CSS 342 Data Structures, Algorithms, and Discrete Mathematics I - PowerPoint Presentation

Slide1

CSS 342

Data

Structures, Algorithms, and Discrete Mathematics I

Lecture 16. 150309.

CUSACK CHAPT 4.

Slide2

Agenda

HW5: Questions / Discussion

Queues: finish.

Trees: BST

Prop Logic Intro. Will be on final.

Slide3

HW5

What should occupy the Queue?

Where should History live?

What should main look like?

Slide4

A Pointer-Based Implementation

2

4

1

7

front

back

NULL

Slide5

Queues with a linked list?

Do we need to overload =, copy constructor?

Why or why not?

Any other overloads?

Slide6

Trees

Slide7

Terminology

FIGURE 15-1 (a) A tree; (b) a subtree of the tree in part a

Root

Parent

Child

Ancestor

Descendent

Height

Subtree

N-

ary

tree

Binary Tree

Slide8

Binary Tree: Algebraic Expressions.

Slide9

Binary Search Tree

For each node n, a binary search tree satisfies the following three properties:n ’s value is greater than all values in its left subtree T L .n ’s value is less than all values in its right subtree T R .Both T L and T R are binary search trees.Well suited for searching

Data Structures and Problem Solving with C++: Walls and Mirrors,

Carrano

and Henry, © 2013

Slide10

Binary Search Tree: Example

Slide11

Binary Search Tree: Example

FIGURE 15-14 Binary search trees with the same data as in Figure 15-13

Slide12

Binary Search Tree: Example

FIGURE 15-14 Binary search trees with the same data as in Figure 15-13

Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

Slide13

Binary Search Tree: Example

FIGURE 15-14 Binary search trees with the same data as in Figure 15-13

Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

Slide14

In-class problem

The maximum number of leaves in a tree can be expressed by the following formula:  h = height of the tree b = max number of branches per node.  Write a recursive function which computes. 

 

Slide15

class

BinarySearchTree

{

public

:

BinarySearchTree

();

~

BinarySearchTree

();

void

Insert(Object *);

bool

Remove(

const

Object &);

bool

Retrieve(

const

Object &, Object * &);

void

Flush();

bool

Contains(

const

Object

&);

void

Display()

const

;

bool

isEmpty

()

const

;

int

Height()

const

;

int

getCount

()

const

;

private

:

struct

Node

{

Object *

pObj

;

Node

*right;

Node

*left;

};

Node

*root;

};

Slide16

Computer Scientist of the week

George Boole

English Mathematician, Philosopher and LogicianWrote: “An Investigation of the Laws of Thought (1854), on Which are Founded the Mathematical Theories of Logic and Probabilities”Found of Boolean Algebra, the Algebra of logicLogical propositions expressed as algebrasbool keyword named after himDied by the fever started by walking to class in the rain to lecture

Slide17

Searching a Binary Search Tree

Data Structures and

Poblem

Solving with C++: Walls and Mirrors,

Carrano

and Henry, © 2013

Slide18

Searching a Binary Search Tree: psudeo-code

Search(

Obj

*root)

{

if

(root ==

NULL)

return

NULL

;

else

if

(target == *(root->

pObj

))

{

return

(root->

pItem

);

}

else

if

(target < *(root->

pObj

))

{

return

Search(root->left);

}

else

{

return

Search(root->right)

}

}

Slide19

Creating a Binary Search Tree

FIGURE 15-16 Empty subtree where the search algorithm terminates when looking for Frank

Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

Slide20

Efficiency of BST Operations

Retrieve:

Insert:

Removal:

Traversal

:

Slide21

Propositional Logic

Slide22

Text Book

An Active Introduction to Discrete Mathematics and Algorithms,

Charles Cusack, David Santos, GNU Free

Software,

2014.

http://www.cs.hope.edu/~cusack/Notes/Notes/Books/Active%20Introduction%20to%20Discrete%20

Mathematics%20and%20Algorithms/ActiveIntroToDiscreteMathAndAlgorithms.pdf

Chapter 4: Logic.

Slide23

Propositions

Proposition

:

a statement that is either true or false, but not both

Examples of propositions

Proposition p

:

All

mathematicians wear sandals

.

Proposition r:

Discrete Math is fun

.

Proposition v:

This is not a proposition.

Proposition s:

The sum of the first n odd integer is equal to n

2

for n >1.

These are not propositions:

What is your name?

Go to the store and get me a steak?

Propositions have a truth value: True or False.

Slide24

Propositions

MathThe only positive integers that divide 7 are 1 and 7 itself: (true)For every positive integer n, there is a prime number larger than n: (true)HistoryAlfred Hitchcock won an Academy Award in 1940 for directing “Rebecca”: (false, he has never won one for directing)Seattle held the World’s Fair, Expo 62: (true )Programming languagesBoolean expressions in if-else, while, and for statementsfor ( index = 0; index < 100; index++ ) { …….;}

A proposition

Not a proposition

Slide25

Compound propositions

Conjunction (AND) of p and qnotations: p ^ q, p && qTrue only if both p and q are trueTruth tableDisjunction (OR) of p or qNotations: p v q, p || qTrue if either p or q or both are truetruth table

pqp ^ qFFFFTFTFFTTT

p

q

p v q

F

F

F

F

T

T

T

F

T

T

T

T

Slide26

Compound Propositions

The negation (NOT) of pNotations: not p, ¬p !pExamplesP: 1 + 1 = 3 (false)!p: !(1 + 1 = 3) ≡ 1 + 1 ≠ 3 (true)Exclusive OR (XOR)Notations: p xor q,

pqp qFFFFTTTFTTTF

p

!p

F

T

F

T

Slide27

Binary Expressions in C++

How do you examine the behavior of if-else?if ( a > = 1 && a <= 100 )true if 1 ≤ a ≤ 100if ( a >= 1 || a <= 100 )always true

a < 1

1 <= a <= 100

100 < a

a >= 1

a <= 100

false

true

true

true

true

false

condition

proposition

a < 1

1 <= a <= 100

100 < a

a >= 1

a <= 100

false

true

true

true

true

false

condition

proposition

Slide28

Commutative Lawp v q = q v pp ^ q = q ^ p Associative Lawp v (q v r) = (p v q) v rp ^ (q ^ r) = (p ^ q) ^ r Distributive Lawp ^ (q v r) = (p ^ q) v (p ^ r)p v (q ^ r) = (p v q) ^ (p v r) Identityp v F = pP ^ T = pDominationp v T = Tp ^ F = F

Theorems for Boolean algebra

Slide29

Complement Law (tautologies)p ^ ¬p = Fp v ¬p = T Square Lawp ^ p = pp v p = p Double Negation¬(¬p) = p Absorptionp ^ (p v q) = p p v (p ^ q) = p

Theorems for Boolean algebra

Slide30

Precedence of Operators

NOT: ¬

AND: ^

XOR

OR: v

Conditional: ->

Slide31

Prove the following Boolean equation

¬p

^

q

v

p

^

¬

q

= (¬

p

v

¬

q

) ^ (

p

v

q

)