Scala algorithm: Fibonacci in purely functional immutable Scala

Published

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

14 lines of Scala (compatible versions 2.13 & 3.0), 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 to 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

How our 100 algorithms look

  1. A description/goal of the algorithm.
  2. An explanation with both Scala and logical parts.
  3. A proof or a derivation, where appropriate.
  4. Links to Scala concepts used in this specific algorithm, also unit-tested.
  5. An implementation in pure-functional immutable Scala, with efficiency in mind (for most algorithms, this is for paid subscribers only).
  6. Unit tests, with a button to run them immediately in our in-browser IDE.
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Fully unit-tested, with explanations and relevant concepts; new algorithms published about once a week.

  1. Compute the length of longest valid parentheses
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  3. Remove duplicates from an unsorted List
  4. Make a queue using stacks (Lists in Scala)
  5. Find height of binary tree
  6. Single-elimination tournament tree
  7. Reverse Polish Notation calculator
  8. Quick Sort sorting algorithm in pure immutable Scala
  9. Find minimum missing positive number in a sequence
  10. Least-recently used cache (LRU)
  11. Count pairs of a given expected sum
  12. Binary heap (min-heap)
  13. Compute a Roman numeral for an Integer, and vice-versa
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  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. Token Bucket Rate Limiter
  20. Leaky Bucket Rate Limiter
  21. Merge Sort: stack-safe, tail-recursive, in pure immutable Scala, N-way
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  23. Reverse first n elements of a queue
  24. Binary search a generic Array
  25. Game of Life
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  27. Make a queue using Maps
  28. Is an Array a permutation?
  29. Count number of contiguous countries by colors
  30. Add numbers without using addition (plus sign)
  31. Tic Tac Toe MinMax solve
  32. Run-length encoding (RLE) Encoder
  33. Print Alphabet Diamond
  34. Find kth largest element in a List
  35. Balanced parentheses algorithm with tail-call recursion optimisation
  36. Reverse a String's words efficiently
  37. Count number of changes (manipulations) needed to make an anagram with an efficient foldLeft
  38. Count passing cars
  39. Establish execution order from dependencies
  40. Counting inversions of a sequence (array) using a Merge Sort
  41. Longest common prefix of strings
  42. Check if an array is a palindrome
  43. Compute missing ranges
  44. Check a directed graph has a routing between two nodes (depth-first search)
  45. Compute nth row of Pascal's triangle
  46. Run-length encoding (RLE) Decoder
  47. Check if a number is a palindrome
  48. In a range of numbers, count the numbers divisible by a specific integer
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  50. Find the index of a substring ('indexOf')
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  60. Rotate Array right in pure-functional Scala - using an unusual immutable efficient approach
  61. Check a binary tree is a search tree
  62. Length of the longest common substring
  63. Sliding Window Rate Limiter
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  66. Check if a String is a palindrome
  67. Count binary gap size of a number using tail recursion
  68. Remove duplicates from a sorted list (Sliding)
  69. Monitor success rate of a process that may fail
  70. Least-recently used cache (MRU)
  71. Find sub-array with the maximum sum
  72. Find the minimum absolute difference of two partitions
  73. Find maximum potential profit from an array of stock price
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  83. Count factors/divisors of an integer
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  88. Find k closest elements to a value in a sorted Array
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  90. 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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