Scala algorithm: Counting inversions of a sequence (array) using a Merge Sort

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Algorithm goal

The number of inversions in a sequence is the number of pairs of elements that are out of order, ie\(|{(i, j) : i < j, A_i > A_j}|\) (count of distinct \((i,j)\) such that \(i < j\) and value of the \(i\)th element is greater than that of the \(j\)th one).

Why is this algorithm useful?

One situation where this is very useful is to see how different preferences between two people are, when ranked. This sort of algorithm, or versions of it, would be useful in eCommerce, or even movie recommendations. While it is not as advanced as machine learning, it has the big advantage of high performance at \(O(n\log{n})\), meaning it can be used as a tactical solution in a highly paced environment, where the cost of implementation of a machine learning solution could only become viable only once the commercial viability of the product has established.

The brute-force solution is \(O(n^2)\) complexity, which eats up computation time very quickly

Examples

\([2,1]\) has 1 inversion, because swapping \(1\) with \(2\) leads to array \([1,2]\) which is sorted.

21
12

Likewise, \([3, 1, 8, 5, 6, 4, 7]\) has 7 inversions: \((1, 3)\), \((8, 5)\), \((8, 6)\), \((8, 4)\), \((8, 7)\), \((5, 4)\).\((6, 4)\).

3185647
1345678

\([1,2,4,3]\) has 1 inversion \((4, 3)\):

1243
1234

What is the most curious about the above diagrams is that the number of crossings between arrows corresponds exactly to the number of inversions! What do you think this would mean?

It will be very helpful to first understand the problem of MergeSort first and compare the two. Although the merging function is different - because this time, we really have to count how many exchanges there would be, as we are sorting it. This is what leads us to a solution that is much more efficient than a typical brute-force solution (which, in fact, we include in the algorithm solution).

The algorithm code here, while a MergeSort, is implemented using the bottom-up approach of the algorithm MergeSortStackSafe, which is more stack-safe.

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Algorithm in Scala

79 lines of Scala (version 2.13).

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Explanation

The problem is actually closely related to the marge sort. In the merge sort, we approach the problem by the divide-and-conquer method, where we process one half of the array, the other half, and then we merge them.

If we divide our input sequence into 2 parts, through example, we will notice that the total number of inversions is equal to the number of inversions on the left-hand side (LHS), plus inversions on RHS, plus the number of inversions across the two sides. Example: (this is © from www.scala-algorithms.com)

\([3,2,1,4]\) has inversion \((2,3)\) on the LHS to make it sorted, no inversion on RHS, and 2 inversions across the half: \((3,1)\) and \((2,1)\), thus 3 inversions in total.

Left hand side with 1 inversion only
32
23Counted 1 inversion
Right hand side with 0 inversions
(it's already sorted)
14
14Counted 0 inversions

As we are sorting the data on a side, we record how many times it was out of order. Even in a sub-problem of 2, we count an inversion of a change between the left side and the right side.

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. For-comprehension

    The for-comprehension is highly important syntatic enhancement in functional programming languages.

    val Multiplier = 10
    
    val result: List[Int] = for {
      num <- List(1, 2, 3)
      anotherNum <-
        List(num * Multiplier - 1, num * Multiplier, num * Multiplier + 1)
    } yield anotherNum + 1
    
    assert(result == List(10, 11, 12, 20, 21, 22, 30, 31, 32))
    
  4. 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').

  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. Range

    The (1 to n) syntax produces a "Range" which is a representation of a sequence of numbers.

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