Design and Analysis Algorithm

Ahmad Afif Supianto, S.Si., M.Kom

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Contoh Algoritma

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m a[1];

for i 2 to size do

if m < a[i] then

m a[i];

end if

end for

return m

1. copy the input a to array t1;

assign n size of input;

2. While n > 1

For i 1 to n/2

t2[i ] min (t1 [ 2*i ], t1[ 2*i + 1] );

copy array t2 to t1;

n n/2;

end for

end while

3. Output t2[1];

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Algoritma

Recipe for getting things done successfully

Recipe

1. .... 2. .... 3. .... 4. .... 5. .... 6. dst

Algoritma

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Algoritma

Recipe for getting things done successfully

"Recipe" - well defined sequence of computational

steps

"things" - computational problems specifying an

input/output relation

"done" - in finite steps and time

"successfully" – correctly

Any special method of solving a certain kind of

problem (Webster Dictionary)

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Algoritma

An algorithm is a sequence of unambiguous

instructions for solving a problem

An algorithm is a sequence of computational

steps that transform the input into the output

An algorithm is a tool for solving a well-specified

computational problem

An algorithm is written in pseudocode

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Contoh : Suatu Algoritma

Problem: Input is a sequence of integers

stored in an array.

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25, 90, 53, 23, 11, 34

INPUT OUTPUT

instance

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ALGORITHM

m a[1];

for i 2 to size do

if m < a[i] then

m a[i];

end if

end for

return m

Output the minimum

Contoh : Algoritma A

Problem: Input is a sequence of integers

stored in an array.

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Output the minimum

Algorithm A

Contoh : Algoritma B

This algorithm uses two temporary arrays.

Visualisasikan ?

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1. copy the input a to array t1;

assign n size of input;

2. While n > 1

For i 1 to n/2

t2[i ] min (t1 [ 2*i ], t1[ 2*i + 1] );

copy array t2 to t1;

n n/2;

end for

end while

3. Output t2[1];

8 9 5 6 11 34 7 20

Visualisasi Algoritma B

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8 9 5 6 11 34 7 20

6 5 8 7

5 7

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Loop 1

Loop 2

Loop 3

Contoh : Algoritma C

Sort the input in increasing order. Return the

first element of the sorted data.

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8 9 5 6 11 34 7 20

5 6 7 8 9 11 20 34

Sorting black

box

Contoh : Algoritma D

For each element, test whether it is the

minimum.

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The Study of Algorithm

How to devise/design algorithms

How to express algorithms

How to validate algorithms

How to analyze algorithms

How to test a program

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Define Problem

Problem:

Description of Input-Output relationship

Algorithm:

A sequence of computational step that transform the

input into the output.

Data Structure:

An organized method of storing and retrieving data.

Our task:

Given a problem, design a correct and good

algorithm that solves it.

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Importance of Analyze Algorithm

Need to recognize limitations of various

algorithms for solving a problem

Need to understand relationship between

problem size and running time When is a running program not good enough?

Need to learn how to analyze an algorithm's

running time without coding it

Need to learn techniques for writing more

efficient code

Need to recognize bottlenecks in code as well

as which parts of code are easiest to optimize

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Why do we analyze about them?

understand their behavior

(Job -- Selection, performance, modify)

improve them

(Research)

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What do we analyze about them?

Correctness

Does the input/output relation match algorithm

requirement?

Amount of work done (aka complexity)

Basic operations to do task

Amount of space used

Memory used

Simplicity, clarity

Verification and implementation.

Optimality

Is it impossible to do better?

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What is the running time of this algorithm?

PUZZLE(x)

while x != 1

if x is even then

x = x / 2

else

x = 3x + 1

end if

Sample run: 7, 22; 11, 34; 17, 52; 26,13; 40, 20; 10, 5; 16, 8; 1

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The Selection Problem (1/2)

Problem: given a group of n numbers,

determine the kth largest

Algorithm 1

Simpan bilangan dalam sebuah array

Urutkan array secara descending

Beri nilai balikan bilangan pada posisi-k

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The Selection Problem(2/2)

Algorithm 2

Simpan bilangan k pertama dalam sebuah array

Urutkan array secara descending

Untuk setiap bilangan yang tersisa, jika bilangan tersebut lebih

besar daripada bilangan ke-k, masukkan bilangan tersebut

pada posisi yang benar di dalam array

Beri nilai balikan bilangan pada posisi-k

Which algorithm is better?

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Which algorithm is better?

The algorithms are correct,

but which is the best?

Measure the running time (number of

operations needed).

Measure the amount of memory used.

Note that the running time of the algorithms

increase as the size of the input increases.

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Time vs. Size of Input

Measurement

parameterized by the

size of the input.

The algorithms

A,B,C are

implemented and run

in a PC.

Algorithms D is

implemented and run

in a supercomputer.

Let Tk( n ) be the

amount of time taken

by the Algorithm

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Input Size

4

0

2

Tc (n)

Runnin

g tim

e

(

second)

Td(n)

What do we need?

Correctness: Whether the algorithm

computes the correct solution for all instances

Efficiency: Resources needed by the algorithm

1. Time: Number of steps.

2. Space: amount of memory used.

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Design dan Analysis Algorithm

Design an algorithm (Correctness) Prove the algorithm is correct.

• Loop invariant.

• Recursive function.

• Formal (mathematical) proof.

Analyze the algorithm (Efficiency) Time

• Worse case, best case, average case.

• For some algorithms, worst case occurs often, average case is often roughly as bad as the worst case. So generally, worse case running time.

Space

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Latihan

Buatlah sebuah algoritma untuk menampilkan N

bilangan pertama secara terbalik.

Contoh : 10, 9, 8, 7, 6, 5, 4, 3, 2, 1

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Latihan

Buat sebuah algoritma yang akan menerima

sebuah bilangan X dari user. Tampilkan pesan

“benar” jika X habis dibagi 2, 3 atau 7 dan

tampilkan “salah” jika tidak habis dibagi.

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