Scala algorithm: Merge Sort: in pure immutable Scala

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Algorithm goal

Merge Sort is a standard merging algorithm. It works by grouping items into pairs, and then merging those pairs by selecting the smallest items in ascending order. Then, it repeats this process until 1 whole array is computed.

Our goal is to achieve a sorting like this:

3214
1234

Merge sort algorithm illustration

The data transformation in Merge Sort looks like this:

Picking of items from two halves
Resulting ListLeft HalfRight Half
Original list (split in half)
3241
Applying a merge+sort function to each of the halves
2314
Then, in the merge function, we begin to extract the smallest elements (as the two halves are sorted)
2314
1234
1234
1234
1234
And now we have solved one level of merging.

In the non-stack-safe version, we achieve this via recursion, where we really say 'our sorted version is the merge of sorting of the two halves of our original input'.

This version is not stack-safe; for stack-safe, see: MergeSortStackSafe

Test cases in Scala

assert(mergeSort(List.empty) == List.empty)
assert(mergeSort(List(1)) == List(1))
assert(mergeSort(List(1, 2)) == List(1, 2))
assert(mergeSort(List(2, 1)) == List(1, 2))
assert(mergeSort(List(2, 1, 3)) == List(1, 2, 3))
assert(mergeSort(List(2, 1, 4, 3)) == List(1, 2, 3, 4))
assert(mergeSort(List(2, 4, 5, 1, 3)) == List(1, 2, 3, 4, 5))
assert(
  {
    val randomArray = scala.util.Random
      .nextBytes(10 + Math.abs(scala.util.Random.nextInt(1000)))
      .map(_.toInt)
      .toList
    mergeSort(randomArray) == randomArray.sorted
  },
  "Random array of any length is sorted"
)

Algorithm in Scala

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

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Explanation

We follow the algorithm: split the items by half, and then perform a merging operation. In this divide-and-conquer algorithm, the more complex part is the merging function:

The merge function

The merge function is more unusual in that its implementation is simpler when the number of inputs is dynamic, rather than fixed. (this is © from www.scala-algorithms.com)

Full explanation is available for subscribers Scala algorithms logo, maze part, which looks quirky

Scala concepts & Hints

  1. 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('-')
    }
    
    assert(exampleDef("test") == "-test-")
    

    It is also frequently used in combination with Tail Recursion.

  2. Drop, Take, dropRight, takeRight

    Scala's `drop` and `take` methods typically remove or select `n` items from a collection.

    assert(List(1, 2, 3).drop(2) == List(3))
    
    assert(List(1, 2, 3).take(2) == List(1, 2))
    
    assert(List(1, 2, 3).dropRight(2) == List(1))
    
    assert(List(1, 2, 3).takeRight(2) == List(2, 3))
    
    assert((1 to 5).take(2) == (1 to 2))
    
  3. Lazy List

    The 'LazyList' type (previously known as 'Stream' in Scala) is used to describe a potentially infinite list that evaluates only when necessary ('lazily').

  4. 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))
    
  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")
    
  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)
    
  7. 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)
    

Scala Algorithms: The most comprehensive library of algorithms in standard pure-functional Scala

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  1. Find minimum missing positive number in a sequence
  2. Longest increasing sub-sequence length
  3. Compute the length of longest valid parentheses
  4. Monitor success rate of a process that may fail
  5. Remove duplicates from an unsorted List
  6. Find combinations adding up to N (unique)
  7. Find k closest elements to a value in a sorted Array
  8. Make a queue using stacks (Lists in Scala)
  9. Single-elimination tournament tree
  10. Quick Sort sorting algorithm in pure immutable Scala
  11. Compute a Roman numeral for an Integer, and vice-versa
  12. Matching parentheses algorithm with foldLeft and a state machine
  13. Traverse a tree Breadth-First, immutably
  14. Read a matrix as a spiral
  15. Remove duplicates from a sorted list (state machine)
  16. Merge Sort: stack-safe, tail-recursive, in pure immutable Scala, N-way
  17. Binary search a generic Array
  18. Merge Sort: in pure immutable Scala
  19. Make a queue using Maps
  20. Is an Array a permutation?
  21. Count number of contiguous countries by colors
  22. Add numbers without using addition (plus sign)
  23. Tic Tac Toe MinMax solve
  24. Run-length encoding (RLE) Encoder
  25. Print Alphabet Diamond
  26. Balanced parentheses algorithm with tail-call recursion optimisation
  27. Reverse a String's words efficiently
  28. Count number of changes (manipulations) needed to make an anagram with foldLeft and a MultiSet
  29. Count passing cars
  30. Counting inversions of a sequence (array) using a Merge Sort
  31. Longest common prefix of strings
  32. Check if an array is a palindrome
  33. Check a directed graph has a routing between two nodes (depth-first search)
  34. Compute nth row of Pascal's triangle
  35. Run-length encoding (RLE) Decoder
  36. Check if a number is a palindrome
  37. In a range of numbers, count the numbers divisible by a specific integer
  38. Find the index of a substring ('indexOf')
  39. Reshape a matrix
  40. Closest pair of coordinates in a 2D plane
  41. Find the contiguous slice with the minimum average
  42. Compute maximum sum of subarray (Kadane's algorithm)
  43. Pure-functional double linked list
  44. Binary search in a rotated sorted array
  45. Check if a directed graph has cycles
  46. Rotate Array right in pure-functional Scala - using an unusual immutable efficient approach
  47. Length of the longest common substring
  48. Tic Tac Toe board check
  49. Find an unpaired number in an array
  50. Check if a String is a palindrome
  51. Count binary gap size of a number using tail recursion
  52. Remove duplicates from a sorted list (Sliding)
  53. Find sub-array with the maximum sum
  54. Find the minimum absolute difference of two partitions
  55. Find maximum potential profit from an array of stock price
  56. Fibonacci in purely functional immutable Scala
  57. Fizz Buzz in purely functional immutable Scala
  58. Find combinations adding up to N (non-unique)
  59. Make a binary search tree (Red-Black tree)
  60. Count factors/divisors of an integer
  61. Compute single-digit sum of digits
  62. Traverse a tree Depth-First
  63. Reverse bits of an integer
  64. QuickSelect Selection Algorithm (kth smallest item/order statistic)
  65. Rotate a matrix by 90 degrees clockwise

Explore the 21 most useful Scala concepts

To save you going through various tutorials, we cherry-picked the most useful Scala concepts in a consistent form.

  1. Class Inside Class
  2. Class Inside Def
  3. Collect
  4. Def Inside Def
  5. Drop, Take, dropRight, takeRight
  6. foldLeft and foldRight
  7. For-comprehension
  8. Lazy List
  9. Option Type
  10. Ordering
  11. Partial Function
  12. Pattern Matching
  13. Range
  14. scanLeft and scanRight
  15. Sliding / Sliding Window
  16. Stack Safety
  17. State machine
  18. Tail Recursion
  19. Type Class
  20. View
  21. Zip

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