algorithms more than just solving the problem

Algorithms More than just solving the problem

Fluency with Information Technology

2012-02-22 Katherine Deibel, Fluency in Information Technology 1

INFO100 and CSE100

Katherine Deibel

What is an algorithm? Algorithms are what computer

scientists study It's not just writing code!

2012-02-22 Katherine Deibel, Fluency in Information Technology 2

Katherine Deibel, Fluency in Information Technology 3

What Computer Scientists DoWe solve problems EFFICIENTLY: Define what the problem is

What inputs are given? What needs to be solved? What needs to be outputted?

Develop a solution process Is it accurate? Is it efficient? Could we improve it?

Implementation How do we organize the data? Does the choice of software/hardware change

the efficiency?

2012-02-22

Katherine Deibel, Fluency in Information Technology 4

Algorithms Everywhere Theory

All about algorithms Graphics

Even more algorithms Architecture/Hardware

Branch prediction, memory schemes, etc. Networks

Ensure transmission, data compression, etc. Human-Computer Interaction

Efficiency of human input/output2012-02-22

What is an algorithm?"[An] algorithm is a procedure and sequence of actions to accomplish some task. The concept of an algorithm is often illustrated by the example of a recipe, although many algorithms are much more complex; algorithms often have steps that repeat (iterate) or require decisions (such as logic or comparison). In most higher level programs, algorithms act in complex patterns, each using smaller and smaller sub-methods which are built up to the program as a whole."

Source: Computer User's online dictionary

2012-02-22 Katherine Deibel, Fluency in Information Technology 5

What is an algorithm? Simply put:

An algorithm is a precise description of the steps and methods used to solve a problem or produce a specified result

2012-02-22 Katherine Deibel, Fluency in Information Technology 6

What is an algorithm? Simply put:

An algorithm is a precise description of the steps and methods used to solve a problem or produce a specified result

2012-02-22 Katherine Deibel, Fluency in Information Technology 7

Properties of Algorithms For an algorithm to be well specified, it

must have the following Inputs specified

Outputs specified

Definiteness

Effectiveness

Finiteness

Form of data to process

Form of results to produce

Agent always knows what to do next

Agent able to do all commands

Will stop with answer, or say ‘none’

2012-02-22 Katherine Deibel, Fluency in Information Technology 8

Properties applied to a recipe Title Ingredients (inputs) Steps

Exceptions When to stop

Servings (outputs)

2012-02-22 Katherine Deibel, Fluency in Information Technology 9

Katherine Deibel, Fluency in Information Technology 10

Context Matters Algorithms are abstract

but how and where you execute them matters

Think about baking bread In an electric oven Brick oven A bread machine

The recipes may need to differ depending on the machine in use

2012-02-22

Algorithms vs Programs The recipe metaphor breaks down here Algorithms are abstract:

Not specific to any hardware Not specific to any programming language

Programs are instantiations of algorithms Put into a specific language Meant to work in specified contexts

2012-02-22 Katherine Deibel, Fluency in Information Technology 11

Previous Algorithms We have already seen several

algorithms (and programs) Constructing a table of Fahrenheit to

Celsius conversions (Ch. 20) Computing the price of espresso drink

(Ch. 18) Computing weight in gold Fingerspelling (parsing of input)

Sorting AlgorithmsEveryday application

2012-02-22 Katherine Deibel, Fluency in Information Technology 13

Elements of Sorting Inputs specified:

A list of items in any order Outputs specified:

The items in ascending order Definiteness: Effectiveness: Finiteness:

Clearly can be done in finite time

2012-02-22 Katherine Deibel, Fluency in Information Technology 14

Turn to your neighbor…

With your neighbor(s), discuss how you would sort a deck of cards into numerical order (As, 2s, 3s,…, Qs, Ks) How many different ways can you

think of? Is any better than the

others? How do you know?

2012-02-22 Katherine Deibel, Fluency in Information Technology 15

Random Sort

Algorithm:Check if deck is sortedIf not sorted, shuffle the deckRepeat

Will it finish? Will it be correct? Is it efficient?

2012-02-22 Katherine Deibel, Fluency in Information Technology 16

Insertion SortAlgorithm:

Take first card and start a sorted pileFor each remaining card

Find the position where it belongs in the sorted pile

Place card in correct spot

2012-02-22 Katherine Deibel, Fluency in Information Technology 17

Insertion SortDeck: 4, 7, 2, 5, J, 8, …

1. Deck: 4 | 7, 2, 5, J, 8, …2. Deck: 4, 7 | 2, 5, J, 8, …3. Deck: 2, 4, 7 | 5, J, 8, …4. Deck: 2, 4, 5, 7 | J, 8, …5. Deck: 2, 4, 5, 7, J | 8, …6. Deck: 2, 4, 5, 7, 8, J | …

2012-02-22 Katherine Deibel, Fluency in Information Technology 18

Insertion Sort Does it end? Yes Will it be correct? Yes Is it efficient?

Fairly efficient in real world Programming has varying performance:▪ Best: Linear O(n)▪ Worst: Quadratic O(n2)

2012-02-22 Katherine Deibel, Fluency in Information Technology 19

Merge Sort

Algorithm:Split deck into two pilesMerge sort first pileMerge sort second pileMerge the two sorted piles This is a recursive function in that it is

continually applied to itself

2012-02-22 Katherine Deibel, Fluency in Information Technology 20

Merge Sort ExampleDeck: 3, 6, 7, 2, 1, 8, 4, 5, 1. 3, 6, 7, 2 | 1, 8, 4, 5 (split)2. 3, 6 | 7, 2 | 1, 8, 4, 5 (split)3. 3, 6 | 7, 2 | 1, 8, 4, 5 (sort left)4. 3, 6 | 2, 7 | 1, 8, 4, 5 (sort right)5. 2, 3, 6, 7 | 1, 8, 4, 5 (merge)6. 2, 3, 6, 7 | 1, 8 | 4, 5 (split)7. 2, 3, 6, 7 | 1, 8 | 4, 5 (sort left)8. 2, 3, 6, 7 | 1, 8 | 4, 5 (sort right)9. 2, 3, 6, 7 | 1, 4, 5, 8 (merge)10. 1, 2, 3, 4, 5, 6, 7, 8 (merge)

2012-02-22 Katherine Deibel, Fluency in Information Technology 21

Merge Sort Does it end? Yes Will it be correct? Yes Is it efficient?

Fairly efficient in real world Programming has constantperformance:▪ Best: O(n log2 n)▪ Worst: O(n log2 n)▪ Worse than linear, better than quadratic

2012-02-22 Katherine Deibel, Fluency in Information Technology 22

Many Sorting Algorithms Multiple approaches for sorting

Some more efficient in time in general Some more efficient depending on

nature of input Examples of sorting performance:

http://www.sorting-algorithms.com/

2012-02-22 Katherine Deibel, Fluency in Information Technology 23

EncryptionKeeping Secrets via Algorithms

2012-02-22 Katherine Deibel, Fluency in Information Technology 24

Encryption / Decryption Encrypt:

Transform data so that it is no longer understandable

Decrypt: Transform encrypted data to be understandable again

2012-02-22 Katherine Deibel, Fluency in Information Technology 25

Encryption and decryptionare algorithms

Caesar Ciper Supposedly used by Julius Caesar Algorithm:

To encrypt:Move each letter n letters ahead

To decrypt:Move each letter n letters back

The movement wraps around the alphabet Y + 3 B

2012-02-22 Katherine Deibel, Fluency in Information Technology 26

Caesar Ciper Example when n = 5

A F B G … M R … Y D Z E

2012-02-22 Katherine Deibel, Fluency in Information Technology 27

N QNPJ HFYX

I LIKE CATS

Katherine Deibel, Fluency in Information Technology 28

Caesar Cipher Simple algorithm Easy to implement Easy to crack

Every play a cryptogram The twelve most common letters in

English (descending): ETAOIN SHRDLU

2012-02-22

RSA Public key cryptography system

invented in 1978 by Ron Rivest, Adi Shamir, and Leonard Adleman

Before we get into the details… Are any of you international students? Is it before 1999?

2012-02-22 Katherine Deibel, Fluency in Information Technology 29

Katherine Deibel, Fluency in Information Technology 30

Encryption is a Munition After WWII, encryption algorithms were deemed

as military equipment Actually listed as munitions

Export to foreign countries tightly controlled Debates about teaching to international students Richard White had the RSA algorithm tattooed on his arm

and was theoretically unable to travel internationally Printing and discussing of encryption algorithms

is now recognized as free speech Bernstein (Pretty Good Privacy) v. United States

2012-02-22

Back to RSAThe basics of RSA:1. Pick two large prime numbers p and q2. Compute n = pq, m = (p-1)(q-1)3. Choose a number e such that 1 < e < m

and e and m's only common divisor is 14. Calculate d = e-1 mod m5. Publish n and e; d and m are kept private

Encryption of integer i: c = ie mod nDecryption of integer c: i = cd mod m

2012-02-22 Katherine Deibel, Fluency in Information Technology 31

Katherine Deibel, Fluency in Information Technology 32

Makes sense, right? RSA involves some fairly complex

number theory What you need to know

Based on prime numbers Factoring numbers into prime

components is computationally difficult Larger prime numbers make RSA really

secure (but only 99.9999999% secure)

2012-02-22

Public key systems RSA is an example of a public key system The encryption involves a numerical

computation with several parameters Some parameters are shared (public keys) Some parameters are not (private keys)

Dominant form of encryption today Scales by increasing the length of parameters

2012-02-22 Katherine Deibel, Fluency in Information Technology 33

Building your own algorithmsSome tools

2012-02-22 Katherine Deibel, Fluency in Information Technology 34

Solving a Large Problem Large problems share many properties:

They are daunting—there’s so much to do! We don’t know were to begin Not sure we know all of the tasks that must be

done to produce a solution Not sure we know how to do all of the parts—

new knowledge may be required Not sure it is within our capability—maybe an

expert is needed

2012-02-22 Katherine Deibel, Fluency in Information Technology 35

Problem Decomposition “Divide and conquer” is a political strategy,

military strategy and IT strategy Top-level Plan for Algorithm Building

1. Describe (in any language) a series of steps that produce a solution

2. For each step, solve it or decompose further3. For steps needing decomposition, repeat 24. Assemble solutions and test correctness5. When solution fully assembled, evaluate

2012-02-22 Katherine Deibel, Fluency in Information Technology 36

1. Give Steps to a Solution Specify (in any language) a series of steps

that produce a solution For a huge problem the steps may at first be

vague, but they can be (and must be) made more precise as the whole picture emerges

The goal is an algorithm(s), so List and describe the inputs List and describe the outputs

Be guided by figuring out how to transform the inputs into the outputs

2012-02-22 Katherine Deibel, Fluency in Information Technology 37

2 and 3. Solve or Decompose For each step, solve it or decompose it

further, i.e. apply same technique Most “top level” steps can’t be brained out, and

need further decomposition “Top level” steps often seem huge, too

The technique allows one to concentrate on only one problem at a time

As before, focus on transforming inputs to intermediary outputs to final outputs

2012-02-22 Katherine Deibel, Fluency in Information Technology 38

Katherine Deibel, Fluency in Information Technology 39

4. Assemble and Test Putting solutions together can be tough

because of different assumptions made while solving the parts

A common practice is to combine parts along the way and to test continuously

Because of the need to test, pick a good order to solve the problems

Mistakes and backtracking are inevitable

2012-02-22

Example: Lab 8 Lab 8 showed some of the basic ideas

of writing an algorithm / program Breaking down parts of the problem Testing along the way Continual effort to just progress forward

2012-02-22 Katherine Deibel, Fluency in Information Technology 40

Example: Chapter 22 Implements Smooth Motion page:

http://preview.tinyurl.com/fit-smooth Fairly big application

125 lines 2500 nonspace characters 6 functions

Chapter details the processof divide-and-conquer

2012-02-22 Katherine Deibel, Fluency in Information Technology 41

Summary Algorithms describe the precise steps to

solve a problem Fundamental concept in computer science Concerned with correctness, efficiency, and

finiteness Building algorithms is about abstracting

and breaking down a problem We do this all the time The challenge of programming is telling a

computer how to do the algorithm

2012-02-22 Katherine Deibel, Fluency in Information Technology 42