Data Representation Methods

1
??????? ? ?????? ??

• ??????? ???? ??

http//www.cise.ufl.edu/sahni/cop3530 ??????? ??

2
???? ???

• ?????? ???? ??????? ???? ??????? ????? ?? ?? ???
  ???
• ???? ??? ???? ??????? ?????? ??? ????? ?? ????
  ??????? ?? ??? ( ????????...)
• class Linear List
• int list-size
• Object Type Memberslist-size
• //Operations
• AddNewMember (Object Type m)
• RemoveMember(int n)

3
????? ???? ??? ?? ?????

• ???? ?? ????? ?? ???? ????? elements ???? ????
  ?? ???

a
b
c
d
e
L (a, b, c, d, e) ??? i ?? ????? ?? ???
elementsi ???? ?? ????
4
????? ???? ?? ??
5
????? ?? ?? ????
6
????? ?????
7
????? ?????
size 5

• ??? i ?? ????? ?? ??? elementsi ???? ?? ????
• ?? ????? size????? ??????? ????? ????? ????
  ??????? ?? ???

8
Add/Remove An Element
add(1,g)
9
????? ???? ????

• ??? ????? ???? ?? ????? ???? ????? ????
• ?? ???? ?? ????? ?? ???? Object ??? ?? ???
• ????? ???? ?? ?? ??? Object ?????? ????
• ?? ??? ???? ??? ???? ??? ????? ( int, char,
  float ,) ?? ??? ???? ?? ???? ???? ???
• Wrapper Classes ? ?????? ????

10
?????? ?????? ????

• ????? ????? ???? ?? ????? ???? ????? ?? ???
  ??????
• ???? ????? ???? ?? ???? ?????? ?? ???? ?? ?????
  ???? ?????? ????? ????

11
?????? ??? ?????

• ??? ??? ????? 6 ???

????? ????? ????? ?? ?????
newArray new Object15
12
?????? ??? ?????

• ??? ????? ????? ??? ?? ?? ?? ??? ?? ????

13
?????? ??? ?????

• ?? ????? ??? ????? ?? ????? ?? ????
• element newArray

element.length 15
14
?????? ??? ?????? ????? ???? ????

• // create a new array of proper length and data
  type
• Object newArray new Object newLength

// copy all elements from old array into new one
System.arraycopy(element, 0, newArray,
0, element.length)
// rename array element newArray
15
??? ????? ???? ???? ???? ?????

• ????? ??? ????? ?? ????? ????!
• ????? ?????? ??? ????? ????? ??? ?? theta(M) ? ??
  ?? ?? M ??? ?????? ??? ???
• ????? n ??? ?????? ??? ?? ????? ???? ?? ?????
  theta(n2) ???

16
????? ???? ????

• element6

newArray new char7
space needed 2 newLength 1
17
??? ???? ??????? ??? ????

• Double the array length.

newArray new char12
????? ?????? n ??? ???? ?? ????? theta(n)
???? ????? ???? ???? 1.5 ??? ???? 3 ???
????
18
??? ????? ???? ???? ???? ?????

• ??? ??? ???? ????? ?? ??? ???? ????
• new length c old length
• ?????? n ??? ????? ?? ?? ????? Theta(n) ?????
  ????

??? ??? ???? ??? ???? ????? ????? ????
new length c old length ?????? n ??? ?????
?? ?? ????? Theta(n2) ????? ????
19
??? ????? ???? ???? ???? ?????

• ??? ??? ???? ????? ?? ??? ???? ????
• new length c old length
• ????? ???? ???? ????? ??? ??
• (1c) (old length)

• ??? ??? ???? ????? ??? ???? ????? ????? ?????
  ????? ???? ???? ????? ??? ??
• (old length) (old length c) 2 (old
  length) c space.

20
?????? ??? ???? ?? ????

• java.util.Vector array doubling
• java.util.ArrayList c 1.5

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????? ???? ????? ??? ?? ???? ?? Java, C, C
a
b
c
d

• 1-dimensional array x a, b, c, d
• map into contiguous memory locations

• location(xi) start i

22
????? ???? ???? ???? ??????? ??????? ???? Overhead

• space overhead 4 bytes for start
• 4 bytes for
  x.length
• 8 bytes

23
????? ??? ?? ????

• ????? ????? ?? ????
• int a new int34
• may be shown as a table
• a00 a01 a02 a03
• a10 a11 a12 a13
• a20 a21 a22 a23

24
?????? ????? ?? ????

• a00 a01 a02 a03
  row 0
• a10 a11 a12 a13
  row 1
• a20 a21 a22 a23
  row 2

25
???? ??? ????? ?? ????

• a00 a01 a02 a03
• a10 a11 a12 a13
• a20 a21 a22 a23

26
????? ???? ????? ??? ?? ???? ?? Java, C, C
2-dimensional array x a, b, c, d e, f, g, h i,
j, k, l

• ????? ?? ????? ????? ?? ????? ??? ???? ???
• x row0, row1, row 2
• row 0 a,b, c, d
• row 1 e, f, g, h
• row 2 i, j, k, l

27
????? ???? ????? ??? ?? ???? ?? Java, C, C

• x.length 3
• x0.length x1.length x2.length 4

28
????? ???? ???? ????

• space overhead overhead for 4 1D arrays
• 4 8 bytes
• 32 bytes
• (number of rows
  1) x 8 bytes

29
????? ???? ????? ??? ?? ???? ?? Java, C, C

• ????? ?? ?? ????? ??
• ?????? ?? ????? ?? ??? ??? 3?4?4?4 ???? ???? ???
• ?? ??? ?????? ???? ??????? ????? ???? ? ?? ????
  ?????? ???? ??????? ????? ???? ???? ???

30
????? ???? Row-Major Mapping

• ????????? 3 4 x
• a b c d
• e f g h
• i j k l
• ????? ??? ?? ????? ?????? ????? ?? ???
• ?????? ????? ?? ???? ?? ?? ?? ???? ? ?? ???? ??
  ????? ?? ??? ????? ???? ?? ?????
• ????? ?? ????? y a, b, c, d, e, f, g, h,
  i, j, k, l

???? ??? ?????????? ?????? ????? ?? ????? ????
???? ???
31
????? xij

• ??? ???? x? ????? r ??? ? c????? ???
• ?? ??? c ??? ????
• ??? ?? ??? i ?? ????? i???? ???? ?????(???? ??
  ???)
• ???????? xij?? ic j? ?? ????? 1 ????
  ????? ?? ???

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Space Overhead

• 4 bytes for start of 1D array
• 4 bytes for length of 1D array
• 4 bytes for c (number of columns)
• 12 bytes
• (number of rows length /c)

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Column-Major Mapping

• a b c d
• e f g h
• i j k l
• ????? ????? ?? ???? ? ???? ?? ???? ?? ?? ?? ????
  ? ?? ???? ?? ????? ?? ????? ?? ???? ???? ?? ?????
• y a, e, i, b, f, j, c, g, k, d, h, l
• ???? ????? ???? ?? ????

34
?????? Matrix

• ?????? ???? ????? ?? ?????? ?? ??? ????? ????.
  ????? ??? ????? ?? 1 ???? ?? ???
• a b c d row 1
• e f g h row 2
• i j k l row 3
• ?? x(i,j) ???? xij. ??????? ?? ???
• ?? ???? ?? ????? ?? ???? ???? ????? ??????
  ??????? ???

35
????? ??????? ?? ????? ?? ???? ???? ????? ??????

• ???? ????? ????? ?? ???
• ????? ??? ???? ?? ????? ???? ?????? ?? ????? ???
  ? ??? ? ???????? ???? ???????? ??? ????
• ???? ??? ?????? ???? ???? ?? ???? ????? ??????
  ????? ???

36
?????? ????

• 1 0 0 0
• 0 2 0 0
• 0 0 3 0
• 0 0 0 4

?????? ????? ??? ?? ????? ??? ??? ???? ?? ???
?????

• x(i,j) is on diagonal iff i j
• number of diagonal elements in an n x n matrix is
  n
• non diagonal elements are zero
• store diagonal only vs n2 whole

37
?????? ????? ?????

• ?????? ????? ????? ?????? ????? ?? ??? ?? ?????
  ??? ???? ? ??? ?? ???? ???? ??? ?????

1 0 0 0 2 3 0 0 4 5 6 0 7 8 9 10

• ????? ????? ??? ??? ?????? ????? ????? 1 2
  n n(n1)/2.
• ????? ???? ????? ?? ???? ?? ??? ????

38
????? ???? ?? ????? ?? ?? ????? ??

• ????? ?????? ?????? ????? ???? ?? ??? ??? ?????
  ???? ??????? ????

39
????? ????? ??? ?????? ?? ????

• // declare a two-dimensional array variable
• // and allocate the desired number of rows
• int irregularArray new int
  numberOfRows
• // now allocate space for the elements in
  each row
• for (int i 0 i lt numberOfRows i)
• irregularArrayi new int sizei
• // use the array like any regular array
• irregularArray23 5
• irregularArray46 irregularArray23
  2
• irregularArray11 3

40
??????? ?? ?? ????? ?? ????

• ?? ??? ???? ???? ???? ???? ??????? ?? ??? ???
  ????? ??? ??? ?? ????
• 1 0 0 0
• 2 3 0 0
• 4 5 6 0
• 7 8 9 10
• 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

41
????? ??? ij

• ????? row 1, row 2, row 3,
• ??? ?? ??? i ?????? 1, 2, , i-1 ???? ?????
• ??? ??? i ????? i. ???
• ????? ?????? ?? ??? ?? ??? i ???? ?????
• 1 2 3 i-1 i(i-1)/2
• ?? ??? (i,j) ?? ???? i(i-1)/2 j -1 ????? 1
  ???? ???? ????.

42
?????? ???? Sparse Matrices

• ?????? ????????? ???? ?????
• ?????? ?????? ????? ?????? ?????
• sparse many elements are zero
• dense few elements are zero
• ???? ?????? ??? ????? ?????? ??? ????? ?????
• ?? ???? ?? ????? ?? ???? ???? ????? ???? ???????
  ???
• ??? ??? ?????? ??? ????? ?? ?????? ???
  ??????????? ???????

43
?????? ???? ???? ?????? ?????

• ???? ???? ????? ??.
• ????????? ?? 1 ?? n? ????? ????? ??? ???
• flight(i,j) ???? ????? ????? ???????? ??????
  ??? ??? ???????????
• ??? ???? n 1000
• n x n array of list references gt 4 million bytes
• ??? ????? ????? ??????? 20,000 ????.
• ?????? 20000 ????? ????? ?? ???

44
?????? ???? ???? ?????? ?????

• ?????? ????? ??.
• ????? ?? 1 ?? n ????? ????? ??? ???
• ????? ?????? ?? ???? i??? ???? web(i,j) j
• ?????? ??.
• authority page page that has many links to it
• hub page links to many authority pages

45
?????? ????? ??

• n 2 billion (and growing by 1 million a day)
• n x n array of ints gt 16 1018 bytes (16 109
  GB)
• ?? ??? ????? ?? ???? ?? 10 ???? ???? ???? ????
• ?? ?? ?? ??? ????? ?? ??? ????? 10 ??? ??? ???
  ???? ????
• ???? ???? ???? ???? ??????? ????? ??? ??? ?????
  ??? ?? 20 billion x 4 bytes 80 billion bytes
  (80 GB)

46
????? ?????? ???? ???? ??????

• ????? ?? ???? ?? ???? ????
• ?? ??? ??? ??? ?? ?? ???? (row, column, value)
  ????? ?? ???
• ??? ?? ???? ?? ?? ???? ?? ????? ??? ?? ???? ??
  ???? ?????? ?? ?? ????? ???? ???

47
????? ?? ??????? ?? ???? ???

• 0 0 3 0 4
• 0 0 5 7 0
• 0 0 0 0 0
• 0 2 6 0 0

48
????? ?? ??????? ?? ???? ???

• row 1 1 2 2 4 4
• list column 3 5 3 4 2 3
• value 3 4 5 7 2 6

49
????? ?? ?????? ?? ?? ???????

• ?????? ?? ?????

50
?? ??????

• row 1 1 2 2 4 4
• list column 3 5 3 4 2 3
• value 3 4 5 7 2 6

51
?? ??? ?? ?? ???? ???????
row1 (3, 3),
(5,4) row2 (3,5), (4,7) row3 row4
(2,2), (3,6)

• 0 0 3 0 4
• 0 0 5 7 0
• 0 0 0 0 0
• 0 2 6 0 0

52
?????? ?? ?? ?????

• ?????? ?? ?????

53
????? ?? ?? ?????? ?????

• 0 0 3 0 4
• 0 0 5 7 0
• 0 0 0 0 0
• 0 2 6 0 0

null
54
????? ?? ??????? ?????? ???? ? ?????

• ?????? ?????

55
???? ?????

• 0 0 3 0 4
• 0 0 5 7 0
• 0 0 0 0 0
• 0 2 6 0 0

56
???? ???? ??

• 0 0 3 0 4
• 0 0 5 7 0
• 0 0 0 0 0
• 0 2 6 0 0

57
Orthogonal Lists

• 0 0 3 0 4
• 0 0 5 7 0
• 0 0 0 0 0
• 0 2 6 0 0

58
????? ???? ????

• 500 x 500 matrix with 1994 nonzero elements
• 2D array 500 x 500 x 4 1million
  bytes
• Single Array List 3 x 1994 x 4 23,928 bytes
• One Chain Per Row 23928 500 x 4 25,928

59
???? ????

• Matrix Transpose
• 500 x 500 matrix with 1994 nonzero elements
• 2D array 210 ms
• Single Array List 6 ms
• One Chain Per Row 12 ms

60
???? ????

• Matrix Addition.
• 500 x 500 matrices with 1994 and 999 nonzero
  elements
• 2D array 880 ms
• Single Array List 18 ms
• One Chain Per Row 29 ms

61
????? ????? ?????? ????

• ?? ADT ???? ????? ?????? ??? ???? ?? ??????? ??
  ???? ??? ??? ??????? ?? ?? ?? ?? ??? ??? ??? ??
  ?? ???? (row, column, value) ???? ???? ?? ???.
• ??? ??? ?? ?????? ?? ???? ??? ??? ????? ?? ????
  ??? ?? ???????
• ????? ?? ??????? ds-cs-sparse ? ??
  ds-math-sparse