Algorithm goal

Make a pure-functional queue using stacks (Lists). It would have the following methods: isEmpty: Boolean, enqueue(T), dequeue: Option[(T, Queue[T])].

Try to make it efficient, where possible.

Test cases in Scala

assert(Queue.empty.isEmpty)
assert(Queue.empty.enqueue("A").dequeue.contains("A" -> Queue.empty))
assert(
  Queue.empty.enqueue("A").enqueue("B").dequeue.map(_._1).contains("A")
)
assert(
  Queue.empty
    .enqueue("A")
    .enqueue("B")
    .dequeue
    .map(_._2)
    .flatMap(_.dequeue)
    .map(_._1)
    .contains("B")
)
assert(
  Queue.empty
    .enqueue("A")
    .enqueue("B")
    .dequeue
    .map(_._2)
    .flatMap(_.dequeue)
    .map(_._2)
    .flatMap(_.dequeue)
    .isEmpty
)
assert(
  Queue.empty
    .enqueue("A")
    .enqueue("B")
    .dequeue
    .map(_._2.enqueue("C"))
    .flatMap(_.dequeue)
    .map(_._1)
    .contains("B")
)
assert(
  Queue.empty
    .enqueue("A")
    .enqueue("B")
    .dequeue
    .map(_._2.enqueue("C"))
    .flatMap(_.dequeue)
    .map(_._2)
    .flatMap(_.dequeue)
    .map(_._1)
    .contains("C")
)
assert(
  Queue.empty
    .enqueue("A")
    .enqueue("B")
    .dequeue
    .map(_._2.enqueue("C"))
    .flatMap(_.dequeue)
    .map(_._2)
    .flatMap(_.dequeue)
    .map(_._2)
    .flatMap(_.dequeue)
    .isEmpty
)

Algorithm in Scala

20 lines of Scala (compatible versions 2.13 & 3.0), showing how concise Scala can be!

Explanation

The most straightforward implementation of a Queue would contain a single List, however in this case, we would have to choose between making enqueue fast, or making dequeue fast, because with a List, to get the tail element is an \(O(n)\) operation.

However, there is a more efficient way, which is to use an 'incoming' and an 'outgoing' List: so long as 'outgoing' is not empty, we can destructure it at O(1) cost, and we can enqueue to 'incoming' at O(1) cost. The only time where we would face a cost would be when 'outgoing' is empty, and 'incoming' is non-empty: then, we would have to convert the 'incoming' to 'outgoing' by reversal, which is an O(n) operation. However, amortized, it would be more efficient than the biased approach. (this is © from www.scala-algorithms.com)

Scala concepts & Hints

1. 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

2. 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

3. 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

4. Tail Recursion

   In Scala, tail recursion enables you to rewrite a mutable structure such as a while-loop, into an immutable algorithm.

   def fibonacci(n: Int): Int = {
     @scala.annotation.tailrec
     def go(i: Int, previous: Int, beforePrevious: Int): Int =
       if (i >= n) previous else go(i + 1, previous + beforePrevious, previous)
   
     go(i = 1, previous = 1, beforePrevious = 0)
   }
   
   assert(fibonacci(8) == 21)
   

   ▶ Run code in the IDE