Struktur Data & Algoritme Objectives Outline Examples

Struktur Data & Algoritme (

Data Structures & Algorithms

)

Denny (denny@cs.ui.ac.id) Suryana Setiawan (setiawan@cs.ui.ac.id)

Fakultas Ilmu Komputer Universitas Indonesia Semester Genap - 2004/2005

Version 2.0 - Internal Use Only

Tree

SDA/TREE/V2.0/2

Objectives

„ understands the definition and terminology of a general tree

„ know applications of the tree

„ know how to traverse of a tree

Outline

„ Tree

„example

„terminology/definition

„ binary tree

„ traversal of trees

„ iterator

Examples

„ a tree represents a hierarchy

„organization structure of a corporation

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Examples

„table of contents of a book

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Examples

„Unix or DOS/Windows file system

Terminology

„ A is the root node

„ B is the parent of D and E

„ C is the sibling of B

„ D and E are the children of B

„ D, E, F, G, I are external nodes, or leaves

„ A, B, C, H are internal nodes

„ The depth/level/path length of E is 2

„ The height of the tree is 3

„ The degree of node B is 2

„ Property: (# edges) = (#nodes) - 1

Tree Viewed Recursively

„ The sub-tree is also a tree!!

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Binary Tree

„ Binary tree: tree with all internal nodes of degree 2

„ Recursive View: Binary tree is either

„empty

„an internal node (the root) and two binary trees(left subtreeand right subtree)

„ Ordered/Search tree: the children of each node are ordered

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Examples of Binary Tree

„ arithmetic expression

Properties of Binary Trees

„ If we restrict that each parent can have two and only two children, then:

„ (# external nodes ) = (# internal nodes) + 1

„ (# nodes at level i) ≤2ⁱ

„ (# external nodes) ≤2 ^(height)

„ (height) ≥log₂(# external nodes)

„ (height) ≥log₂(# nodes) - 1

„ (height) ≤(# internal nodes) = ((# nodes) - 1)/2

Traversing Trees

„ preorder traversal Algorithm preOrder(v)

“visit” node v

for each child w of v do

recursively perform preOrder(w)

„ reading a document from beginning to end

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Traversing Trees

„ postorder traversal Algorithm postOrder(v)

for each child w of v do recursively perform postOrder(w)

“visit” node v

„ du (disk usage) command in Unix

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Traversing Trees

„ Algorithm evaluateExpression(v) if v is an external node

return the variable stored at v else

let o be the operator stored at v x ←evaluateExpression(leftChild(v)) y ←evaluateExpression(rightChild(v)) return x o y

Traversing Trees

„ inorder traversal of a binary tree

Algorithm inOrder(v) recursively perform inOrder(leftChild(v))

“visit” node v recursively perform inOrder(rightChild(v))

„ printing an arithmetic expression

„ print “(“ before traversing the left subtree

„ print “)” after traversing the right subtree

The BinaryNode in Java

„ The tree is a collection of nodes:

class BinaryNode {

Object element; /* Item stored in node */

BinaryNode left;

BinaryNode right;

}

„ The tree stores a reference to the root node, which is the starting point.

public class BinaryTree {

private BinaryNode root;

public BinaryTree( ) {

root = null;

} }

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Think Recursively

„ Computing the height of a tree is complex without recursion.

„ The height of a tree is one more than the maximum of the heights of the subtrees.

„H_T= max (H_L+1, H_R+1)

H_L H_R

H_L+1

H_R+1

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Routine to Compute Height

„ Handle base case (empty tree)

„ Use previous observation for other case.

public static int height (TreeNode t) {

if (t == null) { return 0;

} else {

return 1 + max(height (t.left), height (t.right));

} }

Running Time

„ This strategy is a postorder traversal: information for a tree node is computed after the information for its children is computed.

„ Postorder traversal running time is N times the cost of processing each node.

„ The running time is linear because we do constant work for each node in the tree.

Print Pre-Order

class BinaryNode { void printPreOrder( ) {

System.out.println( element ); // Node if( left != null )

left.printPreOrder( ); // Left if( right != null )

right.printPreOrder( ); // Right }

}

class BinaryTree {

public void printPreOrder( ) {

if( root != null )

root.printPreOrder( );

}

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Print Post-Order

class BinaryNode { void printPostOrder( ) {

if( left != null )

left.printPostOrder( ); // Left if( right != null )

right.printPostOrder( ); // Right System.out.println( element ); // Node }

}

class BinaryTree {

public void printPostOrder( ) {

if( root != null )

root.printPostOrder( );

} }

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Print InOrder

class BinaryNode { void printInOrder( ) {

if( left != null )

left.printInOrder( ); // Left System.out.println( element ); // Node if( right != null )

right.printInOrder( ); // Right }

}

class BinaryTree {

public void printInOrder( ) {

if( root != null ) root.printInOrder( );

} }

Traversing Tree

Pre-Order Post-Order InOrder

Exercise

„ A tree contains Integer objects.

„Find the maximum value

„Find the total value

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Iterator Class

„ Problems with recursive:

„Can not process step-by-step (ie. using loop)

„ How to avoid recursive? How to implement non- recusive traversal?

„Recursion is implemented by using a stack.

„We can traverse nonrecursively by maintaining the stack ourselves (emulate stack of activation records).

„ Is non-recursive approach faster than recursive approach?

„Yes

„ Why?

„We can place only the essentials, while the compiler place an entire activation record.

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Tree Iterator: implementation

„ see online code

http://telaga.cs.ui.ac.id/WebKuliah/IKI101 00/resources/code/DataStructures/TreeItera tor.java

Post-Order Traversal using Stack

„ Use stack to store the current state (nodes we have traversed but not yet completed)

„similar to PC (program counter) in the activation record

„ What are the states for post-order traversal?

0. about to make a recursive call to left subtree 1. about to make a recursive call to right subtree 2. about to process the current node

Post-Order Algorithm/Pseudocode

„ init: push the root into the stack with state 0

„ advance:

„while (true)

• node X = pop from the stack

• switch (state X):

• case 0:

•push node X with state 1;

•push left child node X (if it exists) w/ state 0;

•break;

• case 1:

•push node X with state 2;

•push right child node X (if it exists) w/ state 0;

•break;

• case 2:

•“visit”/”set current to” the node X; return;

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Post-Order traversal: stack states

SDA/TREE/V2.0/30

Post-Order: Implementation

„ see online code

„ http://telaga.cs.ui.ac.id/WebKuliah/IKI10100/res ources/code/DataStructures/PostOrder.java

„ http://telaga.cs.ui.ac.id/WebKuliah/IKI10100/res ources/code/DataStructures/StNode.java

Exercise

„ Create an algorithm/psedo-code for in-order traversal using stack.

„ Create an algorithm/psedo-code for pre-order traversal using stack.

In-Order Traversal using Stack

„ What are the states for in-order traversal?

0. about to make a recursive call to left subtree 1. about to process the current node

2. about to make a recursive call to right subtree

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In-Order Algorithm/Pseudocode

„ init: push the root into the stack with state 0

„ advance:

„while (true)

• node X = pop from the stack

• switch (state X):

• case 0:

•push node X with state 1;

•push left child node X (if it exists) w/ state 0;

•break;

• case 1:

•push node X with state 2;

•“visit”/”set current to” the node X; return;

• case 2:

•push right child node X (if it exists) w/ state 0;

•break;

SDA/TREE/V2.0/34

In-Order Algorithm/Pseudocode

„ init: push the root into the stack with state 0

„ advance (optimize):

„while (true)

• node X = pop from the stack

• switch (state X):

• case 0:

•push node X with state 1;

•push left child node X (if it exists) w/ state 0;

•break;

• case 1:

•“visit”/”set current to” the node X;

•push right child node X (if it exists) w/ state 0;

•return;

In-Order: Implementation

„ see online code

http://telaga.cs.ui.ac.id/WebKuliah/IKI101 00/resources/code/DataStructures/InOrder.j ava

Pre-Order Traversal using Stack

„ What are the states for in-order traversal?

0. about to process the current node

1. about to make a recursive call to left subtree 2. about to make a recursive call to right subtree

SDA/TREE/V2.0/37

Pre-Order Algorithm/Pseudocode

„ init: push the root into the stack with state 0

„ advance:

„node X = pop from the stack

„while (true)

• switch (state X):

• case 0:

•push node X with state 1;

•“visit”/”set current to” the node X; return;

• case 1:

•push right child node X (if it exists) w/ state 0;

•push node X with state 2;

•break;

• case 2:

•push left child node X (if it exists) w/ state 0;

•break;

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Pre-Order Algorithm/Pseudocode

„ init: push the root into the stack

„ advance (optimized):

„node X = pop from the stack

„“visit”/”set current to” the node X;

„push right child node X (if it exists);

„push left child node X (if it exists);

„return

Pre-Order: Implementation

„ see online code

http://telaga.cs.ui.ac.id/WebKuliah/IKI101 00/resources/code/DataStructures/PreOrder.

java

Euler Tour Traversal

„ generic traversal of a binary tree

„ the preorder, inorder, and postorder traversals are special cases of the Euler tour traversal

„ “walk around” the tree and visit each node three times:

„ on the left

„ from below

„ on the right

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Level-order Traversal

„ Visit root followed by its children from left to right and followed by their children. So we go down the tree level by level.

„ Sequence: A - B - C - D - E - F - G - H - I

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Level-order Traversal: idea

„ Using a queue instead of a stack

„ Algorithm (similar to pre-order)

„init: enqueue the root into the queue

„advance:

• node X = dequeue from the queue

• “visit”/”set current to” the node X;

• enqueue left child node X (if it exists);

• enqueue right child node X (if it exists);

Level-order: implementation

„ see online code

http://telaga.cs.ui.ac.id/WebKuliah/IKI101 00/resources/code/DataStructures/LevelOrde r.java

Properties of binary Tree

„ Maximum number of nodes in a binary tree of height k is 2^k+1-1.

„ A full binary tree with height k is a binary tree which has 2^k+1- 1 nodes.

„ A complete binary tree with height k is a binary tree which has maximum number of nodes possible in levels 0 through k -1, and in (k -1)’th level all nodes with children are selected from left to right.

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Properties of binary tree (cont.)

„ Complete binary tree with n nodes can be shown by using an array, then for any node with index i, we have:

„Parent (i) is at ⎣i/2⎦if i ≠1; for i =1, we have no parent.

„Left-child (i ) is at 2i if 2i ≤n. (else no left-child)

„Right-child (i ) is at 2i+1 if 2i +1 ≤n (else no right-child)

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Summary

„ Tree, Binary Tree

„ In order to process the elements of a tree, we consider accessing the elements in certain order

„ Tree traversal is a tree operation that involves "visiting” (or"

processing") all the nodes in a tree.

„ Depth First Search (DFS):

„ Pre-order: Visit node first, pre-order all its subtrees from leftmost to rightmost.

„ Inorder: Inorder the node in left subtree and then visit the root following by inorder traversal of all its right subtrees.

„ Post-order: Post-order the node in left subtree and then post- order the right subtrees followed by visit to the node.

„ Breadth First Search (BFS):

„ Level-order : Visit root followed by its children from left to right and followed by their children. So we go down the tree level by level.

Further Reading

„ http://telaga.cs.ui.ac.id/WebKuliah/IKI101 00/1998/handout/handout10.html

„ http://telaga.cs.ui.ac.id/WebKuliah/IKI101 00/1998/handout/handout15.html

„ Chapter 17

What’s Next

„ Binary Search Tree