Chapter 2 Arrays and Structures

Transcription

1 Data Structure Chapter 2 Arrays and Structures Angela Chih-Wei Tang Department of Communication Engineering National Central University Jhongli, Taiwan 2 Spring

2 Outline Array Structures Polynomial Sparse Matrix Multidimensional Arrays C.E., NCU, Taiwan Angela Chih-Wei Tang, 2 2

3 2. The Array as An Abstract Data Type Array a group of consecutive memory locations (the same type and name) When considering abstract data type operations (create, retrieve, store ) int c[2], *pc[2]; Variable Memory Address c[] base address = a c[] a + sizeof(int) c[2] a + 2*sizeof(int) Name of array (Note that all elements of this array have the same name, c) c[] c[] c[2] c[3] c[4] c[5] c[6] c[7] c[8] c[9] c[] c[] c[3] a + 3*sizeof(int) Position number of c[4] a + 4*sizeof(int) 78 the element within array c 3

4 Example 2.: One-Dimensional Array Addressing int one[ ] = {,, 2, 3, 4}; void print(int *ptr ptr, int rows) { /* print out a one-dimensional array using a pointer */ int i; printf( Address Contents\n ); Address Contents for (i = ; i < rows; i++) printf( %8u%5d\n, %5d\ ptr+i, *(ptr+i)); 228 printf( \n ); 23 } Function call: print(&one[], 5); C.E., NCU, Taiwan Angela Chih-Wei Tang, 2 4

5 Outline Array Structures Polynomial Sparse Matrix Multidimensional Arrays C.E., NCU, Taiwan Angela Chih-Wei Tang, 2 5

6 2.2. Structures Array: collections of data of the same type Structure : collections of related variables (different data types are permitted) under one name An example of create our own structure data types: struct { char name[]; int age; float salary; } human_being; C.E., NCU, Taiwan Angela Chih-Wei Tang, 2 6

7 Case Case Self-Referential Structures An example: item struct list{ a link NULL char data; list *link; item2 }; b link NULL struct list item, item2, item3; Item.data = a ; item3 item2.data = b ; c link NULL item3.data = c ; item.link=item2.link=item3.link=null; Case : null pointer Item.link=&item2; item2.link=&item3; item item2 item3 b link b link c link NULL Case 2: the address in memory 7

8 Outline Array Structurest Polynomial Sparse Matrix Multidimensional Arrays C.E., NCU, Taiwan Angela Chih-Wei Tang, 2 8

9 2.3 The Polynomial Abstract Data Type A ( x) = 2x + and B( x) = x + x + 3x + D( x) = A( x) + B( x) ==> How? MAX_TERMS /*size of terms array*/ struct{ float coef; int expon; } polynomial; struct polynomial terms[max_terms]; int avail = ; starta finisha startb finishb avail coef exp A B D C.E., NCU, Taiwan 9

10 Program 2.5: Function to Add Two Polynomials (/2) void padd(int starta, int finisha, int startb, int finishb, int *started, int *finishd) i { /*add A(x) and B(x) to obtain D(x) */ float coefficient; *startd = avail; while (starta <= finisha && startb <= finishb) switch(compare(terms[starta].expon, terms[startb].expon)) { case -: /* a expon < b expon */ attach(terms[startb].coef, terms[startb. expon); startb++; beak; case : /* equal expontents */ coefficient i = terms[starta]. t t coef + terms[startb].coef; C.E., NCU, Taiwan

11 Program 2.5: Function to Add Two Polynomials (2/2) } if (coefficient) attach(coefficient, terms[starta].expon); expon); starta++; startb++; break; case : /* a expon > b expon */ attach(terms[starta].coef, terms[starta].expon); starta++; t } /* add in remaining terms of A(x) */ for (; starta <= finisha; starta++) attach(terms[starta].coef, terms[starta].expon); /* add in remaining terms of B(x) */ for (; startb <= finishb; startb++) attach(terms[startb].coef, terms[startb].expon); *finishd = avail-; C.E., NCU, Taiwan Angela Chih-Wei Tang, 2

12 Function to Add a New Term void attach(float coefficient, int exponent) { /* add a new term to the polynomial */ if (avail > MAX_TERMS) { fprintf(stderr, Too many terms in the polynomial\n ); exit(); } terms[avail].coef = coefficient; terms[avail++].expon = exponent; } C.E., NCU, Taiwan Angela Chih-Wei Tang, 2 2

13 Outline Array Structurest Polynomial Sparse Matrix Multidimensional Arrays C.E., NCU, Taiwan Angela Chih-Wei Tang, 2 3

14 row row row 2 row 3 row The Sparse Matrix Abstract Data Type col col col a[max_rows][max_cols] OK! row row row 2 row 3 row 4 row 5 col col col2 col3 col4 col Many s a[max_rows][max_cols] inefficient How to efficiently store this matrix? C.E., NCU, Taiwan Angela Chih-Wei Tang, 2 4

15 Abstract Data Type Sparse_Matrix #define MAX_TERMS /* maximum number of terms +*/ row col value struct term{ a[] int col; a[] 5 int row; a[2] 3 22 int value; a[3] 5-5 }term; a[4] struct term a[max_terms]; a[5] 2 3 The number of the rows a[6] The number of the cols The number of nonzero entries a[7] 4 9 a[8]

16 2.4.2 Transposing a Matrix row col value b[] b[] 5 b[2] 4 9 Algorithm: for all elements in column j place element <i, j, value> in element <j, i, value> b[3] The number of the rows b[4] 2 3 b[5] b[6] 3 22 b[7] b[8] 5-5 The number of the cols The number of nonzero entries C.E., NCU, Taiwan Angela Chih-Wei Tang, 2 6

17 void transpose(term a[], term b[]) /* b is set to the transpose of a */ { int n, i, j, currentb; n=a[].value; b[].row = a[].col; b[].value = n; if(n>) {/*non zero matrix */ currentb=l; for(i=; i<a[].col; i++) /*transpose by the columns in a*/ for (j=; j<=n; j++) /* find elements from the current column*/ if (a[j].col == i) { /* element is in current column, add it to b */ b[currentb].row = a[j].col; b[currentb].col col = a[j].row; b[currentb].value = a[j].value; currentb++; } } } Transpose of A Sparse Matrix C.E., NCU, Taiwan Angela Chih-Wei Tang, 2 7

18 Asymptotic Time Complexity of The Transpose Algorithm The original transpose algorithm for(j=; j< columns; j++) for(i=; i<rows;i++) b[j][i]=a[i][j]; O(columns*rows) ( ) The revised transpose algorithm for loop: a[].col times iteration of the inner for loop : a[].value time a[].value: the number of elements in the original matrix O(columns*elements) Less space, more time??? Solution: fast transpose algorithm C.E., NCU, Taiwan Angela Chih-Wei Tang, 2 8

19 Fast Transpose of A Sparse Matrix void fast_transpose(term t a[], term b[]) { /* the transpose of a is placed in b*/ O(columns+elements) O(columns*rows) int row_terms[max_col], starting_pos[max_col]; int i, j, num_cols = a[].col, num_terms = a[].value; b[].row=num_cols; b[].col=a[];. Determine the number of b[].value =num_terms; elements in each row of if(num_terms > ) {/* nonzero matrix */ the transposed matrix for(i=; i< num_ cols; i++) row_terms[i]=; 2. Determine the starting for(i=; i<=num_terms; i++) position of each row in row_terms[a[i].col]++; the transpose matrix starting_pos[]=; 2 for(i=; i< num_cols; i++) starting_pos[i] = starting_pos[i-]+row_terms[i-]; terms[i ]; 3 for(i=; i<=num_terms; i++) { j=starting_pos[a[i].col]++; 3. Moving the elements in b[j].row=a[i].col; [i] b[j].col=a[i].row; [i] the original matrix into b[j].value=a[i].value; their correct position in C.E., NCU, Taiwan } the transpose matrix 9

20 2.4.3 Matrix Multiplication Definition: Given A and B, where A is mxn and B is nxp, the product matrix D has dimension mxp. Its [i][j] element is d ij = n k= for i < m and j < p. Rows a a a2 a a a2 a 2 a 2 a 22 a ik b kj b b b Columns b b b b b b 2 C.E., NCU, Taiwan Angela Chih-Wei Tang, 2 2

21 Multiplication of p Two Sparse Matrices (/3) How to multiply 2 matrices represented as an ordered list? Notice: the product of two sparse matrices may no longer be sparse = C.E., NCU, Taiwan Angela Chih-Wei Tang, 2 2

22 Multiplication of Two Sparse Matrices (2/3) row col value a[] a[] 5 a[2] 3 22 a[3] 5-5 a[4] a[5] 2 3 a[6] a[7] 4 9 a[8] row col value b[] b[] 5 b[2] 4 9 b[3] b[4] 2 3 b[5] b[6] 3 22 b[7] b[8] 5-5 Rows X Columns C.E., NCU, Taiwan Angela Chih-Wei Tang, 2 22

23 Multiplication of Two Sparse Matrices (3/3) C is the transpose matrix of B row col value a[] a[] 5 a[2] 3 22 a[3] 5-5 a[4] a[5] 2 3 a[6] a[7] 4 9 a[8] row col value c[] c[] 5 c[2] 3 22 c[3] 5-5 c[4] c[5] 2 3 c[6] c[7] 4 9 c[8] Angela Chih-Wei Tang, 2 23 Rows X Rows Apply operations similar to polynomial additions!

24 Outline The Array as An Abstract Data Type Structures The Polynomial Abstract Data Type The Sparse Matrix Abstract Data Type Multidimensional Arrays C.E., NCU, Taiwan Angela Chih-Wei Tang, 2 24

25 Representation of Arrays Multidimensional arrays are usually implemented by one dimensional array via either row major order or column major order. Example: One dimensional array α α+ α+2 α+3 α+4 α+5 α+6 α+7 α+8 α+9 α+ α+2 A[] A[] A[2] A[3] A[4] A[5] A[6] A[7] A[8] A[9] A[] A[] C.E., NCU, Taiwan Angela Chih-Wei Tang, 2 25

26 Two Dimensional Array Row Major Order Row Row Col Col Col 2 Col u 2 - X X X X X X X X Row u - X X X X u 2 elements u 2 elements Row Row Row i Row u - i * u 2 element C.E., NCU, Taiwan Angela Chih-Wei Tang, 2 26

27 Generalizing Array Representation The address indexing of Array A[i ][i 2 ],,[i n ] is α+ i u 2 u 3 u n + i 2 u 3 u 4 u n + i 3 u 4 u 5 u n : : + i n- u n + i n =α+ a n n j i ja j where k= j + j= = u k a n = j n C.E., NCU, Taiwan Angela Chih-Wei Tang, 2 27