CSC 172 Data Structures and Algorithms. Fall 2017 TuTh 3:25 pm 4:40 pm Aug 30- Dec 22 Hoyt Auditorium

Transcription

1 CSC 172 Data Structures and Algorithms Fall 2017 TuTh 3:25 pm 4:40 pm Aug 30- Dec 22 Hoyt Auditorium

2 Announcement Coming week: Nov 19 Nov 25 No Quiz No Workshop No New Lab Monday and Tuesday: regular Lab sessions TAs will be there to help you Workshop on 26 th (Sunday) We will let you know

3 Fundamental data structure for - Storing (key, value) pairs - Allowing for efficient insertion, deletion, and search for values given keys BINARY SEARCH TREES

4 Managing (Key, Value) Pairs (username, password) MapReduce framework Domain Name System Dictionary lookup Map (string àint) Binary Search Tree is a good data structure for maintaining (key, value) pairs

5 Binary Search Tree & Its Main Property Key = x Value BST keys x BST keys x

6 Example of BST

7 Example of BST Inorder traversal lists all keys in non-decreasing order!

8 search(tree, key) Basic Operations minimum(tree), maximum(tree) successor(tree, node) predecessor(tree, node) insert(tree, node) node has (key, value) delete(tree, node) node has (key, value)

9 BSTNode in Java class BSTNode <K, V> { K key; V value; BSTNode<K,V> left; BSTNode<K,V> right; BSTNode<K,V> parent; //Note: we can include parent too! };

10 Search in a BST

11 Minimum and Maximum

12 Successor and Predecessor If we perform inorder traversal of a binary tree, the neighbors of a given node are called Predecessor (the node lies behind of given node) Successor (the node lies ahead of given node). Assume: In-order traversal of BST Produces: 1, 3, 4, 6, 7, 8, 9, 12 Then: Predecessor of 8 is 7 Successor of 8 is 9

13 Successor and Predecessor (cont.) Assume: In-order traversal of BST Produces: 1, 3, 4, 6, 7, 8, 9, 12 Then: Predecessor of 8 is 7 Successor of 7 is 8

14 How to find Successor of a Node v If v has a right sub-tree, we will find the successor there Min of Right Subtree If v does not have a right sub-tree: We need to find the node x whose predecessor is v How to find predecessor of node x?: Maximum node of x s left sub-tree Go left, and then go all the way down towards right. Do the reverse: Go up (y à z) until y is the left child of z.

15 Successor If v has a right branch: successor(v) = minimum(right-branch) Else, successor(v) = the first ancestor u with another ancestor as a left child

16 Predecessor If v has a left branch: predecessor(v) = maximum(left-branch) Else, predecessor(v) = the first ancestor u with another ancestor as a right child

17 Insert

18 Delete Node has 1 Child

19 Delete Node Has 2 Children

20 Run Times of Basic Operations search(tree, key) minimum(tree) maximum(tree) successor(tree, node) predecessor(tree, node) insert(tree, node) node has (key, value) delete(tree, node) node has (key, value) All run in time O(h) h is the height of the tree

21 Random BST Consider storing a dictionary using a BST Randomize the word order Insert (word, meaning) pairs into the BST It can be shown that the expected height of a random BST is O(log n)

22 Summary A good data structure for storing (key, value ) pair If the tree is well-balanced, we can perform all the major operations in O(log n) time Our next topic would be: How to create a balanced binary search tree?

23 (Self) Balanced Search Trees

24 Variations of Balanced Binary Search Tree AVL Red-black 2-4 Splay B+ Tree

25 BSTs are Potentially Good In O(h)-time, where h is the height of the tree, we can perform Search Minimum, maximum Predecessor, successor Insert, Delete An n-node binary tree must have height h = Ω(log n) The best we can hope for is h = O(log n)

26 Intuitively, How to Keep a Tree s Height Small? For every internal node v left branch of v right branch of v Exactly the same reason quick sort needs balanced partition AVL trees maintain this property by Keeping the heights of left and the right subtrees roughly equal AVL is more rigid, faster search

27 Named after - Georgy Adelson-Velsky and Evgenii Landis - First self-balancing birany search tree Idea: rebalance the tree after an insert/delete AVL TREES

28 AVL Trees Balanced node: A node v is balanced if its left subtree and right subtree have heights differ by at most 1 An AVL tree is a BST in which every internal node is balanced. Theorem: an AVL tree on n nodes has O(log n)-height

29 Recurrence for n(h) For convenience, heights measured to the NULLs n(1) = 1 n(2) = 2 n(h) = 1+n(h 1) + n(h 2) h h 1 h 2

30 Old Friend: Fibonacci n(h) = h+2 ( 1/ ) h+2 p 5 1 = 1+p n(h) = (1.6 h ), h = O(log n) Not reqd. for Exam or Quiz

31 But how do we maintain AVL property? After an insert One subtree might be taller than the other by 2 Potentially affect the balance of all nodes up to the root Rebalance After a delete One subtree might be shorter than the other by 2 Potentially affect the balance of all nodes up to the root Rebalance

32 Insert unbalanced

33 Balance Let s define the balanceness of a node Balance(v) = height(v.left) height(v.right) We want v s balance to be in {-1, 0, 1} balance = -1 means v is right heavy balance = 1 means v is left heavy After inserting a new node Let a be the first node on the path back to the root that s not balanced, then a s new balance is -2 or 2

34 Example: RR case Single Rotation 0 h T 1 h h +1 h T 1 T 2 h T 3 h +1 T 2 T 3 New node Done!

35 Example: RL case Double Rotation 0 h T 1 h h h 1 T 2 T 3 h h 1 T 2 T 3 T 4 T 1 T 4 New node Done!

36 Picture from Wikipedia 35

37 Delete First, delete as in normal BST But nodes on path to root might become unbalanced Second, fix unbalanced nodes one by one using exactly the same strategy Might require up to O(log n) rotations Insert & delete run in time O(log n)

38 Theorems Insertion: After fixing one node (with a single/double rotation) the tree becomes balanced (i.e. AVL again) why? Deletion: Fixing one node does not necessarily balance the tree Need more fixing up to the root Think of an example AVL-tree for which Ω(log n) fixes are necessary after a deletion!

39 Acknowledgement A lot of the slides are taken from various sources including the Internet and Prof. Hung Ngo slides.