Balanced parentheses algorithm in immutable/pure functional Scala with tail-call recursion optimisation

Algorithm goal

Algorithm to check parentheses in a String are balanced. This problem is also known as:

  • On Codility: Stacks and Queues: Brackets - Determine whether a given string of parentheses (multiple types) is properly nested.
  • On HackerRank: Balanced Brackets - Given strings of brackets, determine whether each sequence of brackets is balanced. If a string is balanced, return YES. Otherwise, return NO.

Parentheses in a String are balanced when an opening bracket is followed by another opening bracket or by a closing bracket of the same time.

For example, ([]) is balanced, but ([) and ([)] are not.

Explanation

Also please find an alternative solution using a foldLeft state machine ParenthesesFoldingStateMachine which makes an advanced streamed implementation possible.

By definition, we need to keep track of the most recent opening bracket, and if the next bracket we approach is a closing bracket that is not the latest bracket, we terminate with a failure, whereas if it is another opening bracket, this is the bracket we consider as the latest, and if it is another closing bracket that matches the latest opening bracket, the new latest opening bracket is 'popped'. This description is enough to suggest to us that we should use a 'stack' structure, and implement our logic as described. (this is © from www.scala-algorithms.com)

In Scala, a List is equivalent to a stack, it is a structure in the form List[T] = ::(head: T, tail: List[T]) | Nil

We could solve this problem in a standard mutable style, but this would not be the immutable Scala way of doing things.

Scala Concepts & Hints

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")

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Def Inside Def

A great aspect of Scala is being able to declare functions inside functions, making it possible to reduce repetition.

def exampleDef(input: String): String = {
  def surroundInputWith(char: Char): String = s"$char$input$char"
  surroundInputWith('-')
}
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)

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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)

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Algorithm in Scala

29 lines of Scala (version 2.13), showing how concise Scala can be!

val OpenToClose: Map[Char, Char] = Map('{' -> '}', '[' -> ']', '(' -> ')')

val CloseToOpen: Map[Char, Char] = OpenToClose.map(_.swap)

def parenthesesAreBalanced(s: String): Boolean = {
  if (s.isEmpty) true
  else {
    @scala.annotation.tailrec
    def go(position: Int, stack: List[Char]): Boolean = {
      if (position == s.length) stack.isEmpty
      else {
        val char = s(position)
        val isOpening = OpenToClose.contains(char)
        val isClosing = CloseToOpen.contains(char)
        if (isOpening) go(position + 1, char :: stack)
        else if (isClosing) {
          val expectedCharForMatching = CloseToOpen(char)
          stack match {
            case `expectedCharForMatching` :: rest =>
              go(position + 1, rest)
            case _ =>
              false
          }
        } else false
      }
    }
    go(position = 0, stack = List.empty)
  }
}

Test cases in Scala

assert(parenthesesAreBalanced("()"))
assert(parenthesesAreBalanced("[()]"))
assert(parenthesesAreBalanced("{[()]}"))
assert(parenthesesAreBalanced("([{{[(())]}}])"))
assert(!parenthesesAreBalanced("{{[]()}}}}"))
assert(!parenthesesAreBalanced("{{[](A}}}}"))
assert(!parenthesesAreBalanced("{[(])}"))
def parenthesesAreBalanced(s: String): Boolean = ???