CHAPTER 1 BASIC CONCEPT · 2016. 10. 20. · d Row-Major: Column-Major: name m row, n column,...

NCUE CSIE Wireless Communications and Networking Laboratory

1

CHAPTER 2

ARRAYS


Arrays

Array A

1 2 3 ……. n

2

Definition:

(1)A set of index and value

(2)Using consecutive memory

(3)Support “Random Access ” and “Sequential Access”

(4) Insert/Delete an element: O(n)


int list[5], *plist[5];

list[5]: five integers

list[0], list[1], list[2], list[3], list[4]

*plist[5]: five pointers to integers

plist[0], plist[1], plist[2], plist[3], plist[4]

implementation of 1-D array

list[0] base address =

list[1] + sizeof(int)

list[2] + 2*sizeof(int)

list[3] + 3*sizeof(int)

list[4] + 4*size(int) 3

Arrays in C

0l

0l0l

0l

0l


Compare int *list1 and int list2[5] in C.

Same: list1 and list2 are pointers.

Difference: list2 reserves five locations.

Notations:

list2 - a pointer to list2[0]

(list2 + i) - a pointer to list2[i]=(&list2[i])

4

Arrays in C (Continued)


Addressing (1-Dimension Arrays)

Declare: A[1~n]

suppose : start address : size of each element

Then

name

Start index ended index

size=n

0l

A

1 2 3 4 5 ….. n

diliA )1(][ 0

d

5



(General case)

Declare: A[l~u]

suppose : start address : size of each element

Then

name

Start index ended index

size=u-l+1

0l

A

l l+1 l+2 l+3 l+4 ….. u

dliliA )(][ 0

d

6



Declare: A[1~m, 1~n] suppose : start address : size of each element Row-Major: Column-Major:

name m row, n column, size=m*n

0l

djniljiA ))1()1((],[ 0

dimjljiA ))1()1((],[ 0

d

7



(General case)

Declare: A[l1~u1, l2~u2] suppose : start address : size of each element Row-Major: Column-Major:

name (u1-l1+1) row, (u2-l2+1) column

0l

dljluliljiA ))()1()((],[ 22210

dliluljljiA ))()1()((],[ 11120

d

8



Declare: A[1~u1, 1~u2, 1-u3] suppose : start address : size of each element Row-Major: Column-Major:

0l

dkujuuilkjiA ))1())1(())1(((],,[ 3320

diujuuklkjiA ))1())1(())1(((],,[ 1120

2u

3u1u

d

9



(General case)

Declare: A[l1~u1, l2~u2, l3-u3] suppose : start address : size of each element Row-Major: Column-Major:

0l

)1()1()((],,[ 332210 lululilkjiA

122 lu

133 lu 111 lu

Row-major

column-major

dlklulj ))()1()( 3332

)1()1()((],,[ 112230 lululklkjiA

dlilulj ))()1()( 1112

d

10


Addressing (n-Dimension Arrays)

Declare: A[1~u1, 1~u2,… 1-un] suppose : start address : size of each element Row-Major:

0l

)...)1(((]...,[ 321021 nn uuualaaaA

)...)1(( 432 nuuua

)...)1(( 543 nuuua

dan ))1(

Row-major

d

11


Addressing (n-Dimension Arrays)

Declare: A[1~u1, 1~u2,… 1-un] suppose : start address : size of each element Column-Major:

0l

)...)1(((]...,[ 121021 uuualaaaA nnnn

)...)1(( 1321 uuua nnn

)...)1(( 1432 uuua nnn

da ))1( 1

column-major

d

12


0002800

0000091

000000

006000

0003110

150220015

col1 col2 col3 col4 col5 col6

row1

row2

row3

row4

row5

row6

(a) (b)

8/36 15/15

sparse matrix data structure?

Sparse Martix

13

5*3 6*6


Sparse Martix

14

Represent:

(1) Represented by a two-dimensional array

→ Sparse matrix wastes space

(2) 3-Tuple (Triples)

→ Each element is characterized by <row, col, value>

(3) Double Link-list

→We will discuss it in chapter 4


Triples

row col value

[0] 6 6 8

[1] 1 1 15

[2] 1 4 22

[3] 1 6 -15

[4] 2 2 11

[5] 2 3 3

[6] 3 4 -6

[7] 5 1 91

[8] 6 3 28

0002800

0000091

000000

006000

0003110

150220015

# of nonzero terms = k # of columns = n

# of rows = m

row1

row2

row3

row4

row5

row6

col1 col2 col3 col4 col5 col6

Row-major

15


Transposing a Matrix

t

mnnm AA t

jiij AA

Ex)

23

81

70

287

310

16


Transposing a Matrix (Algorithm 1)

For i=0 to k put A[i, j, value] into At[j, i, value] End

row col value [0] 6 6 8 [1] 1 1 15 [2] 4 1 22 [3] 6 1 -15 [4] 2 2 11 [5] 3 2 3 [6] 4 3 -6 [7] 1 5 91 [8] 3 6 28

It’s not row-major

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row col value [0] 6 6 8 [1] 1 1 15 [2] 1 5 91 [3] 2 2 11 [4] 3 2 3 [5] 3 6 28 [6] 4 1 22 [7] 4 3 -6 [8] 6 1 -15


For j=1 to n do For i=1 to k do if (A[i].col=j) then put A[i, j, value] into At[j, i, value] End End

→ when it’s not “sparse” anymore… 18

Time Complexity: O(n k)

where k m n


Compared with matrix

transport using 2-D array

For i=1 to m do For j=1 to n do put A[i, j] into At[j, i] End End

0002800

0000091

000000

006000

0003110

150220015

Time Complexity: O(m n)

19



Step1: For i=1 to k do column_element[A[i].col]++ End

Step2: For i=2 to n do row_start[i]=row_start[i-1]+column_element[i-1] End Step3: For i=1 to k do At[row_start[A[i].col]]=A[i] row_start[A[i].col]++ End

Time Complexity: O(k+n) or O(Max(k, n))

2 1 2 2 0 1 column_element

1 2 3 4 5 6

1 3 4 6 8 8 Row_start

1 2 3 4 5 6

20


Lower Triangular Matrix (row-major)

0

nn

Store: count=1 For i=1 to n do For j=1 to i do B[count]=A[i, j] count++ End End

B

1 2 3 4 5 ….. n(n+1)/2

(1,1) (2,1) (2,2) (3,1) (3,2) (n,n)

Access: if (i<j) then return 0 else return B[k]

jii

k

2

)1(

An n, Aij=0 if i<j

21


Lower Triangular Matrix (column-major)

An n, Aij=0 if i<j

B

1 2 3 4 5 ….. n(n+1)/2

(1,1) (2,1) (3,1) (4,1) (n,n)

0

ijj

jn

ijjn

jijjn

2

)1()1(

2

)1)(2(

)1(2

)1)(22(

Access:

22


Upper Triangular Matrix

Row-major:

Column-major:

jii

ink

2

)1()1(

ijj

k

2

)1(

23


Symmetric Matrix

An n, Aij= Aji

24

Definition:

Another ex.: Band Matrix An,a,b

EX:


A[-4~3, -3~2], l0=100, d=1 (a) If Row-major，then A[1, 1]=? (b) If Column-major，then A[1, 1]=?

Question:

25


A[-2~7, -4~10, -2~1, -3~2, 1~10], l0=38, d=8 (a) If Row-major，then A[0, 8, 0, 1, 8]=?

Question:

26


Suppose there is an integer array M[5, 8], the address of M[0, 0], M[0, 3], and M[1, 2] are 100, 106, and 120 respectively. What is the address of M[4, 5]

Question:

27


Assume in a byte machine. A is an array declared as A[-1~m, 2~n], and each element occupied 3 bytes. The address of A[3, 5] is at 180, and A[5, 3] is at 138 (1) Find the address of the element A[-1, 2] (2) Find the value of m or n

Question:

28


A[3, 3]=121, A[6, 4]=159, d=1, then A[4, 5]=?

Question:

29


A[1, 1]=2, A[2, 3]=18, A[3, 2]=28, then A[4, 5]=?

Question:

30


Reference

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Ellis Horowitz, Sartaj Sahni, and Susan Anderson-Freed

〝Fundamentals of Data Structures in C〞, W. H. Freeman & Co

Ltd, 1992.

Ellis Horowitz, Sartaj Sahni, and Dinesh Mehta

〝Fundamentals of Data Structures in C++〞 Silicon Pr, 2006

Richard F.Gilberg, Behrouz A. Forouzan,

〝Data Structures: A Pseudocode Approach with C〞, S

Baker & Taylor Books, 2004

Fred Buckley, and Marty Lewinter 〝A Friendly Introduction to Graph

Theory〞 Prentice Hall, 2002

〝資料結構-使用C語言〞蘇維雅譯，松崗，2004

〝資料結構-使用C語言〞蔡明志編著，全華，2004

〝資料結構(含精選試題)〞洪逸編著，鼎茂，2005

CHAPTER 1 BASIC CONCEPT · 2016. 10. 20. · d Row-Major: Column-Major: name m row, n column,...

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Transcript of CHAPTER 1 BASIC CONCEPT · 2016. 10. 20. · d Row-Major: Column-Major: name m row, n column,...