Design and Analysis Algorithm. Ahmad Afif Supianto, S.Si., M.Kom. Pertemuan 04

Design and Analysis Algorithm Ahmad Afif Supianto, S.Si., M.Kom

Pertemuan 04

Contents

31

Asymptotic Analysis

2

Brute Force Algorithm 1

2

Asymptotic Analysis

Asymptotic Notation  Think of n as the number of records we wish to      

sort with an algorithm that takes f(n) to run. How long will it take to sort n records? What if n is big? We are interested in the range of a function as n gets large. Will f stay bounded? Will f grow linearly? Will f grow exponentially? Our goal is to find out just how fast f grows with respect to n.

Asymptotic Notation Memperkirakan formula untuk run-time

Indikasi kinerja algoritma (untuk jumlah data yang sangat besar)

 Misalkan: T(n) = 5n2 + 6n + 25 T(n) proporsional untuk ordo n2 untuk data yang sangat besar.

5

Asymptotic Notation  Indikator efisiensi algoritma bedasar pada OoG pada basic operation suatu algoritma.  Penggunaan notasi sebagai pembanding urutan OoG:  O (big oh)  Ω (big omega)  Ө (big theta)

 t(n) : algoritma running time (diindikasikan dengan basic operation count (C(n))  g(n) : simple function to compare the count 6

Classifying functions by their Asymptotic Growth Rates (1/2)  asymptotic growth rate, asymptotic order, or order of functions  Comparing and classifying functions that ignores constant factors and small inputs.

 O(g(n)), Big-Oh of g of n, the Asymptotic Upper Bound;  W(g(n)), Omega of g of n, the Asymptotic Lower Bound.  Q(g(n)), Theta of g of n, the Asymptotic Tight Bound; and

Example  Example: f(n) = n2 - 5n + 13.  The constant 13 doesn't change as n grows, so it is not crucial. The low order term, -5n, doesn't have much effect on f compared to the quadratic term, n2. We will show that f(n) = Q(n2) .

 Q: What does it mean to say f(n) = Q(g(n)) ?  A: Intuitively, it means that function f is the same order of magnitude as g.

Example (cont.)  Q: What does it mean to say f1(n) = Q(1)?  A: f1(n) = Q(1) means after a few n, f1 is bounded above & below by a constant.  Q: What does it mean to say f2(n) = Q(n log n)?  A: f2(n) = Q(n log n) means that after a few n, f2 is bounded above and below by a constant times n log n. In other words, f2 is the same order of magnitude as n log n.  More generally, f(n) = Q(g(n)) means that f(n) is a member of Q(g(n)) where Q(g(n)) is a set of functions of the same order of magnitude.

Big-Oh  The O symbol was introduced in 1927 to indicate relative growth of two functions based on asymptotic behavior of the functions now used to classify functions and families of functions

Upper Bound Notation  We say Insertion Sort’s run time is O(n2)  Properly we should say run time is in O(n2)  Read O as “Big-O” (you’ll also hear it as “order”)

 In general a function  f(n) is O(g(n)) if  positive constants c and n0 such that f(n)  c  g(n)  n  n0

 e.g. if f(n)=1000n and g(n)=n2, n0 > 1000 and c = 1 then f(n0) < 1.g(n0) and we say that f(n) = O(g(n))

Asymptotic Upper Bound • f(n)  c g(n) for all n  n0 • g(n) is called an asymptotic upper bound of f(n). • We write f(n)=O(g(n)) • It reads f(n) is big oh of g(n).

c g(n)

f(n)

g(n)

n0

Big-Oh, the Asymptotic Upper Bound  This is the most popular notation for run time since we're usually looking for worst case time.  If Running Time of Algorithm X is O(n2) , then for any input the running time of algorithm X is at most a quadratic function, for sufficiently large n.

 e.g. 2n2 = O(n3) .  From the definition using c = 1 and n0 = 2. O(n2) is tighter than O(n3).

Example 1

g(n) for all n>6, g(n) > 1 f(n). Thus the function f is in the big-O of g. that is, f(n) in O(g(n)).

f(n)

6

Example 2 g(n) There exists a n0 s.t. for all n>n0, f(n) < 1 g(n). Thus, f(n) is in O(g(n)).

f(n)

5

Example 3 3.5 h(n) There exists a n0=5, c=3.5, s.t. for all n>n0, f(n) < c h(n). Thus, f(n) is in O(h(n)).

f(n) h(n) 5

Example of Asymptotic Upper Bound 4g(n)=4n2

4 g(n) = 4n2 = 3n2 + n2  3n2 + 9 for all n  3 > 3n2 + 5 = f(n) Thus, f(n)=O(g(n)).

f(n)=3n2+5

g(n)=n2

3

Exercise on O-notation  Show that 3n2+2n+5 = O(n2) 10 n2 = 3n2 + 2n2 + 5n2  3n2 + 2n + 5 for n  1 c = 10, n0 = 1

Exercise on O-notation    

f1(n) = 10 n + 25 n2 f2(n) = 20 n log n + 5 n f3(n) = 12 n log n + 0.05 n2 f4(n) = n1/2 + 3 n log n

• • • •

O(n2) O(n log n) O(n2) O(n log n)

Classification of Function : BIG O (1/2)  A function f(n) is said to be of at most logarithmic 

  

growth if f(n) = O(log n) A function f(n) is said to be of at most quadratic growth if f(n) = O(n2) A function f(n) is said to be of at most polynomial growth if f(n) = O(nk), for some natural number k > 1 A function f(n) is said to be of at most exponential growth if there is a constant c, such that f(n) = O(cn), and c > 1 A function f(n) is said to be of at most factorial growth if f(n) = O(n!).

Classification of Function : BIG O (2/2)  A function f(n) is said to have constant running time if the size of the input n has no effect on the running time of the algorithm (e.g., assignment of a value to a variable). The equation for this algorithm is f(n) = c  Other logarithmic classifications:  f(n) = O(n log n)  f(n) = O(log log n)

Lower Bound Notation  We say InsertionSort’s run time is W(n)  In general a function  f(n) is W(g(n)) if  positive constants c and n0 such that 0  cg(n)  f(n)  n  n0

 Proof:  Suppose run time is an + b • Assume a and b are positive (what if b is negative?)

 an  an + b

Big W Asymptotic Lower Bound • f(n)  c g(n) for all n  n0 • g(n) is called an asymptotic lower bound of f(n). • We write f(n)=W(g(n)) • It reads f(n) is omega of g(n).

f(n)

c g(n)

n0

Example of Asymptotic Lower Bound 2 g(n)=n 2

g(n)=n

g(n)/4 = n2/4 = n2/2 – n2/4  n2/2 – 9 for all n  6 < n2/2 – 7 Thus, f(n)= W(g(n)).

f(n)=n2/2-7

c g(n)=n2/4

6

Example: Big Omega  Example: n 1/2 = W( log n) . Use the definition with c = 1 and n0 = 16.

Checks OK. Let n > 16 : n 1/2 (1) log n if and only if n = ( log n )2 by squaring both sides. This is an example of polynomial vs. log.

Big Theta Notation  Definition: Two functions f and g are said to be of equal growth, f = Big Theta(g) if and only if both f=Q(g) and g = Q(f).

 Definition: f(n) = O(g(n)) means  positive constants c1, c2, and n0 such that c1 g(n)  f(n)  c2 g(n)  n  n0  If f(n) = O(g(n)) and f(n) = W(g(n)) then f(n) = Q(g(n))

(e.g. f(n) = n2 and g(n) = 2n2)

Theta, the Asymptotic Tight Bound  Theta means that f is bounded above and below by g; BigTheta implies the "best fit".  f(n) does not have to be linear itself in order to be of linear growth; it just has to be between two linear functions,

Asymptotically Tight Bound • f(n) = O(g(n)) and f(n) = W(g(n)) • g(n) is called an asymptotically tight bound of f(n). • We write f(n)=Q(g(n)) • It reads f(n) is theta of g(n).

c2 g(n)

f(n) c1 g(n)

n0

Other Asymptotic Notations  A function f(n) is o(g(n)) if  positive constants c and n0 such that f(n) < c g(n)  n  n0  A function f(n) is (g(n)) if  positive constants c and n0 such that c g(n) < f(n)  n  n0  Intuitively, – o() is like < – O() is like 

– () is like > – W() is like 

– Q() is like =

Examples 1. 2n3 + 3n2 + n = 2n3 + 3n2 + O(n) = 2n3 + O( n2 + n) = 2n3 + O( n2 ) = O(n3 ) = O(n4) 2. 2n3 + 3n2 + n = 2n3 + 3n2 + O(n) = 2n3 + Q(n2 + n) = 2n3 + Q(n2) = Q(n3)

Examples (cont.) 3. Suppose a program P is O(n3), and a program Q is O(3n), and that currently both can solve problems of size 50 in 1 hour. If the programs are run on another system that executes exactly 729 times as fast as the original system, what size problems will they be able to solve?

Example (cont.) n3 = 503 * 729 n = 3 503 * 729 n = 3 503 3 729 n = 50 * 9 n = 50 * 9 = 450

3n = 350 * 729 n = log3 (729 * 350) n = log3(729) + log3 350 n = 6 + log3 350 n = 6 + 50 = 56

 Improvement: problem size increased by 9 times for n3 algorithm but only a slight improvement in problem size (+6) for exponential algorithm.

More Examples (a) 0.5n2 - 5n + 2 = W( n2). Let c = 0.25 and n0 = 25. 0.5 n2 - 5n + 2 = 0.25( n2) for all n = 25

(b) 0.5 n2 - 5n + 2 = O( n2). Let c = 0.5 and n0 = 1. 0.5( n2) = 0.5 n2 - 5n + 2 for all n = 1 (c) 0.5 n2 - 5n + 2 = Q( n2) from (a) and (b) above. Use n0 = 25, c1 = 0.25, c2 = 0.5 in the definition.

More Examples (d) 6 * 2n + n2 = O(2n). Let c = 7 and n0 = 4. Note that 2n = n2 for n = 4. Not a tight upper bound, but it's true. (e) 10 n2 + 2 = O(n4). There's nothing wrong with this, but usually we try to get the closest g(n). Better is to use O(n2 ).

How to show a function f(n) is in/not in O(g(n)). f(n) is in O(g(n)): find a constant c and large n0 such that for all n > n0, f(n) < c g(n). f(n) is not in O(g(n)): for any constant c and any large n0, we can find a m such that f(m) > c g(m).

Usually it is more difficult to proof that a function f(n) is not in the big-O of another function g(n). c,n0 c,n0

n>n0 s.t. n>n0 s.t.

f(n)< c g(n) f(n)>= c g(n)

Big O Again!!!!  O(1) The cost of applying the algorithm can be 

  

bounded independently of the value of n. This is called constant complexity. O(log n) The cost of applying the algorithm to problems of sufficiently large size n can be bounded by a function of the form k log n, where k is a fixed constant. This is called logarithmic complexity. O(n) linear complexity O(n log n) n lg n complexity O(n2) quadratic complexity

Big O Again!!!!        

O(n3) cubic complexity O(n4) quartic complexity O(n32) polynomial complexity O(cn) If constant c 1, then this is called exponential complexity O(2n) exponential complexity O(en) exponential complexity O(n!) factorial complexity O(nn)

Practical Complexity t < 250 250

f(n) = n f(n) = log(n) f(n) = n log(n) f(n) = n^2 f(n) = n^3 f(n) = 2^n

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20



0 1

3

5

7

9

11

13

15

17

19


4000 f(n) = n f(n) = log(n)

3000

f(n) = n log(n) f(n) = n^2 2000

f(n) = n^3 f(n) = 2^n

1000

0 1

3

5

7

9

11

13

15

17

19

Algoritma Brute Force 1

Definisi Brute Force  Brute force : pendekatan straight forward untuk memecahkan suatu masalah  Algoritma brute force memecahkan masalah dengan sangat sederhana, langsung, danjelas (obvious way)

Contoh-contoh (Berdasarkan pernyataan masalah)

 Mencari elemen terbesar (terkecil)  Persoalan: Diberikan sebuah array yang beranggotakan n buah bilangan bulat (a1, a2, ..., an).Carilah elemen terbesar di dalam array tersebut.  Algoritma brute force: bandingkan setiap elemen array untuk menemukan elemen terbesar  Kompleksitas O(n)


 Mencari elemen terbesar (terkecil)


 Pencarian beruntun (Sequential Search)  Persoalan: Diberikan array yang berisi n buah bilangan bulat (a1, a2, ..., an). Carilah nilai x di dalam array tersebut. Jika x ditemukan, maka keluarannya adalah indeks elemen array, jika x tidak ditemukan, maka keluarannya adalah 0.  Algoritma brute force (sequential serach): setiap elemen array dibandingkan dengan x. Pencarian selesai jika x ditemukan atau elemen array sudah habis diperiksa.  Kompleksitas O(n)


 Pencarian beruntun (sequential search)

Contoh-contoh (Berdasarkan definisi konsep yang terlibat)

 Menghitung an (a > 0, n adalah bilangan bulat tak-negatif)  Definisi: an= a x a x ... x a (n kali) , jika n>0 = 1, jika n = 0  Algoritma brute force: kalikan 1 dengan a sebanyak n kali  Kompleksitas O(n)


 Menghitung pangkat


 Menghitung n! (n bilangan bulat tak-negatif) Definisi: n!=1 ×2×3×...×n, jika n>0 = 1, jika n = 0  Algoritma brute force: kalikan n buah bilangan, yaitu 1, 2, 3, ..., n, bersama-sama  Kompleksitas O(n)


 Menghitung faktorial


 Mengalikan dua buah matriks, A dan B Definisi: Misalkan C = A × B dan elemenelemen matrik dinyatakan sebagai cij, aij, dan bij

 Algoritma brute force: hitung setiap elemen hasil perkalian satu per satu, dengan cara mengalikan dua vektor yang panjangnya n.


 Mengalikan dua buah matriks


 Menemukan semua faktor dari bilangan bulat n (selain dari 1 dan n itu sendiri). Definisi: Bilangan bulat a adalah faktor dari bilangan bulat b jika a habis membagi b.  Algoritma brute force: bagi n denga n setiap i = 2, 3, ..., n – 1. Jika n habis membagi i, maka I adalah faktor dari n.


 Menemukan faktor bilangan bulat n


 Uji keprimaan Persoalan: Diberikan sebuah bilangan bilangan bulat positif. Ujilah apakah bilangan tersebut merupakan bilangan prima atau bukan. Definisi: bilangan prima adalah bilangan yang hanya habis dibagi oleh 1 dan dirinya sendiri. Algoritma brute force: bagi n dengan 2 sampai n–1. Jika semuanya tidak habis membagi n, maka n adalah bilangan prima  Perbaikan: cukup membagi dengan 2 sampai √n saja


 Uji bilangan prima


 Algoritma Pengurutan Brute Force, Algoritma apa yang paling lempang dalam memecahkan masalah pengurutan? Bubble sort dan selection sort!  Kedua algoritma ini memperlihatkan teknik brute force dengan jelas sekali.

Contoh-contoh (Berdasarkan definisi konsep yang terlibat) Bubble Sort Mulai dari elemen ke-n:  1. Jika sn< sn-1, pertukarkan  2. Jika sn-1 < sn-2, pertukarkan ...  3. Jika s2< s1, pertukarkan 1 kali pass Ulangi lagi untuk pass ke-i


 Bubble sort

Contoh-contoh (Berdasarkan definisi konsep yang terlibat) Selection Sort Pass ke –1:  1.Cari elemen terbesar mulai di dalam s[1..n]  2.Letakkan elemen terbesar pada posisi n (pertukaran) Pass ke-2:  1.Cari elemen terbesar mulai di dalam s[1..n - 1]  2.Letakkan elemen terbesar pada posisi n - 1 (pertukaran)  Ulangi sampai hanya tersisa 1 elemen


 Selection sort

Contoh-contoh (Berdasarkan definisi konsep yang terlibat) Mengevaluasi polinom  Persoalan: Hitung nilai polinom p(x)=anxn +an-1xn-1 +...+a1x +a0 untuk x = t.  Algoritma brute force: xi dihitung secara brute force (seperti perhitungan an). Kalikan nilai xi dengan ai, lalu jumlahkan dengan suku-suku lainnya.


 Analisa polinom


 Perbaikan (improve)

Karakteristik Algoritma Brute Force

 Algoritma brute force umumnya tidak “cerdas” dan tidak cepat, karena ia membutuhkan jumlah langkah yang besar dalam penyelesaiannya.  Kata “force” mengindikasikan “tenaga” ketimbang “otak”  Kadang-kadang algoritma brute force disebut juga algoritma naif (naïve algorithm).


 Algoritma brute force lebih cocok untuk masalah yang berukuran kecil.  Pertimbangannya:  sederhana,  Implementasinya mudah

 Algoritma brute force sering digunakan sebagai basis pembanding dengan algoritma yang lebih cepat.


 Meskipun bukan metode yang cepat, hampir semua masalah dapat diselesaikan dengan algoritma brute force.  Sukar menunjukkan masalah yang tidak dapat diselesaikan dengan metode brute force. Bahkan, ada masalah yang hanya dapat diselesaikan dengan metode brute force.  Contoh: mencari elemen terbesar di dalam array.

Tugas Buatlah algoritma berikut dengan kompleksitasnya  Pencocokan String (String Matching)  Mencari pasangan titik yang jaraknya terdekat (Closest pairs)  Travelling Salesman Problem (TSP)  Knapsack Problem

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Design and Analysis Algorithm. Ahmad Afif Supianto, S.Si., M.Kom. Pertemuan 04

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