Search Trees. COMPSCI 355 Fall 2016

Transcription

1 Search Trees COMPSCI 355 Fall 2016

2 2-4 Trees Search Trees AVL trees Red-Black trees Splay trees Multiway Search Trees (2, 4) Trees External Search Trees (optimized for reading and writing large blocks) B trees B+ trees

3 Red-Black Trees Same asymptotic complexity as AVL and 2-4 Trees. O(1) time for trinode restructuring O(log n) time to search, insert, delete O(log n) recoloring operations per update But at most 2 structural operations per update. More popular than AVL and 2-4 trees in practice.

4 Defining Properties Root is black. Every leaf is black. Children of a red node are black. Every leaf has the same black depth

5 Red-Black 2-4 Collapse each red child into its parent. Height of red-black tree with n nodes is at most 2lg(n+1)

6 Red-Black 2-4 Collapse each red child into its parent. Height of red-black tree with n nodes is at most 2lg(n+1)

7 Red-Black 2-4 Collapse each red child into its parent. Height of red-black tree with n nodes is at most 2lg(n+1)

8 Red-Black 2-4 Collapse each red child into its parent. Height of red-black tree with n nodes is at most 2lg(n+1)

9 Red-Black 2-4 Collapse each red child into its parent. Height of red-black tree with n nodes is at most 2lg(n+1)

10 Red-Black 2-4 Collapse each red child into its parent. Height of red-black tree with n nodes is at most 2lg(n+1)

11 Red-Black 2-4 Collapse each red child into its parent. Height of red-black tree with n nodes is at most 2lg(n+1)

12 Red-Black 2-4 Collapse each red child into its parent. Height of red-black tree with n nodes is at most 2lg(n+1)

13 2-4 Red-Black Color all nodes black. Keep the children of a 2-node black

14 2-4 Red-Black Color all nodes black. Keep the children of a 2-node black. Split up 3-nodes and 4-nodes

15 2-4 Red-Black Color all nodes black. Keep the children of a 2-node black. Split up 3-nodes and 4-nodes

16 2-4 Red-Black Color all nodes black. Keep the children of a 2-node black. Split up 3-nodes and 4-nodes

17 2-4 Red-Black Color all nodes black. Keep the children of a 2-node black. Split up 3-nodes and 4-nodes

18 2-4 Red-Black Color all nodes black. Keep the children of a 2-node black. Split up 3-nodes and 4-nodes

19 2-4 Red-Black Color all nodes black. Keep the children of a 2-node black. Split up 3-nodes and 4-nodes

20 Insertions Perform BST insertion. If new node is root, color it black; otherwise, color it red insert insert 16

21 Insertions Perform BST insertion. If new node is root, color it black; otherwise, color it red insert insert 16 Simple. Is that all there is to it?

22 Fixing a Double Red (Okasaki's Method) z z x x x D y D A y A z A y x C B z y D B C A B C D B C y Preserves postorder traversal: AxByCzD x z A B C D

23 Fixing a Double Red (Okasaki's Method) z y x z x y x z y x y z x y z A A B A A B B B C C C C D D D D A B C D Another double red might result, but it will be two levels closer to the root. Repeat as needed or until reaching the root, which may then be colored black.

24 Fixing a Double Red (Okasaki's Method) z z x x x D y D A y A z A y x C B z y D B C A B C D B C Another double red might result, but it will be two levels closer to the root. Repeat as needed or until reaching the root, which may then be colored black. y x z A B C D But amazingly, a double red can be fixed with a single trinode restructuring!

25 Fixing a Double Red Parent of new node has black sibling: trinode restructuring.

26 Fixing a Double Red Parent of new node has red sibling: recolor. And if this creates a new double red...?

27 Illustration Insert 5 5 The top node of the double-red has a red sibling, so we recolor.

28 Illustration Insert 5 5 The top node of the double-red has a red sibling, so we recolor.

29 Illustration The top node of the double-red has a black sibling, so we restructure

30 Illustration The top node of the double-red has a black sibling, so we restructure

31 Deletion Remove key Copy successor key to node and delete successor node.

32 Deletion Copy successor key to node and delete successor node. Simple! Now suppose we want to delete key 9. Well, just delete it!

33 Deletion But what if we want to delete key 8, 10, or 11?

34 Deletion The black depth property has been violated. In this case, we can just recolor.

35 Deletion The black depth property has been violated. In this case, we can just recolor.

36 Deletion But suppose we delete key 11 and then we delete key 10.

37 Deletion The black depth property no longer holds.

38 Removing a Double Black The basic idea is to double count the blackness of the problem node and look for a nearby red node to absorb the extra blackness.

39 Removing a Double Black In this case, the sibling of the double black node is black and has a red child, so we can rotate and recolor...

40 Removing a Double Black

41 Removing a Double Black There are two other cases to consider. Double black may propagate up to root. If you really want to delete a key from a RBT by hand without memorizing all the details: Transform the RBT to a (2, 4) tree. Delete the desired key. Transform the (2, 4) tree back to a RBT.