To view this presentation, you'll need to enable Flash.

Show me how

After you enable Flash, refresh this webpage and the presentation should play.

Loading...

PPT – Data Structures and Algorithms PowerPoint presentation | free to download - id: 124f73-MjVhY

The Adobe Flash plugin is needed to view this content

View by Category

Presentations

Products
Sold on our sister site CrystalGraphics.com

About This Presentation

Write a Comment

User Comments (0)

Transcript and Presenter's Notes

Data Structures and Algorithms

Discrete Math Review

Discrete Math review

- Logarithmic Functions
- Sets
- Logic
- Induction
- Counting

Sets

- A set is a collection of individual elements in

the domain D. The universal set U contains

every element in D . The null set ? contains

no element. - If A is a set in the domain D , A must be a

subset of the universal set U , denoted as A ?

U. - If A consists of some but not all elements, A

is then called a proper subset of U , denoted

as - A ? U .

Sets

- A set is a collection of definite and separate

objects. - The cardinality of a set is the number of

elements in the set. - A subset of a set is a set comprised of a

sub-collection of the elements of the original

set. - Example Consider the set S 2, 5, 7. The

subsets of S are the sets 2 , 5, 7, 2,5 ,

2,7 , 5,7 , 2, 5, 7and (the empty set). - Fact If a set has cardinality n , then the

number of subsets 2n .

Sets

- A proper subset of a set is a set comprised of a

sub-collection of some, but not all, of the

elements of the original set. - Example Consider the set S 2, 5, 7. The

proper subsets of S are the sets 2 , 5, 7,

2,5 , 2,7 , 5,7 and (the empty set). - S cardinality of 3 ? 23 ? 7 ?

Sets

- There are only two different types of sets
- 1. An Infinite Set is a set that can be placed in

a one-to-one correspondence to a proper subset of

itself. - The Basic Infinite Set is the set of positive

whole numbers 1, 2, 3, 4, 5,. New Notation

The Basic Infinite Set has cardinality . - A Finite Set is a set that cannot be placed in a

one-to-one correspondence to a proper subset of

itself. - Something to think about.2 subsets ?

Logarithmic Functions

- Logarithms were invented about 1614 by John

Napier. The word logarithm was derived from two

Greek words, logos, which means "ratio," and

arithmos, which means "number." - Before calculators and computers were available,

common logarithms were used to do certain kinds

of calculations - Growth and decay
- Big O notation growth of computational complexity

Logarithmic Functions

- Logarithms are exponents
- A logarithmic function is the inverse of an

exponential function - One way to describe a logarithmic function is to

interchange variables in the equation y ax - Thus, x ay is logarithmic

Logarithmic Functions

- For logarithmic functions we use the notation

loga x - Therefore, The following are equivalent.
- x ay and
- y loga x

Logarithmic Functions

- Definition of Logarithm
- Suppose bgt 0 and . For n gt 0, there is a number

p such that logb n p if and only if bp n. - log2 256 ? ? 2p 256
- log10 1000 ? ? 10p 1000
- log7 2401 ? ? 7p 2401

Logic

- Examples Boolean functions NOT, AND, OR, XOR, .

. . - a NOT( a)
- 0 1
- 1 0

Logic

Examples Boolean functions NOT, AND, OR

Logic

- Examples Boolean functions XOR, XNOR, NOR

Logic

Boolean Simplification

Express F in terms of A and B

F AB AB

Using the uniting theorem

? A ( B B ) F

Series

- Arithmetic
- One common type of series is the arithmetic

series (also called an arithmetic progression).

Each new term in an arithmetic series is the

previous term plus a given number. For example

this is an arithmetic series 1471013.... - In this case each term is the previous term plus

3. The difference between each term (the 3 in

this case) is called the "common difference" and

is generally denoted by the letter d. - There are many arithmetic series that have d3.

To specify which series we mean, we need to know

one more piece of information the value of the

first term (usually called "a").

Series

- Arithmetic
- If we're given a and d, then, that specifies a

unique arithmetic series. All arithmetic series

therefore have the following form

a(ad)(a2d)(a3d).... - So in general we can say that the value of the

nth term is a(n-1)d

Series

- Geometric
- Another common type of series is the geometric

series (also called a geometric progression). In

this case, each term is the previous term

multiplied by a given number. - Here's a geometric series, for example
- 261854...
- In this case, each term is the previous term

multiplied by 3. The number you multiply by (in

this case 3) is called the "common ratio" and is

generally denoted by r. - There are many geometric series that have r3. To

specify which series we mean, again we need to

know one more piece of information the value of

the first term (usually called "a").

Series

- Geometric
- A geometric series is uniquely specified by the

values of a and r. Every geometric series has the

following form - aarar2ar3.......
- Writing the series in this way we can see the

formula for working out the value of any term.

The first term is just a. The second term is a

multiplied by r once. The third term is a

multiplied by r twice, and so on. So the nth term

must be a multiplied by r (n-1) times, that's

arn-1.

Series

- Are there others?

The P word.Induction

- The Principle of Mathematical Induction
- Suppose we have an assertion P(n) about the

positive integers. - Then if we show both of (i) and (ii) below, then

P(n) is true for all n gt 1. - (i). P(1) is true
- (ii). For each k gt 1 If P(k) is true, then

P(k1) is true.

The P word.Induction

Prove that 123 n

- Prove that 135 (2n 1) n2

Counting

- Product Rule n ways to do m tasks nm ways.
- How many different bit strings are there of

length 7?

Counting

- Pigenhole Principle If N objects are placed into

k boxes, then there is at least one box

containing at least N/k objects. - Ten persons were born on the 1st, 11th and the

27th of the months of May, August and November.

How many share the exact same birthday? - Tells us nothing of the date. Asserts only that

one exists!

Counting

- Permutations and Combinations
- Permutations Given that position is important,

if one has 4 different objects (e.g. A, B, C and

D), how many unique ways can they be placed in 3

positions (e.g. ABD, ACD, BDA, DBA, BAD, ADB,

ADC, DCA, DAC, CDA etc.) - An ordered arrandement of r elements of a set is

called a r-permutation.

Counting

- Permutations and Combinations
- Combinations If one has 4 different objects

(e.g. A, B, C and D) how many ways can they be

grouped as 2 objects when position does not

matter (e.g. AB, AC, AD are correct but DA is not

ok as is equal to AD.) - An r-combination is simply a subset of the set

with r elements.

Base Conversion

- The base value of a number system is the number

of different values the set has before repeating

itself. For example, decimal has a base of ten

values, 0 to 9. - Binary 2 (0, 1)
- Octal 8 (0 - 7)
- Decimal 10 (0 - 9)
- Duodecimal 12 (used for some purposes by the

Romans) - Hexadecimal 16 (0 - 9, A-F)
- Vigesimal 20 (used by the Mayans)
- Sexagesimal 60 (used by the Babylonians)

Base Conversion

- Successive Division
- 3710 base 6
- 37/6 6 r 1
- 6/6 1 r 0
- 1/6 0 r 1
- ?3710 base 6 101

Base Conversion

- 1016 base 10
- 1x600x611x62 1036 37
- 5810base4
- 11110base13

About PowerShow.com

PowerShow.com is a leading presentation/slideshow sharing website. Whether your application is business, how-to, education, medicine, school, church, sales, marketing, online training or just for fun, PowerShow.com is a great resource. And, best of all, most of its cool features are free and easy to use.

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

Recommended

«

/ »

Page of

«

/ »

Promoted Presentations

Related Presentations

Page of

Home About Us Terms and Conditions Privacy Policy Presentation Removal Request Contact Us Send Us Feedback

Copyright 2018 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

Copyright 2018 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

The PowerPoint PPT presentation: "Data Structures and Algorithms" is the property of its rightful owner.

Do you have PowerPoint slides to share? If so, share your PPT presentation slides online with PowerShow.com. It's FREE!