Scala algorithm: Fibonacci in purely functional immutable Scala

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Algorithm goal

The Fibonacci sequence is \(0, 1, 1, 2, 3, 5, 8, 13, 21, ...\), ie \(F(n + 2) = F(n + 1) + F(n)\), with \(F(1) = 1\) and \(F(0) = 1\).

  • \(F(0) = 0\)
  • \(F(0) = 1\)
  • \(F(2) = F(0) + F(1) = 0 + 1 = 1\)
  • \(F(3) = F(1) + F(2) = 1 + 1 = 2\)
  • \(F(4) = F(2) + F(3) = 1 + 2 = 3\)
  • \(F(5) = F(3) + F(4) = 2 + 3 = 5\)
  • \(F(6) = F(4) + F(5) = 3 + 5 = 8\)
  • \(...\)

The Fibonacci sequence ("Fibonacci numbers") is hugely important in mathematics, aesthetics and nature.

Goal is to compute it in an immutable and pure-functional fashion in Scala.

Test cases in Scala

assert(
  FibonacciNumbers.take(8).toList.map(_.toInt) ==
    List(0, 1, 1, 2, 3, 5, 8, 13)
)
assert(
  FibonacciNumbers2.take(8).toList.map(_.toInt) ==
    List(0, 1, 1, 2, 3, 5, 8, 13)
)
assert(FibonacciNumbers.apply(2000) > 0, "There is no Stack Overflow")
assert(
  FibonacciNumbers2.apply(2000) > 0,
  "There is no Stack Overflow with version 2"
)

Algorithm in Scala

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

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Explanation

We present 2 solutions using LazyList, both of which are stack-safe (Stack Safety).

First solution using laziness/deferred evaluation

In the first solution, the computation is defined recursively using LazyList - which means the definition of the next item is defined in terms of the recursion. computeFollowing(1, 1) = 1 #:: <lazy sequence of computeFollowing(1 + 1, 1)> = 1 #:: 1 #:: <lazy sequence of computeFollowing(1 + 2, 2)> = ... (this is © from www.scala-algorithms.com)

This way of defining it is indeed quite unusual and is a forward-looking computation.

Second solution using the memoisation property

The second solution is using a different property of LazyLists, which is that they memoise the items (in the previous solution, we do not strictly utilise this property; it could be implemented with an Iterator-like function that does not memoise).

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. 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').

  3. 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")
    
  4. Zip

    'zip' allows you to combine two lists pair-wise (meaning turn a pair of lists, into a list of pairs)

    It can be used over Arrays, Lists, Views, Iterators and other collections.

    assert(List(1, 2, 3).zip(List(5, 6, 7)) == List(1 -> 5, 2 -> 6, 3 -> 7))
    
    assert(List(1, 2).zip(List(5, 6, 7)) == List(1 -> 5, 2 -> 6))
    
    assert(List(5, 6).zipWithIndex == List(5 -> 0, 6 -> 1))
    
    assert(List(5, 6).zipAll(List('A'), 9, 'Z') == List(5 -> 'A', 6 -> 'Z'))
    
    assert(List(5).zipAll(List('A', 'B'), 1, 'Z') == List(5 -> 'A', 1 -> 'B'))
    

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