B-Tree & SortedSet in Swift Collections

B-Tree & SortedSet in Swift Collections by Yu-Che Cheng @ COSCUP 2023

https://coscup.org/2023/en/session/PZSRQ8

Transcript

1. Follow 「iamhands0me」on 鄭宇哲, Software Engineer @ B-Tree & SortedSet in Swift Collections 1
2. Swift Collections 2 • Deque, OrderedSet • Heap, BitArray • SparseSet, SortedSet An open-source package of data structure implementations for the Swift 1.0.4 Latest 1.1 under development
3. class Library { var books: SomeCollection<Int> /// let book = library.lendBook(10000) func lendBook(_ id: Int) -> Int? { } /// library.returnBook(10000) func returnBook(_ id: Int) { } /// let books = library.lendBooks(10001..<11000) func lendBooks(_ ids: Range<Int>) -> [Int] { } } Scenario 3 Design a Library class that can lend/return a book and lend a series of books
4. class Library { var books: Set<Int> /// let book = library.lendBook(10000) func lendBook(_ id: Int) -> Int? { return books.remove(id) } /// library.returnBook(10000) func returnBook(_ id: Int) { books.insert(id) } /// let books = library.lendBooks(10000..<11000) func lendBooks(_ ids: Range<Int>) -> [Int] { var existedBooks: [Int] = [] for id in ids { if let book = books.remove(id) { existedBooks.append(book) } } return existedBooks } } 4 Set<Int> Set 10000 12300 10050 700 𝑂(1) 𝑂(1) 𝑂 𝐾 𝐾: Number of ids in the range for id in ids {
5. 5 We need a collection that can 1. Store unique keys 2. Keep the keys being sorted
6. 6 We need a collection that can 1. Store unique keys 2. Keep the keys being sorted => Binary Search Tree (Sorted Binary Tree/Sorted Set) Binary Search Tree 10000 12300 10050 700
7. class Library { var books: SortedSet<Int> /// let book = library.lendBook(10000) func lendBook(_ id: Int) -> Int? { return books.remove(id) } /// library.returnBook(10000) func returnBook(_ id: Int) { books.insert(id) } /// let books = library.lendBooks(10000..<11000) func lendBooks(_ ids: Range<Int>) -> [Int] { var subSequence = books[ids.lowerBound ..< ids.upperBound] let existedBooks = Array(subSequence) for id in existedBooks { books.remove(id) } return existedBooks } } Binary Search Tree 7 SortedSet<Int> 𝑂(log 𝑛) 𝑂(log 𝑛) 𝑂 log 𝑛 + 𝑘 𝑘: Number of ids in the subSequence 10000 12300 10050 700 var subSequence = books[ids.lowerBound ..< ids.upperBound] for id in existedBooks {
8. Red-black Tree AVL Tree Self-balancing binary search tree 8 B-Tree Source: https://en.wikipedia.org/wiki/Red-black_tree Source: https://en.wikipedia.org/wiki/AVL_tree Source: https://en.wikipedia.org/wiki/B-tree
9. Why B-Tree? 9
10. Why not Array? 10 (sorted Array)
11. class Library { var books: [Int] /// let book = library.lendBook(10000) func lendBook(_ id: Int) -> Int? { guard let index = books.firstIndex(of: id) else { return nil } return books.remove(at: index) } /// library.returnBook(10000) func returnBook(_ id: Int) { let index = books.firstIndex { $0 > id } ?? books.endIndex if !books.contains(id) { books.insert(id, at: index) } } /// let books = library.lendBooks(10000..<11000) func lendBooks(_ ids: Range<Int>) -> [Int] { let startIndex = books.firstIndex { $0 >= ids.lowerBound } ?? books.endIndex let endIndex = books.firstIndex { $0 >= ids.upperBound } ?? books.endIndex defer { books.removeSubrange(startIndex ..< endIndex) } return Array(books[startIndex ..< endIndex]) } } Array 11 [Int] 𝑂(𝑛) 𝑂(𝑛) 𝑂 𝑛 0 700 1 10000 2 10050 3 12300 books.removeSubrange(startIndex ..< endIndex)
12. 12 Source: https://github.com/attaswift/BTree Array vs. red-black tree 10!
13. 13 Source: https://github.com/attaswift/BTree Array vs. red-black tree 10000 12300 10050 700 Array 0 700 1 10000 2 10050 3 12300 0x42B7 0x1C3E 0x54D9 0x877A 0x0800 0x0802 0x0804 0x0806 Tree
14. Small Array is better for sorted list 14
15. What if the list gets too long? 15
16. Split into two Arrays 16 B-Tree Source: https://en.wikipedia.org/wiki/B-tree
17. 17 Source: https://github.com/attaswift/BTree B-tree vs. Array vs. red-black tree
18. 18 SortedSet<Element: Comparable> _root: BTree<Element> Node<Key: Comparable> count BTree<Key: Comparable> root: Node<Key> public func contains(_ member: Element) -> Bool public func insert(_ newMember: Element) public func remove(_ member: Element) -> Element? capacity depth children: [Node<Key>] keys: [Key]
19. 1. Search for the key X. Start at the root 2. Find the slot of first key which equal or greater than X 3. If it’s equal, return True 4. Else, go to the child node at the slot 5. Repeat the steps from 2 till no more traversal 6. Return False 19 public func contains(_ member: Element) -> Bool 80 35 63 60 40 71 . . . . . . 84 63 contains( ) slot 1 slot 2
20. slot 3 1. Insert for the key X 2. Find the slot of first key which greater than X till no more traversal 3. If not full, then insert 20 public func insert(_ newMember: Element) 80 35 63 60 40 71 . . . . . . 84 insert( ) slot 1 67
21. slot 3 1. Insert for the key X 2. Find the slot of first key which greater than X till no more traversal 3. If not full, then insert 21 public func insert(_ newMember: Element) 80 35 60 40 71 . . . . . . 84 67 slot 1 63
22. 1. Insert for the key X 2. Find the slot of first key which greater than X till no more traversal 3. If not full, then insert 4. Else, split! 22 public func insert(_ newMember: Element) 80 35 60 40 71 . . . . . . 84 67 63 55 insert( ) slot 1 slot 1
23. Splinter key rightChild 1. Insert for the key X 2. Find the slot of first key which greater than X till no more traversal 3. If not full, then insert 4. Else, split! 23 public func insert(_ newMember: Element) 80 35 60 40 . . . . . . 84 55 insert( ) slot 1 slot 1 71 67 63
24. Splinter key rightChild 1. Insert for the key X 2. Find the slot of first key which greater than X till no more traversal 3. If not full, then insert 4. Else, split! 24 public func insert(_ newMember: Element) 80 35 60 40 . . . . . . 84 55 insert( ) slot 1 slot 1 71 67 63
25. Splinter 63 key rightChild 1. Insert for the key X 2. Find the slot of first key which greater than X till no more traversal 3. If not full, then insert 4. Else, split! 25 public func insert(_ newMember: Element) 80 35 . . . . . . 84 71 67 60 40 55 slot 1
26. Splinter key rightChild 1. Insert for the key X 2. Find the slot of first key which greater than X till no more traversal 3. If not full, then insert 4. Else, split! 26 public func insert(_ newMember: Element) 80 35 71 . . . . . . 84 67 63 60 40 55 slot 1
27. 1. Insert for the key X 2. Find the slot of first key which greater than X till no more traversal 3. If not full, then insert 4. Else, split! 27 public func insert(_ newMember: Element) 80 35 60 40 71 . . . 84 67 63 insert( ) slot 3 slot 1 28 20 55
28. Splinter 63 key rightChild 1. Insert for the key X 2. Find the slot of first key which greater than X till no more traversal 3. If not full, then insert 4. Else, split! 28 public func insert(_ newMember: Element) . . . 84 71 67 60 40 slot 3 28 20 80 35 55
29. Splinter 35 key rightChild 1. Insert for the key X 2. Find the slot of first key which greater than X till no more traversal 3. If not full, then insert 4. Else, split! 29 public func insert(_ newMember: Element) . . . 84 60 40 55 67 71 28 20 80 63
30. BTree<Key: Comparable> Splinter 35 key rightChild 1. Insert for the key X 2. Find the slot of first key which greater than X till no more traversal 3. If not full, then insert 4. Else, split! 30 public func insert(_ newMember: Element) . . . 84 80 63 60 40 55 67 71 28 20 root
31. BTree<Key: Comparable> 1. Insert for the key X 2. Find the slot of first key which greater than X till no more traversal 3. If not full, then insert 4. Else, split! 31 public func insert(_ newMember: Element) . . . 84 80 63 60 40 55 67 71 28 20 root 35
32. 1. Remove for the key X 2. Find the slot of first key which equal or greater than X till no more traversal 3. Pop last key from the child node at the slot, replace X with the popped key 32 public func remove(_ member: Element) -> Element? 84 80 63 60 40 55 67 71 28 20 35 80 remove( ) slot 1 slot 1 90
33. 1. Remove for the key X 2. Find the slot of first key which equal or greater than X till no more traversal 3. Pop last key from the child node at the slot, replace X with the popped key 33 public func remove(_ member: Element) -> Element? 84 63 60 40 55 67 71 28 20 35 80 remove( ) slot 1 slot 1 90
34. 1. Remove for the key X 2. Find the slot of first key which equal or greater than X till no more traversal 3. Pop last key from the child node at the slot, replace X with the popped key 34 public func remove(_ member: Element) -> Element? 84 63 60 40 55 67 71 28 20 35 80 remove( ) slot 1 slot 1 90
35. 1. Remove for the key X 2. Find the slot of first key which equal or greater than X till no more traversal 3. Pop last key from the child node at the slot, replace X with the popped key 4. Balance the child node if needed 35 public func remove(_ member: Element) -> Element? 84 80 63 60 40 55 67 71 28 20 35 80 remove( ) slot 1 slot 1 90
36. 1. If of child node equal or greater than , return 2. If of left child node greater than , rotate right from left 3. If of right child node greater than , rotate left from right 36 func balance(atSlot slot: Int) 84 80 40 55 67 71 28 20 35 slot 1 slot 1 90 count capacity / 2 count capacity / 2 count capacity / 2 60 63
37. 1. If of child node equal or greater than , return 2. If of left child node greater than , rotate right from left 3. If of right child node greater than , rotate left from right 37 func balance(atSlot slot: Int) 84 80 40 55 67 71 28 20 35 slot 1 slot 1 90 count capacity / 2 count capacity / 2 count capacity / 2 63 60
38. 1. If of child node equal or greater than , return 2. If of left child node greater than , rotate right from left 3. If of right child node greater than , rotate left from right 38 func balance(atSlot slot: Int) 84 40 55 71 28 20 35 slot 1 slot 1 90 count capacity / 2 count capacity / 2 count capacity / 2 63 60 71 remove( ) 67
39. 1. If of child node equal or greater than , return 2. If of left child node greater than , rotate right from left 3. If of right child node greater than , rotate left from right 39 func balance(atSlot slot: Int) 84 40 55 28 20 35 slot 1 slot 1 90 count capacity / 2 count capacity / 2 count capacity / 2 63 60 71 remove( ) 67
40. 1. If of child node equal or greater than , return 2. If of left child node greater than , rotate right from left 3. If of right child node greater than , rotate left from right 4. Else, collapse the child 40 func balance(atSlot slot: Int) 40 55 28 20 35 slot 1 slot 1 90 count capacity / 2 count capacity / 2 count capacity / 2 63 60 71 remove( ) 67 84
41. 1. If of child node equal or greater than , return 2. If of left child node greater than , rotate right from left 3. If of right child node greater than , rotate left from right 4. Else, collapse the child 41 func balance(atSlot slot: Int) 40 55 28 20 slot 1 90 count capacity / 2 count capacity / 2 count capacity / 2 63 60 67 84 35
42. 1. If of child node equal or greater than , return 2. If of left child node greater than , rotate right from left 3. If of right child node greater than , rotate left from right 4. Else, collapse the child 42 func balance(atSlot slot: Int) 40 55 28 20 90 count capacity / 2 count capacity / 2 count capacity / 2 63 60 67 84 35
43. BTree<Key: Comparable> 1. If of child node equal or greater than , return 2. If of left child node greater than , rotate right from left 3. If of right child node greater than , rotate left from right 4. Else, collapse the child 43 func balance(atSlot slot: Int) 40 55 90 count capacity / 2 count capacity / 2 count capacity / 2 63 67 84 28 20 35 60 root
44. BTree<Key: Comparable> 1. If of child node equal or greater than , return 2. If of left child node greater than , rotate right from left 3. If of right child node greater than , rotate left from right 4. Else, collapse the child 44 func balance(atSlot slot: Int) 40 55 90 count capacity / 2 count capacity / 2 count capacity / 2 63 67 84 28 20 35 60 root
45. Reference 45 • https://github.com/attaswift/BTree • https://github.com/vihanb/swift-collections/tree/sorted-collections/Sources/SortedCollections