Csci20 Prelim Lecture

8/6/2019 Csci20 Prelim Lecture

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CSCI20Prelim

References:

Introduction to Algorithms, Thomas H. Cormen, 2nd edition,

MIT Press

Instructor: Priyanka S


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What are the Algorithms?

Algorithm:

Is any well-defined computational procedure that takes somevalue, or set of values, as input and produces some value, or set

of values, as output.

It is a tool forsolving a well-specified computational problem.

They are written in apseudo code which can be implemented inthe language of programmers choice.


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Example: sorting numbers.

Sorting is a fundamental operation.

Many algorithms existed for that purpose.

The best algorithm to use depends on: The number of items to be sorted.

possible restrictions on the item values

kind of storage device to be used: main memory, disks, or tapes.


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Correct and incorrect algorithms

Algorithm is correct if, for every input instance, itends with the correct output. We say that a correct

algorithm solves the given computational problem.

An incorrect algorithm might not end at all on some

input instances, or it might end with an answer other

than the desired one.

We shall be concerned only with correct algorithms.


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Practical Examples

Internet and Networks.

The need to access large amount of information with theshortest time.

Problems of finding thebest routs for the data to travel. Algorithms forsearching this large amount of data toquickly find the pages on which particular informationresides.

Electronic Commerce. The ability ofkeeping the information (credit card

numbers, passwords, bank statements) private, safe, andsecure.

Algorithms involves encryption/decryption techniques.


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These lists are far from ex

haustive, but ex

hibittow characteristics that are common to many

interesting algorithms.

(1)There are many candidate solutions, most of

which are not we want. Finding one that we do

want can present quite a challenge.

(2)There are practical applications.


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Hard problems

We can identify the Efficiency of analgorithm from its speed (how long does the

algorithm take toproduce the result).

Some problems have unknown efficient

solution.

These problems are called NP-complete

problems.


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Hard problems

Nobody has proven that an efficientalgorithm for one cannot exist.

Several NP-complete problems are similar, but not

identical.

If we can show that the problem is NP-complete, we

can spend our time developing an efficient algorithm

that gives a good, but not the best possible solution.

Example of NP-complete problem: traveling

salesman.


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Why do we need the algorithms?

One always need to proofthat his solution method terminateand does the correct answer.

Computers may be fastbut they are not infinitely fast, andmemory may be cheap but it is not free. This resources should

be used wisely.

As an example:

There is two algorithms for sorting; merge sort algorithm,

and insertion sort algorithm

Insertion sort takes an execution time equal c1*n2 to sort nitems.

Merge sort takes an execution time equal c2*n log2n to sort n

items.


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Algorithms as a technology

Suppose computers were infinitely fast and computermemory was free. Would you have any reason to studyalgorithms?

The answer is yes, if for no other reason than that you

would still like to demonstrate that your solution methodterminates and does so with the correct answer.

If computers were infinitely fast, any correct method forsolving a problem would do. You would probably wantyour implementation to be within the bounds of goodsoftware engineering practice (i.e., well designed anddocumented), but you would most often use whichevermethod was the easiest to implement


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Why do we need algorithms?- cont.

c1 and c2 are constants.

Insertion sort usually has a smallerconstant factor thanmerge sort, so that c1< c2

Two computers; computer A running insertion sort, andcomputer B running merge sort.

Assuming Computer A executes one billion instructions per

second and computer B executes only ten million instructionsper second, so that computer A is 100 times faster thancomputer B.


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Why do we need algorithms?- cont.

Suppose (c1 =2, and c2 =50).

To sort one million number, computer A takes,

2 x (106 ) 2 instruction /109 instruction/sec = 2000 sec.

While computer B takes, 50 x 106x log2 106 instruction /107

instruction/sec = 100 sec.


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Algorithms and other technologies

Algorithms are truly that important oncontemporary computers in light of otheradvanced technologies, such as

Hardware with high clock rates, pipelining, andsuperscalar architectures,

Easy-to-use, intuitive graphical user interfaces

(GUIs),Object-oriented systems, and

Local-area and wide-area networking.


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Expressing algorithms

Algorithms can be expressed in many

kinds of notation, including

natural languages, pseudocode,

flowcharts,

programming languages or

control tables (processed by interpreters).


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Algorithm vs Pseudo-code vs

Program-code Algorithm:

Step 1: Declare two integers and variable that

computes the sum. (num1, num2, sum)

Step 2: Accept the first number.

Step 3: Accept the second number.

Step 4: Computes for the sum of the first and

second numbers.Step 5: Display the computed sum of the two

numbers.


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Pseudo-code:

declare num1 as int

declare num2 as intdeclare sum as double

accept num1

accept num2

sum = num1 + num2

display sum


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Flowchart: Is an algorithm that uses symbols

to describe a method or a program. Used

usually instead ofpseudo-code.


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Program

Import java.util.Scanner;

Public class sample{

public static void main(String[] args)

{

int num1, num2, sum;

Scanner sc = new Scanner(System.in);

num1 = sc.nextInt();

num2 = sc.nextInt();

sum = num1 + num2;

System.out.println(The sum is +sum);

}

}


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Comparing algorithms

Time complexity

The amount of time that an algorithm needs torun to completion.

Space complexity The amount of memory an algorithm needs to

run

The better algorithm is the one whichruns faster (has smaller timecomplexity)


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Coding an Algorithm

Coding an algorithm consists of

implementing the algorithm using a

computerprogramming language.


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Algorithm examples

To find the sum of two numbers

1. declare n1,n2 as integers

2. declare sum as integers3. input values for n1,n2

4. sum n1+n2

5. print Sum is, sum6. halt


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Write an algorithm to calculate the final price

as the product of itemprice and tax.

declare itemprice,tax as double

declare finalprice as double

input values for itemprice, tax

finalprice = itemprice + tax

print final price is , finalpricehalt


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To find the largest of two numbers

1. Declare a, b as integers

2. Input values for a,b3. if a> b then

3.1 print a is larger than b

4. else

4.1 print b is larger than a

5. halt


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Loops

Write an algorithm to display the first

20 natural numbers

declare I as integerfor I=1;I


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To find the sum of first n natural numbers.

Declare i, n,sum as integers

Input value for n

Assign sum =0

for i=1;i


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Euclids Algorithms

The Euclidean algorithm (also called Euclid's

algorithm) is an algorithm to determine the

greatest common divisor(gcd) of two integers.

In mathematics, the Euclidean algorithm(alsocalled Euclid's algorithm) is an efficient method

for computing the greatest common divisor

(GCD), also known as the greatest common

factor (GCF) or highest common factor (HCF). Itis named after the Greek mathematician Euclid.


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GCD

3 divides 15 (15/3 is a whole number), and is called adivisor of 15. 1, 3, 5, and 15 are all of the divisors of15.

Prime numbers have only themselves and 1 asdivisors.

Example 7 has 1 and 7 as divisors Example: Take number 36 and 15

Divisors of 15 are 1,3,5,15

Divisors of 36 are 1,2,3,4,6,9,12,18,36

36 and 15 have some "common divisors," 1, and 3. You canlist the divisors of each, and you will find that 1 and 3 are onboth lists.

Sometimes you need to know the greatest common divisor(gcd), the greatest number that is a common divisor of two ormore whole numbers.


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Euclids Algorithm

In Euclid's Elements we find a way ofcalculating the gcd of two numbers, withoutlisting the divisors of either number. It is nowcalled Euclid's Algorithm.

[An algorithm is a step by stepprocess (orrecipe) for doing something.]

We will find the gcd of 36 and 15. Divide 36 by 15 (the greater by the smaller), getting 2

with a remainder of 6. Then we divide 15 by 6 (the previous remainder) andwe get 2 and a remainder of 3.

Then we divide 6 by 3 (the previous remainder) andwe get 2 with no remainder.

The last non-zero remainder (3) is our gcd.


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Euclids Algorithm for computing

gcd(m,n)

Step1: if n=0, return the value of m as the

answer and stop;otherwise,proceed to

Step2.

Step2: Divide m by n and assign the value

of the remainder to r.

Step3: Assign the value of n to m and the

value of r to n. Go to Step1.


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//Computes gcd(m,n)

//Input: Two nonnegative, not-both-zero integersm and n

//output: Greatest common divisor of m and n While n 0 do

rm mod n

mn

nr

return m


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Example

Using Euclids algorithm find the gcd of 24, 33

Euclid's algorithm comes in handy withcomputers, because listing divisors is moredifficult than the above algorithm. Largenumbers are difficult to factor, while they arerelatively easy to divide.

Euclids algorithm is considered to be one of thebest examples of an efficient algorithm.

33/24 =1 remainder 9 24/9 = 2 remainder 6

9/6=1 remainder 3

6/3=2 remainder 0

The last non-zero remainder is 3


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1. Write an algorithm to check if a given number is even orodd

2. Write an algorithm to find the smallest of 3 numbers.

3. To find the smallest of 3 numbers

4. To check if a given number is a positive or negativenumber

5. To check if a number is divisible by 5

6. Write an algorithm to print the sum and count of non-

negative numbers out of a list of 50 numbers7. Write an algorithm to read the gender and age of 50

voters and count how many male voters of age above18 are present in the list

8. Write an algorithm to find the largest number out of a list

of 100 numbers.

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Csci20 Prelim Lecture

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