a < b, "We can order tuples using Scala-provided type classes")
   

   ▶ Run code in the IDE