< C >1

2 .

1.

2.

3.

< C >2

1. . . (Algorithms are used for calculation, data processing, and automated reasoning

(Definition)

( a finite set of instructions to accomplish a particular task )

(Criteria) (1) : zero or more inputs

(2) : at least one output

(3) : clear and unambiguous instruction

(4) : terminates after a finite number of steps

3 (, , ) .

?a step-by-step procedure for calculations.

< C >3

1.1

3

(sequence) (repetition) .(condition) .

1) LA

.

.

LA

.

< C >4

2)

.

1.1

??

.

LA

.

.

Yes

No

< C >5

1.2

( )

n

.

Yes

No

Big x

(x)

Big

Big > x

[ C ]6

2.

(Sorting) (Searching).

(, )

.

( , )

.

.

( 1 , 1 ).

< C >7

2.1

: n .

: n .

.

(n-1) .

( )

10 13 25 15 4 20 5 29 14 21

4 5 10

A[10]

B[10]

< C >8

/* A B */

=> => ( 1 ) =>

void selection_sort(int *A, int*B){

int i, j, small, temp; //int A[10], B[10];

small=0; for(i=0; i A[i]) small=i; }B[0]=A[small]; A[small]=99;

/*small=0;for(i=1; i A[i]) small=i; }B[1]=A[small]; A[small]=99;

small=0;for(i=1; i A[i]) small=i; }B[2]=A[small]; A[small]=99;

}

[] sortingsearching.txt

< C >9

() A .

10 13 25 15 4 20 5 29 14 21

4 13 25 15 10 20 5 29 14 21

2.1

( )

A[10]

A[10]

( )

< C >10

/* A A */

void selection_sort(int *A, int*B){

int i, j, small, temp; //int A[10], B[10];

for(j=0; j

< C >11

( )

: n .

for (i=0; i

< C >12

( )

for (i=0; i

< C >13

2.2

(Search) : n .

(Sequential Search) .

n

( )

(Binary Search) .

/

( .)

9 15 16 19 21 39 51 65 76 85 99

9 15 16 19 21 39

< C >14

( )

* 65

9 15 16 19 21 39 51 65 76 85 99

51 65

51 65 76 85 99

.

39 65 . 65 39

.

76 65 . 65 76

.

2.2

< C >15

( ) =>

left right

- left right ,

- left=0, right=n-1

list

middle = (left+right)/2

list[middle] searchnum

1) searchnum < list[middle]

right middle (right=middle-1)

2) searchnum = list[middle]

middle (return middle)

3) searchnum > list[middle]

left middle+1 (left=middle+1)

middle

2.2

< C >16

/* , . */

int binsearch(int list[],int searchnum, int left,int right)

{

/*searchnum list [0]

Q/A 8 ?

(1) 55 (2) 45 (3) 9 (4) 10

7 ?(1) 55 (2) 45 (3) 9 (4) 10

int data[]={20, 1, 50, 55, 34, 21, 4, 66, 71, 8};

void selection_sort(int data[])

{ int i, j, t, min;

for(i=0; i17

Q/A

binsearch(list, 15, 0, 10)

3 ?

1 2 3 4

int list[]={9, 15, 16, 19, 21, 39, 51, 55, 76, 85, 99 };

int binsearch(int list[], int searchnum, int left, int right)

{ /*searchnum list [0]

< C >19

3.

(performance)

-

:

: (Complexity Theory)

-

: .

.

(Space Complexity)

:

(Time Complexity)

:

< C >20

3.1 (Space Complexity)

-

.

)

-

.

n 2 x n, 3 x n, n x n .

. ( !)

< C >21

- n

list[], n, tempsum, i => n+3

float sum(float list[], int n) { float tempsum = 0;

int i;for(i = 0; i < n; i++)

tempsum += list[i];return tempsum;

}

-

= = n + 3

< C >22

3.2 (Time complexity)

P t . 1GHz 2GHz ?

t T(P) ! T(P) = count the number of operations the program performs

< C >23

1 : float sum(int n) { 2 : int count=0;3 : int i;4 : for(i = 0; i < n; i++)5 : count+=2;6 : count += 3;7 : return count;8 : }

n=0 :for loop

4 = 2, 4, 6, 7

n=0 :

2 x n + 4 = 2 + 4, 5 n + 4, 6, 7

: (n 2)2-4-5-4-5-4-6-7(n 3 )2-4-5-4-5-4-5-4-6-7

3.2 (Time complexity)

< C >24

. .

void add(int a[][M_SIZE],int b[][M_SIZE], int c[][M_SIZE],int rows,int cols)

{int i, j;for(i = 0; i < rows; i++)for(j = 0; j < cols; j++)c[i][j] = a[i][j] + b[i][j];

}

= rows + 1 + (rows) * (cols + cols + 1))

2(rows)(cols) + 2(rows) + 1

rows cols n = 2n2 + 2n + 1

3.2 (Time complexity)

< C >25

-

2rows x cols + 2rows + 1

0 0 0

0 0 0

1 rows+1 rows+1

1 rows x (cols+1) rows x cols+rows

1 rows x cols rows x cols

0 0 0

void add(int a[ ], [Max_SIZE])

{

int i,j;

for(i=0;i

< C >26

3.3

n .

?

1 2

f(n) g(n)

: (time complexity)1) (input size)

- n .T(n) = ?

2)

- : A(n) = ?

: W(n) = ?- .- .

< C >27

n , n

Big O ) A f(n) O(g(n)) .

=> c n0 , n0 n f(n) cg(n) . , n g(n) f(n) . f(n) g(n) ..

) f(n) = 25n, g(n) = n2

if let c = 1,

25n n2 for all n 25 (n0 =25) f1(n) = 25n c n0 n

f(n) cg(n)

f(n) O(g(n)) .

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

( O-)

3.3

n f(n) = 25n g(n) = n2

1 25 1

2 50 4

25 625 625

26 650 676

< C >28

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

n, n n0 g(n) (upper bound). .

n = O(n2), n = O(n2.5)

n = O(n3), n = O(2n)

g(n) f(n) .

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

) if f(n) = amnm + ... + a1n + a0, then f(n) = O(n

m)

)

f(n) |ak|nk + |ak-1|nk-1 +...+ |a1|n + |a0|= {|ak| + |ak-1|/n +...+ |a1|/n

k-1+ |a0|/nk}nk

{|ak| + |ak-1| +...+ |a1| + |a0|}nk= cnk (c = |ak|+|ak-1|+...+|a1|+|a0|)= O(nk)

( O-)

3.3

< C >29

[] O-

1) f(n) = 3n + 4

3n + 4

< C >30

[] .

Omega : (lower bound) .

) f(n) = (g(n)) c n0 , n0 n f(n) cg(n)

. , n , g(n) f(n) . f(n) g(n) .

Theta :

) f(n) = (g(n)) c1, c2 n0 , n0 n c1g(n)

f(n) c2g(n). , n g(n) f(n) . O . .

- more precise than both the big oh and omega notations- g(n) is both an upper and lower bound on f(n)

( O-)

3.3

< C >31

class of time complexities

O(1): constant

O(log2n): logarithmic

O(n): linear

O(nlog2n): log-linear O(n2): quadratic

O(n3): cubic

O(2n): exponential

O(n!): factorial

(polynomial time)

(exponential time)

( O(f(n)) )

3.3

< C >32

( O(f(n)) )

n 1 2 4 8 16 32

1log n

nn log n

n2

n3

constantlogarithmiclinearlog linearquadraticcubic

101011

112248

1248

1664

138

2464

512

14

1664

2564096

15

32160

102432768

2nn!

exponentialfactorial

21

42

1624

25640326

6553620922789888000

429496729626313X 10 33

.

3.3

Q/A

1. (O- ) O- .

1) 5n4 + 6n + 3

2) nlogn + 2n + 7

3) 10

4) nlogn + 1000n

2. (O- ) - O- Order .

n nlogn n2logn n! 2n n2 n1/2 logn

< C >33

< C >34

Review

.

:

: , ,

.

n .

O- .

O(f(n)), O(g(n)) f(n) > g(n) O(g(n)) .

O(f(n)) f(n) factorial .