Chapter 2: Arrays

1
Chapter 2 Arrays
2
Arrays

• An array is a finite, ordered set of same data
  type element.
• Also called tables.
• An array might be comprised of elements that are
  all strings another might be comprised of
  elements that are all integers.

3
One-Dimensional Arrays

• Also called vector
• A notation A(LU) indicates that array named A
  with element of same data type having subscripts
  or index which identify the position of each
  element from L through U.
• For example, Score(120), Name(1040)
• The number of elements in an array is called its
  range. The range of array A(LU) is U-L1
• For example, The range of array Score(120) is
  20.
• The range of array Name(1040) is 40-101 31.

position
String name
4
Multi-Dimensional Arrays

• Two-Dimensional Array
• is an array in which each element is itself an
  array.
• For example, B(1M,1N) is an array named B which
  has row from 1 to M and column from 1 to N.

• To identify an individual element, two subscript
  values are needed.
• The first subscript refers to row of array,
  second subscript refers to a column of the array.
• Total number of elements in array B is MN

5
Multi-Dimensional Arrays

• A two-dimensional array B with its first
  subscript having lower bound L1 and upper bound
  U1 and is second subscript having lower bound L2
  and upper bound U2 is defined as B(L1U1,L2U2)
• The number of element in a row of B is U2 - L2
  1
• The number of element in a column of B is U1 -
  L1 1
• The total number of elements in array B is
• (U2 - L2 1)(U1 - L1 1)

6
Multi-Dimensional Arrays

• Cross-Section
• A Cross-Section of a two dimensional array is
  obtained by holding one of the subscripts
  constant while varying the other through its
  entire range of values.
• The asterisk () is used for the subscript value
  that is to be allowed to take on its entire rage
  of value.
• For example, B(,4) GRADE(2,)

GRADE(120,14)
7
Multi-Dimensional Arrays

• Transpose
• The transpose of a two-dimensional array is
  obtained by reversing the subscript position.
• The transpose of M-by-N array is an N-by-M array.
• The transpose of array B is commonly demoted BT
• For example, the transpose of array B(3,8) is
  denoted as BT(8,3)

8
Multi-dimensional Array

• An array can be defined to be three-dimensional,
  four-dimensional, or N-dimensional arrays.

J
N
M
9
Multi-dimensional Array

• An N-dimensional array A could be identified as
• A( L1U1 ,L2U2 ,L3U3 , LNUN )
• And the total number of elements in array A
  (N-dimensional array) is
• (U1 L1 1)(U2 L2 1)(UN LN 1)
• the cross-section normally is formed by holding
  one or more index values constant and varying
  each of the other indexes through its entire
  range of values. i.e. A(5, , 3)

10
Mapping to StorageOne-Dimensional Arrays

• To calculate the starting address (i.e., the
  location) of element A(I)
• B is arrays starting address of base location
• I is the ith element of array
• L is the lower bound of the array which can be
  positive, negative, or zero
• S is size of each element in the array

B (I L) S
11
Mapping to Storage Multi-Dimensional Arrays

• Row-major Order
• Because computer memory is linear, a
  muti-dimensional array must be linearized as it
  is mapped to storage.
• Row-major order stores the first row of the
  array, then the second row, then the third row,
  and so forth.

Row 0
Row 1
Row 2
Row -1
Row -2
A(0,6)
A(0,6)
12
Mapping to Storage Multi-Dimensional Arrays

• The element A(I,J) of array defined by
  A(L1U1,L2U2) is at location
• For example, the starting location of A(0,6) in
  an array A(-22,46) is at
• B (0 (-2)) (6 4 1) S (6 - 4) S
• There are two rows (row -2 and row-1), each of
  length 3S, preceding row 0.
• In row 0, there are 2 (i.e., 6-4) elements, each
  of length S, preceding element A(0,6)

B (I L1) (U2 L2 1) S (J L2) S
13
Mapping to Storage Multi-Dimensional Arrays

• Column-major Order
• Column-major order stores first the first column
  of the array, then the second column of the
  array, then the third column and so forth.

1
2
3
4
5
Col 3
Col 4
Col 5
Col 2
Col 1
Rate(2,4)
1
2
3
Rate(2,4)

• The element ARRAY(I,J) of the array defined by
  ARRAY(L1U1,L2U2), with column-major storage, is
  at location

B (J L2) (U1 L1 1) S (I L1) S
14
Mapping to Storage Multi-Dimensional Arrays

• Column-major Order
• For example, the array rate(13,15) is stored in
  memory by column. If base location B 0 and
  element size S 1, the starting address of
  element rate(2,4) is
• 0 (4 1) (3 1 1) 1 (2 1) 1 10

15
Triangular Arrays

• Two types of triangular array
• Upper triangular all elements below the
  diagonal are zero.
• Lower triangular all elements above the
  diagonal are zero.
• An array is called strictly upper-(lower)
  triangular if the elements of the diagonal are
  also zero.

Lower--triangular array
Upper-triangular array
16
Triangular Arrays

• In a lower-triangular array with N rows, the
  maximum number of nonzero elements in the ith row
  is I.
• The total number of nonzero elements (both
  upper-triangular array and lower triangular
  array) is no more than

17
Triangular Arrays

• Linearization
• Lenearize the array and store only the elements
  of the nonzero triangular portion.
• For example,

18
Sparse Arrays

• An array that has high density of zero elements

8 by 10 sparse array
19
Sparse Arrays

• Vector Representation
• Each nonzero element in a two-dimensional sparse
  array can be represented as a triple with the
  format (row-subscript, column-subscript, value).