CSE 12 – Basic Data Structures Cynthia Bailey Lee Some slides and figures adapted from Paul Kube’s CSE 12 CS2 in Java Peer Instruction Materials by Cynthia Lee is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. Based on a work at Permissions beyond the scope of this license may be available at LeeCreative Commons Attribution-NonCommercial 4.0 International Licensehttp://peerinstruction4cs.org

Today’s Topics 1. Wrap up traversals from Tuesday  Level-order traversal  Also known as Breadth-First Search (BFS) 2. Binary Search Trees (BSTs)  Note: These are different from plain vanilla Binary Trees! They are a special kind of Binary Tree! 2

Level-order traversal AKA Breadth-first search

Level-order traversal  Also commonly called Breadth-First Search or BFS  As opposed to something like pre-order or post-order, which are Depth-First Search (DFS) algorithms

Level-order challenges  How do we know where to go next?  1, 2, 3—but there is no edge from 2 to 3!  While we’re still at 1, we must “remember” to come back to 3 after we’re done with 2  At 2, remember to come back to 4 and 5 after 3  At 8, remember to come back to 16 and 17 after 15  Etc…

How to “remember”  When we visit a node, add references (pointers) to its children to a queue  To know which node to visit next, remove next element from the queue

Tracing the queue in BFS Which shows the state of the queue at the end of “visiting” node #10? (“head” of the queue, i.e. the next element that will be removed, is the leftmost) A. 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11 B. 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 C. 20, 21 D. 1, 2, 5 E. Other/none/more

BFS code BFS(node) { Queue q = new Queue (); q.add(node); while (!q.isEmpty()) { current = q.remove(); System.out.println(current.getData()); } A. BFS(current.getLeftChild()); B. BFS(current.getRightChild()); C. q.add(current.getLeftChild()); D. q.add(current.getRighChild()); E. Other/none/more What goes here?

Reading quiz!

1. A total order must apply to data elements in a Binary Search Tree. A. TRUE B. FALSE Reading quiz!

2. An inorder traversal of a binary search tree visits the tree’s elements in sorted order. A. TRUE B. FALSE Reading quiz!

Binary Search Trees

Binary Search Tree (BST) RULE:  For every node:  Everything in its left subtree is less than it  Everything in its right subtree is greater than it

Binary Search Tree (BST) FACT:  The “in-order” traversal method, when applied to a BST, gives sorted order FACT:  Easy to find things: just recursively check if you should go left or right based on > or <

BST add() Create two BSTs by inserting the elements, one by one, in the order given below for the first letter of your last name :  A-F: {18, 9, 34, 3, 22}  G-L: {3, 18, 22, 9, 34}  M-R: {22, 9, 34, 3, 18}  S-Z: {34, 3, 9, 18, 22}  EVERYBODY do this for a 2 nd BST: {3, 9, 18, 22, 34}

What is the BEST CASE cost for doing find() in BST? A. O(1) B. O(log n) C. O(n) D. O(n log n) E. O(n 2 )

What is the WORST CASE cost for doing find() in BST? A. O(1) B. O(log n) C. O(n) D. O(n log n) E. O(n 2 )

What is the WORST CASE cost for doing find() in BST if the BST is complete ? A. O(1) B. O(log n) C. O(n) D. O(n log n) E. O(n 2 )

BALANCE!!!  The #1 issue to remember with BSTs is that they are great when balanced (O(log n) operations), and horrible when unbalanced (O(n) operations)  Balance depends on order of insert of elements  Over the years, people have devised many ways of making sure BSTs stay balanced no matter what order the elements are inserted

BST Balance Strategies

Red-Black trees  One of the most famous (and most tricky) strategies for keeping a BST balanced  Not guaranteed to be perfectly balanced, but “close enough” to keep O(log n) guarantee on operations

Red-Black trees  In addition to the requirements imposed on a binary search trees, red–black trees must meet these:  A node is either red or black.  The root is black.  All leaves (null children) are black.  Both children of every red node are black.  Every simple path from a given node to any of its descendant leaves contains the same number of black nodes.  (This is what guarantees “close” to balance)

Red-Black trees  Every simple path from a given node to any of its descendant leaves contains the same number of black nodes.  (This is what guarantees “close” to balance)

Insert procedure must maintain the invariants (this gets tricky)  Video:

Other BST balance strategies  Red-Black tree  AVL tree  Treap (BST + heap in one tree!) Other fun types of BST:  Splay tree  Rather than only worrying about balance, Splay Tree dynamically readjusts based on how often a particular item is searched for  Commonly-searched items move to the top, saving time  B-Tree  Like BST, but a node can have many children, not just 2

More practice: find()

 Practice finding each of these: 6, 25, 11

More practice: find()  How do we know that 12 is NOT in the tree?