Arrays
• Array: a set of pairs (index and value)
• data structure
• For each index, there is a value
associated with that index.
• Representation
• implemented by using consecutive
memory.
Objects: A set of pairs <index, value> where for each value of index there is a value from the set item.
Index is a finite ordered set of one or more dimensions, for example,
{0, … , n-1} for one dimension,
{(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2)} for two
dimensions, etc.
The Array as ADT
Methods: for all A Array, i index, x item, j, size integer
Array Create(j, list) ::= return an array of j dimensions where list is a j-tuple (a structure containing multiple parts) whose kth element is the size of the kth dimension. Items are undefined.
Item Retrieve(A, i) ::= if (i index) return the item associated with index value i in array A else return error
Array Store(A, i, x) ::= if (i in index) return an array that is identical to array A except the new pair <i, x> has been inserted else return error
The Array ADT
Arrays in C 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 = α list[1] α + sizeof(int)
list[2] α + 2*sizeof(int)
list[3] α + 3*sizeof(int)
list[4] α + 4*size(int)
Arrays in C (cont’d)
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] i.e (&list2[i])
*(list2 + i) = list2[i]
Address Contents
1228 0
1230 1
1232 2
1234 3
1236 4
Example:
int one[] = {0, 1, 2, 3, 4}; //Goal: print out
address and value
void print1(int *ptr, int rows)
{
printf(“Address Contents\n”);
for (i=0; i < rows; i++)
printf(“%8u%5d\n”, ptr+i, *(ptr+i));
printf(“\n”);
}
Example
Other Data Structures Based on Arrays
•Note
• Arrays:
•Basic data structure
•May store any type of elements
Other Data Structures Based on Arrays:
Polynomial ADT
Polynomials: defined by a list of coefficients and exponents
-degree of polynomial = the largest exponent in a polynomial
-In a MONOMIAL Term of the form
Axⁿ
A is the Coefficient of the term, x is the base and n is the exponent
For example, given the term 10x³ it is a monomial term of degree 3
with exponent 3 base x and coefficient 10
-Note: a monomial is a product of powers of variables with
nonnegative integer exponents,
Polynomials A(X)=3X20+2X5+4,
B(X)=X4+10X3+3X2+1
Other Data Structures Based on Arrays:
Polynomial ADT
-An example of a single variable polynomial:
Remark: the order/degree of this polynomial is 20
(look for highest exponent)
• Polynomial ADT (continued)
…This sum can be expanded to:
anxn + a(n-1)x(n-1) + … + a1x
1 + a0
If you like, CExE + C(E-1)x(E-1) + … + C1x
1 + C0
Notice the two visible data sets namely: (C
and E), where
C is the coefficient object [Real #].
and E is the exponent object [Integer #].
Polynomial ADT
By definition of a data types:
Why call it an Abstract Data Type (ADT)?
A set of values and a set of allowable
operations on those values.
We can now operate on this polynomial
the way we like…
What kinds of operations?
Here are the most common operations on a
polynomial:
• Add & Subtract
• Multiply
• Differentiate
• Integrate
• etc…
Polynomial ADT
Polynomial ADT
Why implement this?
• Calculating polynomial operations by hand
can be very cumbersome. Take
differentiation as an example:
d(23x9 + 18x7 + 41x6 + 163x4 + 5x + 3)/dx
= (23*9)x(9-1) + (18*7)x(7-1) + (41*6)x(6-1) + …
Polynomial ADT
How to implement this?
There are different ways of implementing the
polynomial ADT:
• Array (not recommended)
• Linked List (preferred and recommended)
• Watch out for the reason---
Polynomial ADT
•Array Implementation:
• p1(x) = 8x3 + 3x2 + 2x + 6
• p2(x) = 23x4 + 18x - 3
6 2 3 8
0 2
Index represents
exponents
-3 18 0 0 23
0 4 2
p1(x) p2(x)
3 1 3 1
This is why arrays aren’t good to represent
polynomials:
• p3(x) = 16x21 - 3x5 + 2x + 6
Polynomial ADT
6 2 0 0 -3 0 0 16 …………
WASTE OF SPACE!
Polynomial ADT
Advantages of using an Array:
• only good for non-sparse polynomials.
• ease of storage and retrieval.
Disadvantages of using an Array:
• have to allocate array size ahead of time.
• huge array size required for sparse
polynomials. Waste of space and runtime.
Polynomial ADT
21
Object: Polynomial.
Operations: Boolean IsZero(poly)
::= return FALSE or TRUE.
Coefficient Coeff(poly, expon)
::= return coefficient of xexpon
Polynomial Add(poly1, poly2)
::= return poly1 + poly2
Polynomial Subtract(poly1, poly2)
::= return poly1 - poly2
Polynomial ADT Operations
Additions
#define MAXDEGREE 101
typedef struct {
int degree;
int coef[MAXDEGREE]
} polynomial;
Polynomial ADT Operations
Additions
polynomial addp(polynomial a, polynomial b)
{ polynomial c;
c.degree = max(a.degree, b.degree)
for (i=0; i<=MAXDEGREE; i++)
c.coef[i] = a.coef[i] + b.coef[i];
return c;
}
Running time?
advantage: easy implementation
disadvantage: waste space when sparse
2nd Representation As Arrays
24
Only represent non-zero terms
Need to represent non-zero exponents and its corresponding coefficients
• Use one global array to store all polynomials
Polynomial Addition (2)
2 1 1 10 3 1
1000 0 4 3 2 0
coef
exp
Start_a finish_a start_b finishb avail
0 1 2 3 4 5 6
A(X)=2X1000+1
B(X)=X4+10X3+3X2+1
2nd Representation As Arrays
26
In C:
Comparisons of two representations: If polynomial sparse, 2nd repre. is better.
If polynomial full, 2nd repre. is double size of 1st.
typedef struct {
int coeff, exp;
} polynomial;
polynomial terms[MAX_TERMS];
int avail = 0;
Polynomial ADT
Methods: functions:
for all poly, poly1, poly2 Polynomial, coef Coefficients, expon Exponents
Polynomial Zero( ) ::= return the polynomial,
p(x) = 0
Boolean IsZero(poly) ::= if (poly) return FALSE
else return TRUE
Coefficient Coef(poly, expon) ::= if (expon poly) return its
coefficient else return Zero
Exponent Lead_Exp(poly) ::= return the largest exponent in
poly
Polynomial Attach(poly,coef, expon) ::= if (expon poly) return error
else return the polynomial poly
with the term <coef, expon>
inserted
Polyomial ADT (cont’d)
Polynomial Remove(poly, expon) ::= if (expon poly) return the
polynomial poly with the term whose exponent is
expon deleted else return error
Polynomial SingleMult(poly, coef, expon) ::= return the polynomial
poly • coef • xexpon
Polynomial Add(poly1, poly2) ::= return the polynomial
poly1 +poly2
Polynomial Mult(poly1, poly2) ::= return the polynomial
poly1 • poly2
Matrix – Abstract Data Type (ADT)
29
Object: Matrix. dimension = #rows x #cols
Operations: Matrix Transpose(matrixA)
::= return transpose of matrixA
Matrix Add(matrixA, matrixB)
::= return (matrixA + matrix B)
Matrix Multiply(matrixA, matrixB)
::= return (matrixA * matrixB)
-27 3 4
6 82 -2
109 -64 11
12 8 9
48 27 47
col. 0 col. 1 col. 2
row 0
A =
row 1
row 2
row 3 row 4
Matrix Representation
30
We use arrays to represent matrices.
Use array a[M][N] to store a matrix A(M, N).
Use a[i][j] to store A(i, j).
Operations: Transpose & Add
31
c transpose(a) // a: m x n matrix
Running time = O(mn)
c add(a, b) // a, b: m x n matrices
Running time = O(mn)
for (i=0; i<rowA; i++) // O(m)
for (j=0; j<colA; j++) // O(n)
c[j][i]=a[i][j];
for (i=0; i<rowA; i++) // O(m)
for (j=0; j<colA; j++) // O(n)
c[i][j]=a[i][j]+b[i][j];
Operations: Multiply
32
c multiply(a, b) //a: m x n mat., b: n x p mat.
Running time = O(mnp)
for (i=0; i<rowA; i++) { // O(m)
for (j=0; j<colB; j++) { // O(p)
sum=0;
for (k=0; k<colA; k++) // O(n)
sum += a[i][k]*b[k][j];
c[i][j]=sum;
}
}
=
x c: m x p mat.
Sparse Matrices
An example sparse matrix:
A lot of “zero” entries. Thus large memory space is wasted.
Could we use other representation to save memory space ??
15 0 0 22 0 -15
0 11 3 0 0 0
0 0 0 -6 0 0
0 0 0 0 0 0
91 0 0 0 0 0
0 0 28 0 0 0
A =
34
Representation for Sparse Matrices
35
Use triple <row, col, value> to characterize an element in the matrix.
Use array of triples a[] to represent a matrix. row by row
within a row,
column by column a[0] 6 6 8
a[1 ] 0 0 15
a[2] 0 3 22
a[3] 0 5 -15
a[4] 1 1 11
a[5] 1 2 3
a[6] 2 3 -6
a[7] 4 0 91
a[8] 5 2 28
row col value
Representation for Sparse Matrices
36
In C:
typedef struct {
int col, row, value;
} term;
term a[MAX_TERMS];
Operations: Transpose
37
c transpose(a) // a: m x n matrix
Eg.
//Algorithm 1:
for each row i {
take element (i, j, value) and
store it as (j, i, value).
} row col valuae
a[0] 6 6 8
a[1 ] 0 0 15
a[2] 0 3 22
a[3] 0 5 -15
a[4] 1 1 11
a[5] 1 2 3
a[6] 2 3 -6
a[7] 4 0 91
a[8] 5 2 28
row col
value c[0] 6 6 8
c[1 ] 0 0 15
c[2] 3 0 22
c[3] 5 0 -15
c[4] 1 1 11
c[5] 2 1 3
c[6] 3 2 -6
c[7] 0 4 91
c[8] 2 5 28
[15 0 0 22 0 − 15
0 11 3 0 0 0
0 0 0 − 6 0 0
0 0 0 0 0 0
91 0 0 0 0 0
0 0 28 0 0 0]
col1 col2 col3 col4 col5 col6 row0
row1
row2
row3
row4
row5
8/36
6*6 5*3
15/15
sparse matrix data structure?
Sparse Matrices
Sparse Matrix ADT
Objects: a set of triples, <row, column, value>, where row and column are integers and form a unique combination, and value comes from the set item.
Sparse Matrix ADT
Methods:
for all a, b Sparse_Matrix, x item, i, j, max_col, max_row index Sparse_Marix Create(max_row, max_col) ::= return a Sparse_matrix that can hold up to max_items = max _row x max_col and whose maximum row size is max_row and whose maximum column size is max_col.
Sparse Matrix ADT (cont’d)
Sparse_Matrix Transpose(a) ::= return the matrix produced by interchanging the row and column value of every triple. Sparse_Matrix Add(a, b) ::= if the dimensions of a and b are the same return the matrix produced by adding corresponding items, namely those with identical row and column values. else return error Sparse_Matrix Multiply(a, b) ::= if number of columns in a equals number of rows in b return the matrix d produced by multiplying a by b according to the formula: d [i] [j] = (a[i][k]•b[k][j]) where d (i, j) is the (i,j)th element else return error.
(1) Represented by a two-dimensional array.
Sparse matrix wastes space.
(2) Each element is characterized by <row, col, value>.
Sparse Matrix Representation
Sparse_matrix Create(max_row, max_col) ::= #define MAX_TERMS 101 /* maximum number of terms +1*/ typedef struct { int col; int row; int value; } term; term A[MAX_TERMS]
The terms in A should be ordered
based on <row, col>
Sparse Matrix Operations • Transpose of a sparse matrix.
• What is the transpose of a matrix?
row col value row col value a[0] 6 6 8 b[0] 6 6 8 [1] 0 0 15 [1] 0 0 15 [2] 0 3 22 [2] 0 4 91 [3] 0 5 -15 [3] 1 1 11 [4] 1 1 11 [4] 2 1 3 [5] 1 2 3 [5] 2 5 28 [6] 2 3 -6 [6] 3 0 22 [7] 4 0 91 [7] 3 2 -6 [8] 5 2 28 [8] 5 0 -15
transpose
(1) for each row i
take element <i, j, value> and store it
in element <j, i, value> of the transpose.
difficulty: where to put <j, i, value>?
(0, 0, 15) ====> (0, 0, 15)
(0, 3, 22) ====> (3, 0, 22)
(0, 5, -15) ====> (5, 0, -15)
(1, 1, 11) ====> (1, 1, 11)
Move elements down very often.
(2) For all elements in column j,
place element <i, j, value> in element <j, i, value>
Transpose a Sparse Matrix
Transpose of a Sparse Matrix (cont’d)
void transpose (term a[], term b[]) /* b is set to the transpose of a */ { int n, i, j, currentb; n = a[0].value; /* total number of elements */ b[0].row = a[0].col; /* rows in b = columns in a */ b[0].col = a[0].row; /*columns in b = rows in a */ b[0].value = n; if (n > 0) { /*non zero matrix */ currentb = 1; for (i = 0; i < a[0].col; i++) /* transpose by columns in a */ for( j = 1; j <= n; j++) /* find elements from the current column */ if (a[j].col == i) { /* element is in current column, add it to b */