stacks, queues and hierarchical collections · queues and stacks •linear collections •can be...

Programming III April 2007

Copyright © 2002-2007 by René Hexel. Allrights reserved. 1

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Stacks, Queues and HierarchicalCollections

2501ICTNathan

P3 Lecture April2007

Copyright © 2002-2007 René Hexel.All rights reserved.

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Contents

• Linked Data Structures Revisited• Stacks• Queues

• Trees• Binary Trees• Generic Trees

• Implementations





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Queues and Stacks

• Linear Collections• Can be implemented in different ways, e.g.

• Arrays• Linked Lists

• Are often used in Operating Systemsand run time environments• Function/Method calling Stacks• Process priority Queues



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Queues

• FIFO collections• First element enqueued is first to be dequeued• Insertions occur at one end, the rear• Removals occur at the other end, the front

• Priority Queue Extensions• Reordering according to priority• Few priorities: multiple Queues• Many priorities: insert according to priority





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Queues using Lists

• Use the head and tail Pointers

• Enqueue: Insert from the tail (rear)• Dequeue: Remove from the head (front)

data

NULLhead

rearfrontNULL

tail



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Priority Queues

• Implementation #1: Multiple Lists• One list per priority• Enqueue on the corresponding list• Dequeue starting at the highest priority list

• Go to next lower priority if empty, etc.• Does not require search operation to enqueue• Requires two pointers (head/tail) per priority:

• Faster enqueue, but slower dequeue operation¬ Useful if number of different priorities is small





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Priority Queues (2)

• #2: Priority Ordering• Only one list• Enqueue according to priority

• Higher priority entries “overtake” lower ones• Dequeue always from the head• Requires linear search to enqueue: O(n)

• Dequeue doesn’t require search: O(1)• Useful if number of different priorities is large



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Queue Applications

• First-Come-First Serve Algorithms→ OS Process Scheduling→ Event Handling→ Modelling and Simulation of Real-

World Processes• E.g. checkout queue in a supermarket





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Stacks

• LIFO collection• The last element pushed onto the stack is the first to

be retrieved• Both push (insertion) and pop (deletion) operations

occur at the top of the stack

• Implementation• Arrays• Singly linked lists



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Array Implementation

• Index: i = 0;• Push: stack[i++] = element;

• Pop: element = stack[--i];

• Peek: element = stack[i-1];• “Sneak preview,” without changing i

• Watch Preconditions!





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Stack Example

• Parsing Matching Parentheses• expression = { letter } | “(” expression “)”

| “[” expression “]”

• “either a letter or an expression in brackets”

• Legal examples• a ab (a) (ab) [(a)] [a(b)cd]

• Illegal examples• a) )( a( [[(]) [a)a]



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Matching () Algorithm

• For each character c in the String• If c is (, [, or {

• Push c onto the stack

• Else if c is ), ], or }• If the stack is empty: return false• Else pop the top opening bracket and check if it

matches the corresponding closing bracket

• Return true if the stack is empty





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Hierarchical Collections

• Tree definition• Types of Trees• Binary Expressions

• Expression Trees• Tree traversals: pre-, in-, postorder

• Examples• Generating Postfix, Parsing



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Tree Definition

• Each node has at most onepredecessor• Parent

• Many Successors• Children

• Siblings• Nodes sharing the same parent (eg, D2 and D3)

D2

D1

D3





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Tree Definition (2)

• Topmost Node:root

• Successors• Children, Children of

Children• Also called

descendants• All nodes are successors

of root

D2

D1

D3

D4



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Tree Definition (3)

• Leaf Nodes:• Nodes without

Successors• E.g. D3 and D4

• Frontier:• The set of all leaf nodes

D2

D1

D3

D4





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Tree Definition (4)

• Interior Nodes:• Nodes with at least one

successor• E.g. D1 and D2

• Ancestors:• Immediate or indirect

predecessors• E.g, D1 is an ancestor of

D2, D3, and D4

D2

D1

D3

D4



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Tree Definition (5)

• Levels arenumbered from 0• Level 0 is always the

root• This tree has 3 Levels

• Level 0: D1• Level 1: D2 and D3• Level 2: D4

D2

D1

D3

D4

Level 0

Level 2

Level 1





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Binary Trees

• Binary Trees allowat most twochildren per Node

• Generic Treesallow any numberof children perNode

D2

D1

D3

D4



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Generic Trees

• Order of the Tree:• Maximum Number of

successors allowed forany given Node

• E.g., Order: 3• Generic Trees are

sometimes calledGeneral Trees

D2

D1

D3

D4

D5





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Tree Applications

• Parsing Languages• Computer Languages, Mathematical Formula• Natural Languages

• Searchable Data Structures• Databases (e.g., B-Trees)

• Heaps and Balanced Trees• Sorting and organising Data



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Parsers

• Read in Expressions• E.g., (2 + 3) * 5

• Check Syntactical Correctness• Is everything where it should be?

• Create Parse Tree• Evaluator checks semantic meaning and processes

the data in the Tree to produce meaningful output





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Binary Expressions

• Stored in Binary Trees: e.g. 3+5• Numbers

• Leaf Nodes

• Operators• Interior Nodes

• Operands• Contained in Subtree of the Expression

3

+

5



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Example Expression

• 3 * 4 + 5

*

+

5

3 4





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Example Expression (2)

• 3 * (4 + 5)

3

*

+

4 5



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Operator Precedence

• 3 * (4 + 5)

• The higher theprecedence, the lowerin the tree!

• Overridden byparentheses!

3

*

+

4 5





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Operator Precedence (2)

• 3 + 4 + 5• If operators have equal

precedence, the oneson the left appearlower in the tree whenparsed from left toright!

5

+

+

3 4



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Evaluating an Expression Tree

• Begin at the root Node• If a number, return it, otherwise• Run the operator w/ the results of• Evaluating its left and right subtrees,

and• Return this value.





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Evaluating Example

• 3 * (4 + 5)

3

*

+

4 5



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Evaluating Example (2)

• 3 * (4 + 5)

• * is an operator

¬Evaluate left and rightsubtrees first!

3

*

+

4 5





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• 3 * (4 + 5)

• 3 is a number

¬Return 33

*

+

4 5



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• 3 * (4 + 5)

• + is an operator

¬Evaluate left and rightsubtrees first!

3

*

+

4 5





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• 3 * (4 + 5)

• 4 is a number

¬Return 43

*

+

4 5



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• 3 * (4 + 5)

• 5 is a number

¬Return 53

*

+

4 5





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• 3 * (4 + 5)

• Add subtree results

¬Return 4 + 5 = 93

*

+

4 5

9



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• 3 * (4 + 5)

• Multiply subtreeresults

¬Return 3 * 9 = 27

3

*

+

4 5

9

27





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Evaluate Pseudocode

evaluate(node){ if node is a number return number; else { left = evaluate(node.left); right = evaluate(node.right); return compute(node, left, right); }}



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Binary Tree Traversals

• Preorder• Visit node, then go left, then go right

• Inorder• Go left, then visit node then go right

• Postorder: Depth First• Go left, then go right, then visit node

• Breadth First• Level 0, then Level 1, then Level 2, etc.





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Binary Tree Traversals (2)

• Preorder, Inorder, and Postorder• Correspond with Prefix, Infix, and Postfix Notations

of an Expression• Infix: 3 + 5• Prefix = Polish Notation (PN) : +(3,5)• Postfix = Reverse Polish Notation (RPN) : 3 5 +

• Use the Same Generic RecursiveAlgorithm!



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Prefix Pseudocode

String prefix(node){ if (node == NULL) return ””; else return node + prefix(node.left) + prefix (node.right);}





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Infix Pseudocode

String infix(node){ if (node == NULL) return ””; else return infix(node.left) + node + infix (node.right);}



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Postfix Pseudocode

String postfix(node){ if (node == NULL) return ””; else return postfix(node.left) + postfix (node.right) + node;}





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A Binary Tree Structure

@interface BTNode: NSObject;{ BTNode *left; // left child BTNode *right; // right child

id value; // the actual data}// here go the usual access methods:// -setLeft, -setRight, …@end



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A Binary Tree Constructor

- initWithValue: v left: (BTNode *) lright: (BTNode *) r

{if (![self init]) return nil;

value = v;left = l;right = r;

return self;}





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Creating a Binary Tree

BTNode *left = [[BTNode alloc] initWithValue: @”3” left: nil right: nil];

3

nil nil



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BTNode *left = [[BTNode alloc] initWithValue: @”3”left: nil right: nil];

BTNode *right = [[BTNode alloc] initWithValue: @”5”left: nil right: nil];

3 5

nil nil





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BTNode *root = [[BTNode alloc] initWithValue: @”+” left: left right: right];

3

+

5



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BTNode *root = [[BTNode alloc] initWithValue: @”+” left: left right: right];

• Infix: 3 + 5

• Postfix: 3 5 +

• Prefix: +(3, 5)add(3, 5)Functional notation

3

+

5





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Grammar Parsing

• Recursive Definition, e.g Infix:Expression = Term { + | - Term }Term = Factor { * | / Factor }Factor = number | ( Expression )

• Represents standard math formulas, e.g.• 3 + 4 * ( 5 - ( 6 / 7 ) )

• Can be used to create a parse tree!



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Recursive Descent Parsing

Expression(){ Term(); while (token == '+' || token == '-') { get_token(); Term(); }}

Expression = Term { + | - Term }Term = Factor { * | / Factor }Factor = number | ( Expression )





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Term(){ Factor(); while (token == '*' || token == '/') { get_token(); Factor(); }}




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Factor(){ switch (token) { case number: get_token(); break;

case '(': get_token(); Expression(); if (token != ')')

error(”No closing ')'"); get_token(); break;

default: error(Error '%s'\n", token); }}






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References

• Lambert, K. A., & Osborne, M. (2004): A Framework forProgram Design and Data Structures: Brooks/Cole.Chapters 8-10

• http://en.wikipedia.org/wiki/Tree_data_structure• http://en.wikipedia.org/wiki/Binary_tree• http://en.wikipedia.org/wiki/Recursive_descent_parser• http://mathworld.wolfram.com/WeightedTree.html• Week 6 example code: SimpleTree.zip

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Appendix

C Language Examples





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A Binary Tree Structure (C)

typedef struct BinTreeNode btNode;

struct BinTreeNode{ btNode *left; /* left child */ btNode *right; /* right child */

void *value; /* the actual data */};



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A Binary Tree Constructor (C)

btNode *newBTNode(void *data, btNode *left, btNode *right)

{ btNode *this = malloc(sizeof btNode);

this->value = data; this->left = left; this->right = right;

return this;}





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Creating a Binary Tree (C)

btNode *left = newBTNode(”3”, NULL, NULL);

3

NULL NULL



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btNode *left = newBTNode(”3”, NULL, NULL);btNode *right = newBTNode(”5”, NULL, NULL);

3 5

NULL NULL





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btNode *left = newBTNode(”3”, NULL, NULL);btNode *right = newBTNode(”5”, NULL, NULL);btNode *root = newBTNode(”+”, left, right);

3

+

5



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btNode *left = newBTNode(”3”, NULL, NULL);btNode *right = newBTNode(”5”, NULL, NULL);btNode *root = newBTNode(”+”, left, right);

• Infix: 3 + 5

• Postfix: 3 5 +

• Prefix: +(3, 5)add(3, 5)

Functional Notation

3

+

5

stacks, queues and hierarchical collections · queues and stacks •linear collections •can be...

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