Inverting a Binary Tree in One Line of Elm

Joël Quenneville talks about Inverting a binary tree with 1 line of Elm at the Elm Online Meetup on September 1st 2021.

https://www.youtube.com/watch?v=dSMB3rsufC8

Transcript

1. Folding Inverting a binary tree with a single line of Elm Joël Quenneville @joelquen
2. Joël Quenneville @joelquen
3. Invert a binary tree... Joël Quenneville @joelquen
4. ... on a whiteboard! Joël Quenneville @joelquen
5. Joël Quenneville @joelquen
6. fold (\v l r -> Node v r l) Empty Joël Quenneville @joelquen
7. 33 chars (61 idiomatic) Joël Quenneville @joelquen
8. How does this work? Joël Quenneville @joelquen
9. Tree overview Joël Quenneville @joelquen
10. Defining a binary tree type Tree a = Empty | Node a (Tree a) (Tree a) Joël Quenneville @joelquen
11. myTree : Tree Int myTree = Empty Joël Quenneville @joelquen
12. myTree : Tree Int myTree = Node 1 Empty Empty Joël Quenneville @joelquen
13. myTree : Tree Int myTree = Node 1 (Node 2 Empty Empty) (Node 3 Empty Empty) Joël Quenneville @joelquen
14. myTree : Tree Int myTree = Node 1 (Node 2 (Node 4 Empty Empty) (Node 5 Empty Empty) ) (Node 3 Empty (Node 6 Empty Empty) ) Joël Quenneville @joelquen
15. Joël Quenneville @joelquen
16. What is "inverting"? Joël Quenneville @joelquen
17. Flip horizontally Joël Quenneville @joelquen
18. Joël Quenneville @joelquen
19. → Left children are red Joël Quenneville @joelquen
20. → Left children are red → Right children are green Joël Quenneville @joelquen
21. 6 ↔ Empty Joël Quenneville @joelquen
22. 5 ↔ 4 Joël Quenneville @joelquen
23. 3 ↔ 2 Joël Quenneville @joelquen
24. Back to the one-liner Joël Quenneville @joelquen
25. What is fold? fold (\v l r -> Node v r l) Empty yourTree Joël Quenneville @joelquen
26. fold : (a -> b -> b -> b) -> b -> Tree a -> b Joël Quenneville @joelquen
27. fold : (a -> b -> b -> b) -- ??? -> b -- ??? -> Tree a -- tree being folded -> b -- return value Joël Quenneville @joelquen
28. Mental models Joël Quenneville @joelquen
29. Reducing Joël Quenneville @joelquen
30. Reducing Constructor Replacement Joël Quenneville @joelquen
31. Reducing Constructor Replacement Abstraction over Recursion Joël Quenneville @joelquen
32. Reducing Many → Aggregate Joël Quenneville @joelquen
33. Sum → 21 Joël Quenneville @joelquen
34. Count → 6 Joël Quenneville @joelquen
35. Min → 1 Joël Quenneville @joelquen
36. fold : (a -> b -> b -> b) -- aggregation -> b -- start accumulator -> Tree a -> b Joël Quenneville @joelquen
37. Aggregating function add3 : Int -> Int -> Int -> Int add3 currentVal leftAcc rightAcc = currentVal + leftAcc + rightAcc Joël Quenneville @joelquen
38. Summing a tree fold add3 -- aggregation 0 -- start accumulator tree Joël Quenneville @joelquen
39. Further discussion → What other common operations can be described as aggregations of a collection? Joël Quenneville @joelquen
40. Tree → Tree? Joël Quenneville @joelquen
41. Constructor Replacement Joël Quenneville @joelquen
42. type Tree a = Node a (Tree a) (Tree a) | Empty Joël Quenneville @joelquen
43. Signatures type Tree a = Node a (Tree a) (Tree a) -- a -> Tree a -> Tree a -> Tree a | Empty -- Tree a Joël Quenneville @joelquen
44. Replace Tree a with b type Tree a = Node a (Tree a) (Tree a) -- a -> b -> b -> b | Empty -- b Joël Quenneville @joelquen
45. fold : (a -> b -> b -> b) -> b -> Tree a -> b Joël Quenneville @joelquen
46. Coincidence? Joël Quenneville @joelquen
47. fold : (a -> b -> b -> b) -- Replace Node -> b -- Replace Empty -> Tree a -> b Joël Quenneville @joelquen
48. add3 : Int -> Int -> Int -> Int add3 a b c = a + b + c Joël Quenneville @joelquen
49. Summing a tree Tree.fold add3 -- replace Node with add3 0 -- replace Empty with 0 someTree Joël Quenneville @joelquen
50. Node 1 (Node 2 (Node 4 Empty Empty) Empty ) (Node 3 Empty Empty) Joël Quenneville @joelquen
51. add3 1 (add3 2 (add3 4 Empty Empty) Empty ) (add3 3 Empty Empty) Joël Quenneville @joelquen
52. add3 1 (add3 2 (add3 4 0 0) 0 ) (add3 3 0 0) Joël Quenneville @joelquen
53. add3 1 (add3 2 (add3 4 0 0) -- 4 0 ) (add3 3 0 0) Joël Quenneville @joelquen
54. add3 1 (add3 2 -- | 4 -- | 6 0 -- | ) (add3 3 0 0) Joël Quenneville @joelquen
55. add3 1 6 (add3 3 0 0) -- 3 Joël Quenneville @joelquen
56. add3 1 --| 6 --| 10 3 --| Joël Quenneville @joelquen
57. Folding new trees Joël Quenneville @joelquen
58. fold Node -- replace Node with Node Empty -- replace Empty with Empty someTree Joël Quenneville @joelquen
59. Lesson: We can use fold to disassemble and rebuild a tree Joël Quenneville @joelquen
60. node : a -> Tree a -> Tree a -> Tree a node value left right = Node value left right Joël Quenneville @joelquen
61. inverseNode : a -> Tree a -> Tree a -> Tree a inverseNode value left right = Node value right left Joël Quenneville @joelquen
62. Tree.fold inverseNode -- replace Node with inverseNode Empty -- replace Empty with Empty someTree Joël Quenneville @joelquen
63. Node 1 (Node 2 (Node 4 Empty Empty) Empty ) (Node 3 Empty Empty) Joël Quenneville @joelquen
64. inverseNode 1 (inverseNode 2 (inverseNode 4 Empty Empty) Empty ) (inverseNode 3 Empty Empty) Joël Quenneville @joelquen
65. inverseNode 1 (inverseNode 2 (inverseNode 4 Empty Empty) -- swap two empties Empty ) (inverseNode 3 Empty Empty) -- swap two empties Joël Quenneville @joelquen
66. inverseNode 1 (inverseNode 2 (Node 4 Empty Empty) -- swap this Empty -- with this ) (Node 3 Empty Empty) Joël Quenneville @joelquen
67. inverseNode 1 (Node 2 Empty (Node 4 Empty Empty) ) (Node 3 Empty Empty) Joël Quenneville @joelquen
68. inverseNode 1 (Node 2 -- | Empty -- | swap this (Node 4 Empty Empty) -- | ) (Node 3 Empty Empty) -- with this Joël Quenneville @joelquen
69. Node 1 (Node 3 Empty Empty) (Node 2 Empty (Node 4 Empty Empty) ) Joël Quenneville @joelquen
70. Success! Joël Quenneville @joelquen
71. Further discussion → How can this technique be applied to other data structures? Joël Quenneville @joelquen
72. Recursion Joël Quenneville @joelquen
73. Pseudocode IF EMPTY your_base_case ELSE RECURSE LEFT RECURSE RIGHT your_action Joël Quenneville @joelquen
74. Abstraction over recursion fold : (a -> b -> b -> b) -- action to do recursively -> b -- base case -> Tree a -- tree -> b -- return Joël Quenneville @joelquen
75. fold inverseNode -- recursively inverse node Empty -- base case tree Joël Quenneville @joelquen
76. fold : (a -> b -> b -> b) -> b -> Tree a -> b fold fn baseCase tree = case tree of Node val left right -> fn val (fold fn baseCase left) (fold fn baseCase right) Empty -> baseCase Joël Quenneville @joelquen
77. Layers of abstraction Joël Quenneville @joelquen
78. Further discussion → What other abstractions might built on top of recursion? Joël Quenneville @joelquen
79. Summary Joël Quenneville @joelquen
80. 3 Perspectives on those args fold : (a -> b -> b -> b) -- ??? -> b -- ??? -> Tree a -> b Joël Quenneville @joelquen
81. Reducing 1. Aggregation 2. Initial accumulator Joël Quenneville @joelquen
82. Constructor replacement 1. Replaces Node 2. Replaces Empty Joël Quenneville @joelquen
83. Recursion 1. Recursive case 2. Base case Joël Quenneville @joelquen
84. Invert a binary tree in 1 line of Elm fold (\v l r -> Node v r l) Empty tree Joël Quenneville @joelquen