02_The Role of Algorithms in Computing

8/8/2019 02_The Role of Algorithms in Computing

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What is an Algorithm

An algorithm is all about the procedure with

the input and output

1



The Sort Problems

Input

A sequence of numbers <a1, a2, ,an>

Output A permutation (reordering) of the input sequence

such that a1a2an

2010/11/30The Role of Algorithms in Computing 2



Correct Algorithms

A correct algorithm

Always produces the correct output for any input

For example: Input <31,41,59,26,41,58>

>>>>>A correct algorithm<<<<<

<26,31,41,41,58,59>




What is an Algorithm?

A detail step by step instruction

Describe the instruction in words

What kind of computer programming languages? Doesnt matter

What kind of human spoken languages?

The language you like, even Chinese, Martian

Language




How to Solve the Sort

Algorithms? Selection Sortvoid selectionSort(int numbers[], int array_size)

{

int i, j, min;

for (i = 0; i < array_size-1; i++)

{

min = i;

for (j = i+1; j < array_size; j++)

{

if (numbers[j] < numbers[min])

min = j;

}

if(min!=i) swap(numbers[x],numbers[i]);

}

}2010/11/30The Role of Algorithms in Computing 5



How to Solve the Sort

Algorithms? II MergeSort

Separate the list into halves

Sort the two halves recursively Merge the two sorted halves into one sorted list




Which one is better?

Better means

Time

Space Generally focusing on the worst-case number

of comparisons required to solve the

problem.




Time Needed

Selection Sort

n-1, n-2, n-3, ., 1

O(c1n2) MergeSort

T(n)=T(n/2) +T(n/2) + cn

T(n)=O(c2nlog2n)




Time Needed

MergeSort Wins

And Space?




Think about

Two machine

Machine A: 109 per second instructions

MachineB: 106

per seconds instructions Supposed c1=2, c2=50

We want to sort 102 integers

Which one runs faster? We want to sort 106 integers

Which one runs faster?




Problems in the Computer?

The Human Genome Project

Identify 100,000 genes

Determine 3 billion chemical based pairs Genes search between different human





Internet and the Web

IP route method.

Search Engine Compress the data

How to store the password (RSA)

NetworkSecurity

Lots of problems





Manufacturing

Resources price

Human power price Transportation price

Market

The place to set the

Factories

Warehouse

Stores




Applications in Computer

Hardware

GUI

Object-OrientedSystem WAN, LAN




Thinks About

Program P1 solves a problem in n days.

Program P2 solves the same problem in 2n

seconds.

Q: Which one you will use?

A: Program P1 runs faster forn

> 20.




Motivation

The rate of growth affects more.

We need notation to capture the concept of

rate of growth when we measure the timeand space efficiency of algorithms




Asymptotic Notation

Edmund Landau

1877~1938

Inventor of the asymptotic notation Donald E. Knuth

1938 ~

Turing Award, 1974.

Father of the analysis of algorithms

Popularizing the asymptotic notation




Asymptotic Notation




Using Asymptotic Notation in

Sentences Let n be the number of input integers

Program 1 takes time O(n)

It takes time O(n) for program 1 to solve theproblem

The time required by program 2 is O(2n)

O(n) reads?

Big-Oh of n

Order n




Meaning

Sentence

Program p2 takes time O(n) to solve the problem.

Means The time needed by p2 is a function f(n) with

f(n)=O(n)




Meaning 2

Sentence

The comparison-based sort methods can not be

solved in O(n) time. Means

The comparison-based sort problems can not besolved in h(n) time, for any function h(n) with

h(n)=O(n)




f(n)=O(g(n))

Intuitive meaning

f(n) does not grow faster then g(n)

Comments f(n)=O( g(n) ) roughly means f(n) g(n) in terms of

rate of growth

= is not the meaning of equation. Its more like

We do not write O(g(n))=f(n)




Formal Definition of O

For any two functions f(n) and g(n), we write

f(n)=O(g(n))

If there exists positive constants n0 and c suchthat the inequality f(n) c g(n)

holds for each integer n n0




In Other Words

The definition of f(n)=O(g(n)) states that

there exists a positive constant c such that

the value of f(n) is upper-bounded by cg(n)for all sufficiently large positive integers n.




Questions

n=O(n)?

999n=O(n)?

5n2=O(n3-n2)? n2=O(n)?




n=O(n)?

Yes, for n0=1 and c=1

n 1n , for n 1




999n=O(n)?

Yes, for n0=1, c=999

999n 999n , for n 1




5n2=O(n3-n2)?

Yes, let n0=2, c=5

5n25(n3-n2)

5n2 5n3-5n2

10n2 5n3

2 n




n2=O(n)?

Is n can be the upper-bound of n2

Instinctive feeling: NO

How to proof? Contradiction Method




Proof

Assume for a contradiction that there exists

constants c and n0 such that

n2

cn holds for any integer n with n n0. n is an

arbitrary integer strictly larger than

max(n0,c ).

For instance, let n = 1+max(n0,c)

n2>cn (n>c), contradiction.




Other Notations

Big-Oh only gives the upper bounds. We need

other asymptotic notations.

f(n)=O(g(n)) f(n) g(n) in rate of growth

f(n)=(g(n)) f(n) g(n) in rate of growth

f(n)=(g(n)) f(n) g(n) in rate of growth

f(n)=o(g(n)) f(n) g(n) in rate of growth

f(n)= (g(n)) f(n) g(n) in rate of growth




Omega

Definition [Omega]

f(n) = (g(n)) iff there exist positive constants

c and n0

such that f(n) cg(n) for all n, n n0.

The function g(n) is only a lower bound on f(n).

f(n) = (g(n)) to be informative, g(n) should be

as large a function of n as one can come up

with for which f(n) =

(g(n)) is true.

2010/11/30CH1: Basic Concepts 32



Theta

Definition [Theta]:

f(n) = (g(n)) iff there exist positive constants

c1, c

2and n

0such that c

1g(n) f(n) c

2g(n) for

all n, n n0.

Lower bound and upper bound on f(n)

The theta notation is more precise than

both ³big oh´ and omega notations.

2010/11/30CH1: Basic Concepts 33



Comparing Two Algorithms

We say that Algorithm A is no worse than

Algorithm B in terms of worst-case time

complexity if there exists a positive functionf(n) such that

Algorithm A runs in time O(f(n)) and

AlgorithmB runs in time(f(n)) in the worst case




Comparing Two Algorithms (2)

Algorithm A is strictly better than algorithm B

in terms of worst-case time complexity if

there exists a positive function f(n) such that Algorithm A runs in time O(f(n)) and

AlgorithmB runs in time (f(n)) in the worst case

OR

Algorithm A runs in time o(f(n)) and

AlgorithmB runs in time (f(n)) in the worst case




Tightness in Analysis

Supposed we figure out that the time

complexity of a algorithm is O(f(n)). We say

that the analysis is tight if the algorithm runsin(f(n)) in the worst case.

In others words, if the time complexity of the

algorithm is O(f(n)) and the analysis is tight,

then the time complexity of the algorithm is(f(n))




Time Optimal Algorithm

We say that Algorithm A is a optimal

algorithm for a problem P in terms of worst-

case time complexity if Algorithm A runs in time O(f(n)) and

Any algorithm that solves the problem P requirestime(f(n)) in the worst case.


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02_The Role of Algorithms in Computing