Algorithms and Complexity: Searching, Sorting, and Growth of Functions, Study notes of Discrete Mathematics

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The Fundamentals

Definition 13.1 Algorithms

An algorithm is a definite procedure for solving a

Problem: Searching Algorithmsproblem using a finite number of steps.

Locate an element

x

in a list of distinct elements

a 1 , a

2 ,... , a

n

or determine that it is not in the list.

Binary Search Algorithm procedure binary

search(

x :integer,

a 1 , a

2 ,... , a

n

integers in in-

1. creasing order)

i := 1

j :=

n

3. while

i < j

5. 4. begin

m

:=

i (^) +

(^) j ) / 2 ⌋

middle element

if

x > a

m

then

i :=

m

else

j

:=

m

9. if 8. end

x

=

a i then

location

i

10. else

location

Problem:Sorting Algorithms

Put a list

a 1 , a

2 ,... , a

n

in the increasing order.

The Bubble Sort procedure bubblesort(

a 1 ,... , a

n )

1. for

i := 1 to

n

−

for

j := 1 to

n

−

i

if

a j

a

j

then interchange

a j

with

a j

Definition 2 3.2 The Growth of Functionswhen solving an optimization problem. Greedy algorithm makes the ”best” decision for the current stepGreedy Algorithms

Let

f

and

g

be two functions from the set of integer

(or real) numbers. We say that

f (^) ( x )

is

O

g ( x ))

(which we write

as

f (^) ( x ) =

O

g ( x ))

) if there are two constants

C

and

k

such that

f (^) ( x ) | ≤

C | g ( x ) | ,

whenever

x > k

Theorem 1

Let

f (^) ( x ) =

a n x n + a n − 1 x n − 1 +

a 0

where

a 0 ,... , a

n

are real numbers. Then

f (^) ( x ) =

O ( x n ).

Fact 2

1. Suppose that

f 1 ( x ) =

O

g ( x ))

and

f 2 ( x ) =

O

g ( x ))

.

Then

( f 1 + f 2

x ) =

O

g ( x ))

.

2. Suppose that

f 1 ( x ) =

O ( g 1 ( x

and

f 2 ( x ) =

O ( g 2 ( x ).

Then

f 1 f 2 )(

x ) =

O ( g 1 ( x ) g 2 ( x

3.3 Complexity

1. Time used to solve a problem (

time complexity

2. Memory space used (

space complexity

Time analysis:

Worst-case analysis.

Average-case analysis.

3. Best-case analysis.

Definition 6Modular Arithmetic

Let

a

be an integer and

m

a positive integer.

We

denote by

a

mod

m

the remainder when

a

is divided by

m

.

Definition 7

If

a

and

b

are integers and

m

is a positive integer

then

a

is congruent to

b

modulo

m

if

m | ( a − b ).

Fact 4

Let

m

be a positive integer and let

a

and

b

be two inte-

gers.

a

≡

b

(mod

m

)

if and only if

a

mod

m

b

mod

m

.

a

≡

b

(mod

m

)

if and only if there is an integer

k

such that

a

=

b

mk

Theorem 6The Division Algorithm

Let

a

be an integer and let

d

be a positive integer.

Then there exist unique integers

q

and

r

such that

a

=

dq

r

and

r < d

Definition 9Greatest Common Divisor and Least Common Multiple

Let

a , b be two integers not both zero. The largest

integer

d

such that

d

| a

and

d

| b

is called the greatest common

divisor of

a

and

b

. It is denoted by

gcd

a, b

Definition 10

The least common multiple of the positive inte-

gers

a

and

b

is the smallest positive integer that is divisible by

both

a

and

b

. It is denoted by

lcm

a, b

Theorem 7

Let

a , b

be positive integers. Then

a (^) ·

b

=

gcd

a, b

lcm

a, b

Euclidean Algorithm3.6 Integers and Algorithms Goal:

Find

gcd

a, b

a

≥

b .

Method:

Let

r 0

=

a, r

1

=

b

. Then

r 0 = r 1 q 1 + r 2

where 0

< r

2

< r

1

r 1 = r 2 q 2 + r 3

where 0

< r

3

< r

2

r... n − 2 = r n 1 q n − 1 + r n

where 0

< r

n

< r

n − 1

r n − 1 = r n q n

procedure gcd(Euclidean Algorithm

a, b

: positive integers)

x

:=

a

y

:=

b

3. while(

y

6 = 0)

5. 4. begin

r

:=

x

mod

y

x

:=

y

y

:=

r

9. return 8. end

gcd

a, b

x

Lemma 9

Let

a

=

bq

r

where

a, b, q

and

r

are integers.

Then

gcd

a, b

gcd

b, r