Design and Analysis of Algorithm

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Design and Analysis of Algorithm

Week 1: Introduction

Dr. Putu Harry Gunawan1

1_{Department of Computational Science} School of Computing

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Outline

1 Introduction

About this course Beginning of Algorithm The efficiency of algorithm

2 Exercise Rules Skilled Group Careful Group Smart Group 3 Homeworks

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Outline

1 Introduction

About this course

Beginning of Algorithm The efficiency of algorithm

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Goals

Sebelum UTS:

1 _{Mahasiswa mampu untuk mejelaskan kompleksitas waktu pada suatu}

algoritma

2 _{Mahasiswa mampu menganalisis kompleksitas suatu algoritma.}

Setelah UTS:

1 Mahasiswa mampu membedakan tipe dan karakteristik

masing-masing algoritma.

2 Mahasiswa mampu merancang suatu algoritma berdasarkan

contoh-contoh algoritma yang sudah diberikan.

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References

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Grading

TUGAS 15% KUIS 20% UTS 25% TUGAS BESAR 15% UAS 25%

NB: Bonus jika absensi mencapai 100%

(Membuat buku tugas yang akan dikumpul setiap pengumpulan tugas)

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Outline

1 Introduction

About this course

Beginning of Algorithm

The efficiency of algorithm

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What is algorithm?

An algorithm is a sequence of unambiguous instructions for solving a problem, i.e., for obtaining a required output for any legitimate input in a finite amount of time.

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Algorithm

An algorithm is a sequence of unambiguous instructions for solving a problem, i.e., for obtaining a required output for any legitimate input in a finite amount of time.

Can be represented various forms Unambiguity/clearness

Effectiveness

Finiteness/termination Correctness

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Example of computational domain

Statement of problem:

Input: A sequence of n numbers<a1,a2,· · · ,an>

Output: A reordering of the input sequence<a0₁,a₂0,· · ·,a0_n>so that a0_i ≤a0_j wheneveri<j

Instance: The sequence<5,3,2,8,3>

Algorithms:

Selection sort Insertion sort Merge sort (many others)

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Some well-known of computational domain

Sorting Searching

Shortest paths in a graph Minimum spanning tree Primality testing

Traveling salesman problem Knapsack problem

Chess

Towers of Hanoi Program termination

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Basic Issues Related to Algorithms

How to design algorithms How to express algorithms Proving correctness

Efficiency (or complexity) analysis

Theoretical analysis Empirical analysis

Optimality

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Algorithms and design strategies

Brute force

Divide and conquer Decrease and conquer Transform and conquer Greedy approach Dynamic programming

Backtracking and branch-and-bound Space and time tradeoffs

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Analysis algorithms

How good is the algorithm?

Correctness Time efficiency Space efficiency

Does there exist a better algorithm?

Lower bounds Optimality

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Euclid’s algorithm

Problem: Find gcd(m,n), the greatest common divisor of two nonnegative, not both zero integers mand n

Examples: gcd(60,24) = 12, gcd(60,0) = 60, gcd(0,0) = ?

Euclids algorithm is based on repeated application of equality

gcd(m,n) =gcd(n,m mod n) until the second number becomes 0, which makes the problem trivial.

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Euclid’s algorithm

Examples: gcd(60,24) = 12, gcd(60,0) = 60, gcd(0,0) = ? Euclids algorithm is based on repeated application of equality

Example: gcd(60,24)

= gcd(24,12) = gcd(12,0) = 12

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Euclid’s algorithm

Examples: gcd(60,24) = 12, gcd(60,0) = 60, gcd(0,0) = ? Euclids algorithm is based on repeated application of equality

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Euclid’s algorithm

Exercise: Make an algorithm for computing the gcd(m,n)!

Step 1 If n = 0, return m and stop; otherwise go to Step 2 Step 2 Divide m by n and assign the value of the remainder to r

Step 3 Assign the value of n to m and the value of r to n. Go to Step 1.

while n6= 0 do

r ←m mod n m←n n ←r

return m

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Euclid’s algorithm

Exercise: Make an algorithm for computing the gcd(m,n)! Step 1 If n = 0, return m and stop; otherwise go to Step 2 Step 2 Divide m by n and assign the value of the remainder to r

Step 3 Assign the value of n to m and the value of r to n. Go to Step 1.

while n6= 0 do

r ←m mod n m←n n ←r

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Another method to compute gcd

Consecutive integer checking algorithm Step 1 Assign the value of min{m,n}to t

Step 2 Divide m by t. If the remainder is 0, go to Step 3; otherwise, go to Step 4

Step 3 Divide n by t. If the remainder is 0, return t and stop; otherwise, go to Step 4

Step 4 Decrease t by 1 and go to Step 2

Is this slower than Euclids algorithm? How much slower?

O(n), if n≤m , vsO(logn)

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Another method to compute gcd

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Another method to compute gcd

O(n), if n≤m , vsO(logn)

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Outline

1 Introduction

About this course Beginning of Algorithm

The efficiency of algorithm

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Efficiency

Pertimbangan Memilih algoritma : Kebenaran

Tepat guna (efektif)

output sesuai dengan inputnya sesuai dengan permasalahan

Kemudahan/ kesederhanaan

Untuk dipahami

Untuk diprogram (proses coding)

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Efficiency

Pertimbangan Memilih algoritma : Kecepatan Algoritma:

Berkaitan dengan kecepatan eksekusi program Hemat Biaya:

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Efficiency

Keempatnya sulit dicapai bersamaan, dan biasanya yang diutamakan adalah efisiensi (cepat dan hemat)

Analisis Algoritma : menganalisis efisiensi algoritma, yang mencakup :

efisiensi waktu (kecepatan)→banyaknya operasi yang dilakukan efisiensi memori→struktur data dan variabels yang digunakan

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Efficiency

Algoritma yang bagus adalah algoritma yang efisien

Algoritma yang efisien ialah algoritma yang meminimumkan kebutuhan waktu dan ruang/memori.

Kebutuhan waktu dan ruang suatu algoritma bergantung pada ukuran masukan (n), yang menyatakan jumlah data yang diproses.

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Efficiency

Question:

Mana yang lebih baik : menggunakan algoritma yang waktu eksekusinya cepat dengan komputer standard atau menggunakan algoritma yang waktunya tidak cepat tetapi dengan komputer yang cepat?

¡Watch Video¿

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Efficiency

Misal dipunyai :

Algoritma dengan waktu eksekusi dalam orde 2n Sebuah komputer dengan kecepatan 10−4×2n

n= 10→1/10 detik

n= 20→2 menit

n= 30→lebih dari satu hari

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Efficiency

Misal dipunyai :

Algoritma dengan waktu eksekusi dalam orde 2n Sebuah komputer dengan kecepatan 10−6x2n

n= 45→1 tahun

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Efficiency

Misal dipunyai :

Algoritma dengan waktu eksekusi dalam orde n3

Sebuah komputer dengan kecepatan 10−4xn3 n= 900→1 hari

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Efficiency

Misal dipunyai :

Mengapa kita memerlukan algoritma yang efisien? Lihat grafik di bawah ini.

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Outline

1 Introduction

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Rules

Make a group

Each group consists of 5 or more peoples Choose a leader

Prepare the answer sheet

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Outline

1 Introduction

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Problem 1

Buatlah algoritma untuk membuat Jus Nanas!

Usahakan menggunakan sintax-syntax berikut sebanyak mungkin Comment Assignment Logical relational while loop for loop repeat loop conditional case input output procedure

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Problem 1

Buatlah algoritma untuk membuat kopi!

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Problem 2

Buatlah algoritma untuk membuat Sambal Balado!

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Problem 2

Buatlah algoritma untuk membuat Lumpia Basah!

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Outline

1 Introduction

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Problem 3

Given the following algorithm:

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Outline

1 Introduction

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Problem 4

Design a algorithm that reads as its inputs the (x,y) coordinates of the endpoints of two line segments P1,Q1 and P2,Q2 and determines

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Problem 5

Design an algorithm for the following problem: Given a set ofn points in the Cartesian plane, determine whether all of them lie on the same circumference.

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Homeworks

Compare two algorithms of shorting problem, give an analysis and a comment for the each algorithms.

Compare two algorithms of searching problem, give an analysis and a comment for the each algorithms.

Prove the following relations

1 _{1 + 2 + 3 +}· · ·₊_n₌n(n+1)

2

2 _{1 + 3 + 5 +}· · ·_{+ (2n}−_{1) =}_n2 3 _if_n∈ Z _and_n≥_{0, then}Pn

i=0i·i! = (n+ 1)!−1

Compute the following sums.

1 Pn+1 i=3 1 2 Pn+1 i=3 i 3 Pn−1 i=0 i(i+ 1) 4 Pn i=03j+1 5 Pn i=1 Pn j=1ij

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The end of week 1

Thank you for your attention!