1. Data Structure Sang Yong Han http://ec.cse.cau.ac.kr/
Chung-Ang University Spring 2011

2. Arrays Array: a set of pairs (index and value) data structure
For each index, there is a value associated with
that index.
representation (possible)
implemented by using consecutive memory.

3. The Array ADT
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. 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 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

4. 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 = 
list[1]  + sizeof(int)
list[2]  + 2*sizeof(int)
list[3]  + 3*sizeof(int)
list[4]  + 4*size(int)

5. 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] (&list2[i])
*(list2 + i) - list2[i]

6. Example
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”); }

7. 2D Arrays The elements of a 2-dimensional array a declared as:
int a[3][4];
may be shown as a table
a[0][0] a[0][1] a[0][2] a[0][3]
a[1][0] a[1][1] a[1][2] a[1][3]
a[2][0] a[2][1] a[2][2] a[2][3]

8. Rows Of A 2D Array a[0][0] a[0][1] a[0][2] a[0][3] row 0

9. Columns Of A 2D Array column 0 column 1 column 2 column 3
a[0][0] a[0][1] a[0][2] a[0][3]
a[1][0] a[1][1] a[1][2] a[1][3]
a[2][0] a[2][1] a[2][2] a[2][3]
column 0
column 1
column 2
column 3

10. 2D Array Representation In C
2-dimensional array x
a, b, c, d
e, f, g, h
i, j, k, l
view 2D array as a 1D array of rows
x = [row0, row1, row 2]
row 0 = [a,b, c, d]
row 1 = [e, f, g, h]
row 2 = [i, j, k, l]
and store as 1D arrays

11. Array Representation In Cb c d e f g h i j k l
x[]
This representation is called the array-of-arrays representation.
Contiguous space required for this representation ?

12. Other Data Structures Based on Arrays
Basic data structure
May store any type of elements
Polynomials: defined by a list of coefficients and exponents
- degree of polynomial = the largest exponent in a polynomial
Polynomials A(X)=3X20+2X5+4, B(X)=X4+10X3+3X2+1

13. Polynomial ADT
Objects: a set of ordered pairs of <ei,ai> where ai in Coefficients and ei in Exponents, ei are integers >= 0 Methods: 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

14. 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

15. Polynomial Addition (1)
#define MAX_DEGREE 101
typedef struct {
int degree;
float coef[MAX_DEGREE];
} polynomial;
Running time?
Addition(polynomial * a, polynomial * b, polynomial* c) { … }
advantage: easy implementation
disadvantage: waste space when sparse

16. Polynomial Addition (2)
Use one global array to store all polynomials
A(X)=2X1000+1
B(X)=X4+10X3+3X2+1
coef
exp
starta finisha startb finishb avail

17. Polynomial Addition (2) (cont’d)
#define MAX_TERMS 100
typedef struct {
int exp;
float coef;
} polynomial;
polynomial terms[MAX_TERMS];
Running time?
Addition(int starta, int enda, int startb, int endb, int startc, int endc) { … }
advantage: less space
disadvantage: longer code
Time Complexity ?

18. Sparse Matrices 5*3 6*6 15/15 8/36 sparse matrix data structure?
col1 col2 col3 col4 col5 col6
row0
row1
row2
row3
row4
row5
5*3
6*6
15/15
8/36
sparse matrix
data structure?

19. Sparse Matrix - Airline flight
Airline flight matrix.
airports are numbered 1 through n
flight(i,j) = list of nonstop flights from airport i to airport j
n = 1000 (say)
n x n array of list pointers => 4 million bytes
total number of nonempty flight lists = 20,000 (say)
Assume that flight(i,j) = null if there is no flight from i to j. Of the 1 million entries in the flight matrix at most 20,000 can be nonnull as each nonnull entry represents at least 1 different flight. So we need to store at most 20,000 nonnull entries.

20. Sparse Matrix - Web page matrix
web pages are numbered 1 through n
web(i,j) = number of links from page i to page j
Web analysis.
authority page … page that has many links to it
hub page … links to many authority pages
Web page analyses are done using a matrix such as web(*,*). See IEEE Computer, August 1999, p 60, for a paper that describes Web analysis based on such a matrix. Operations such as matrix transpose and multiply are used.

21. Web Page Matrix n = 2 billion (and growing by 1 million a day)
n x n array of ints => 16 * 1018 bytes (16 * 109 GB)
each page links to 10 (say) other pages on average
on average there are 10 nonzero entries per row
space needed for nonzero elements is approximately 2billion x 10 x 4 bytes = 80 billion bytes (80 GB)
On average, 10 entries in each row are nonzero.

22. 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. 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  max_col and whose maximum row size is max_row and whose maximum column size is max_col.

23. 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.

24. Sparse Matrix Representation
(1) Represented by a two-dimensional array.
Sparse matrix wastes space.
(2) Each element is characterized by <row, col, value>.
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>

25. Matrix Transpose

26. Transpose of a Sparse Matrix in 2D array representation
for (j = 0; j < columns; j++) for( i = 0; i <rows; i++)
b[j][i] = a[i][j];
Time and Space Complexity ?

27. Sparse Matrix Operations
Transpose of a sparse matrix.
row col value row col value a[0] b[0] [1] [1] [2] [2] [3] [3] [4] [4] [5] [5] [6] [6] [7] [7] [8] [8]
transpose

28. Transpose a Sparse Matrix
(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>

29. Transpose of a Sparse Matrix
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 */

30. Time Complexity Analysis of Matrix Transpose

31. Fast Matrix Transpose Step 1: #nonzero in each row of transpose.
= #nonzero in each column of original matrix
= [2, 1, 2, 2, 0, 1]
Step2: Calculate Starting position of each row of transpose
= [1, 3, 4, 6, 8, 8]
Step 3: Move elements from original list to transpose list.
Complexity
m x n original matrix
t nonzero elements
Step 1: O(n+t)
Step 2: O(n)
S: O(t)
Overall O(n+t)
Step 1: initialize array rowSize to 0, n values then stream through nonzero list of original matrix left to right
Step 2: O(n)
Step 3: O(t)

32. Time Complexity Analysis of Fast Matrix Transpose
m x n original matrix
t nonzero elements
Step 1: O(n+t)
Step 2: O(n)
Step 3: O(t)
Overall O(n+t)

33. Runtime Performance - Transpose
500 x 500 matrix with 1994 nonzero elements
Run time measured on a 300MHz Pentium II PC
2D array ms
SparseMatrix ms

34. Runtime Performance - Addition
Matrix Addition.
500 x 500 matrices with 1994 and 999 nonzero elements
2D array ms
SparseMatrix ms