Variance, a Scala language concept

Covariance, Invariance and Contravariance (will refer to as CIC) in Scala determine the relationship between generic types and subtyping. We noticed that people find it tricky to get their head around CIC, but in our view it is mostly an issue in the terminology used, such as "covariant positions".

We've thought long about this, and found that examples of Cat and Dog being sub-types of Animal is too concrete, and make CIC more difficult to understand. Often, describing a generic thing in general terms, is just more straightforward.

The subtyping assumption

When A2 is a subtype of A (syntax: A2 <: A), the assignments between A and A2 we can try are:

✅ compiles ❌ does not make sense
Conversion Convert an A2 to an A Convert an A to an A2
Example def convert[A2 <: A](a2: A2): A = a2 def convert[A2 <: A](a: A): A2 = a
Why? A2 is an A An A is not an A2, so we cannot convert an A to an A2 as it may be an A3 or some other subtype that does not conform to A2.

Subtyping generic types

Upon understanding the material out there, we concluded that the function I => O, or Function1[I, O] type, where Function1 is a generic type, is the best example for demonstrating subtyping. First, we use functions all the time; second, it is already predefined for us.

Subtyping over the return type

We first focus on the case of a return type:

Converting I => O2 to I => O

Example def convert[I, O, O2 <: O](f: I => O2): I => O = f
Outcome ✅ compiles
Why? The O2 that is returned from f is also an O.

Converting I => O to I => O2

Example def convert[I, O, O2 <: O](f: I => O): I => O2 = f
Outcome ❌ does not make sense
Why? An O is not necessarily an O2 when returning.

Therefore, if O2 <: O, then (I => O2) <: (I => O) . A function with a more specific return type is a subtype of a less specific return type.

Subtyping over the input type

Let's try to do the above but on the input type:

Converting I2 => O to I => O

Example def convert[I, I2 <: I, O](f: I2 => O2): I => O = f
Outcome ❌ does not make sense
Why? f only accepts I2, and does not support I values.

Converting I => O to I2 => O

Example def convert[I, I2 <: I, O](f: I => O): I2 => O = f
Outcome ✅ compiles
Why? f accepts all I2 values, as it accepts all I values.

Then, I2 <: I means that (I => O) <: (I2 => O), which is the other way round compared to the relationship we had for the output type.

What we learned

We ended up with 2 cases:

When inner subtype Then generic/outer subtype
O2 <: O (I => O2) <: (I => O)
I2 <: I (I => O) <: (I2 => O)

Before we jump onto the next step, make sure you have really understood the above. In case you haven't, write it out on a piece of paper, spend a good 15 minutes going back and forth.

What we learned, in terminology

Now that we've built an intuition around this, let's name what we have just done:

Terminology applied to Function1

Function1 is defined as:
trait Function1[-T1, +R] {
  def apply(v1: T1): R

The - and the + modifiers on the type parameters of Function1 signifies a demand to make Function1 possible to subtype based on its input (-) and output (+) type parameters; and that is is all that is meant by "contravariant position" and "covariant position" respectively.

Enabling subtyping brings the user additional powers, but what if we don't add these modifiers?

The most basic generic type

By default in Scala, a generic type such as trait X[A] does not possess subtyping. Let's define a simple subtype relationship and a generic type:

trait Z
// Z2 <: Z
trait Z2 extends Z
trait X[A]

Now try to convert X[Z2] to X[Z], and we get:

def convert(x2: X[Z2]): X[Z] = x2
-- [E007] Type Mismatch Error: -------------------------------------------------
1 |def x: X[Z] = x2
  |              ^^
  |              Found:    X[Z2]
  |              Required: X[Z]

The compiler sees no way to see X[Z2] as X[Z]. This is what is meant by Invariance: there is no relationship between X[Z2] and X[Z].

Change trait X[A] to trait X[+A], and now this function compiles!

def convert(x2: X[Z2]): X[Z] = x2

Change it to trait X[-A], and this compiles:

def convert(x: X[Z]): X[Z2] = x

Finalising CIC

Lastly, we ought to understand the relationship between Covariance, Invariance and Contravariance. The way I think about it is that "Invariance is the intersection of Contravariance and Covariance". Function1 is a great example to use because saying A => A we say Function1[-A, +A], combining both contravariance and covariance.

Then we try to convert A => A to A2 => A2:

Example def convert[A, A2 <: A](f: A => A): A2 => A2 = f
Outcome ❌ does not make sense
Why? f can return A which is more than A2 that is expected in the conversion.

When we try to convert A2 => A2 to A => A:

Example def convert[A, A2 <: A](f: A2 => A2): A => A = f
Outcome ❌ does not make sense
Why? f only takes A2 as inputs, yet we require a function that will take any A.

The only case where these examples work is if A2 = A, taking us back to the idea of invariance: regardless of a subtyping relationship of the inner types, there is no subtyping relationship of the generic type if it is invariant. To complete your understanding, try the following in a Scala console:

type X[A] = A => A
type X[-A] = A => A
type X[+A] = A => A

As next steps, read through the code of List and Function1 to learn about context bounds.

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

How our 100 algorithms look

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  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.
Screenshot of an example algorithm demonstrating the listed features

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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
  2. Check a binary tree is balanced
  3. Print a binary tree
  4. Remove duplicates from an unsorted List
  5. Make a queue using stacks (Lists in Scala)
  6. Find height of binary tree
  7. Single-elimination tournament tree
  8. Reverse Polish Notation calculator
  9. Quick Sort sorting algorithm in pure immutable Scala
  10. Check word in grid (depth-first search)
  11. Maximum wait at a fuel station
  12. Find minimum missing positive number in a sequence
  13. Least-recently used cache (LRU)
  14. Count pairs of a given expected sum
  15. Binary heap (min-heap)
  16. Compute a Roman numeral for an Integer, and vice-versa
  17. Compute keypad possibilities
  18. Matching parentheses algorithm with foldLeft and a state machine
  19. Traverse a tree Breadth-First, immutably
  20. Read a matrix as a spiral
  21. Remove duplicates from a sorted list (state machine)
  22. Token Bucket Rate Limiter
  23. Check word in grid (stack-safe)
  24. Leaky Bucket Rate Limiter
  25. Merge Sort: stack-safe, tail-recursive, in pure immutable Scala, N-way
  26. Median of two sorted arrays
  27. Longest increasing sub-sequence length
  28. Reverse first n elements of a queue
  29. Binary search a generic Array
  30. Game of Life
  31. Merge Sort: in pure immutable Scala
  32. Make a queue using Maps
  33. Is an Array a permutation?
  34. Count number of contiguous countries by colors
  35. Add numbers without using addition (plus sign)
  36. Tic Tac Toe MinMax solve
  37. Run-length encoding (RLE) Encoder
  38. Print Alphabet Diamond
  39. Find kth largest element in a List
  40. Balanced parentheses algorithm with tail-call recursion optimisation
  41. Reverse a String's words efficiently
  42. Count number of changes (manipulations) needed to make an anagram with an efficient foldLeft
  43. Count passing cars
  44. Count dist intersections
  45. Establish execution order from dependencies
  46. Counting inversions of a sequence (array) using a Merge Sort
  47. Longest common prefix of strings
  48. Check if an array is a palindrome
  49. Compute missing ranges
  50. Check a directed graph has a routing between two nodes (depth-first search)
  51. Compute nth row of Pascal's triangle
  52. Run-length encoding (RLE) Decoder
  53. Check if a number is a palindrome
  54. In a range of numbers, count the numbers divisible by a specific integer
  55. Merge intervals
  56. Compute minimum number of Fibonacci numbers to reach sum
  57. Find the longest palindrome within a string
  58. Find the index of a substring ('indexOf')
  59. Reshape a matrix
  60. Compute the steps to transform an anagram only using swaps
  61. Compute modulo of an exponent without exponentiation
  62. Closest pair of coordinates in a 2D plane
  63. Find the contiguous slice with the minimum average
  64. Compute maximum sum of subarray (Kadane's algorithm)
  65. Pure-functional double linked list
  66. Binary search in a rotated sorted array
  67. Check if a directed graph has cycles
  68. Rotate Array right in pure-functional Scala - using an unusual immutable efficient approach
  69. Check a binary tree is a search tree
  70. Length of the longest common substring
  71. Sliding Window Rate Limiter
  72. Tic Tac Toe board check
  73. Find an unpaired number in an array
  74. Check if a String is a palindrome
  75. Count binary gap size of a number using tail recursion
  76. Remove duplicates from a sorted list (Sliding)
  77. Monitor success rate of a process that may fail
  78. Least-recently used cache (MRU)
  79. Find sub-array with the maximum sum
  80. Find the minimum absolute difference of two partitions
  81. Find maximum potential profit from an array of stock price
  82. Fibonacci in purely functional immutable Scala
  83. Fizz Buzz in purely functional immutable Scala
  84. Find triplets that sum to a target ('3Sum')
  85. Find combinations adding up to N (non-unique)
  86. Find the minimum item in a rotated sorted array
  87. Make a binary search tree (Red-Black tree)
  88. Mars Rover
  89. Find combinations adding up to N (unique)
  90. Find indices of tuples that sum to a target (Two Sum)
  91. Count factors/divisors of an integer
  92. Compute single-digit sum of digits
  93. Fixed Window Rate Limiter
  94. Traverse a tree Depth-First
  95. Reverse bits of an integer
  96. Check Sudoku board
  97. Find k closest elements to a value in a sorted Array
  98. Print a binary tree vertically
  99. QuickSelect Selection Algorithm (kth smallest item/order statistic)
  100. Rotate a matrix by 90 degrees clockwise

Explore the 22 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. Variance
  21. View
  22. Zip

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