Data Structures and Algorithms(12)

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1 Ming Zhang "Data s and Algorithms" Data s and Algorithms(12) Instructor: Ming Zhang Textbook Authors: Ming Zhang, Tengjiao Wang and Haiyan Zhao Higher Education Press, (the "Eleventh Five-Year" national planning textbook)

2 12.1 Multidimensional array Basic Concepts of Array Storage of Array Declaration of Array Special Matrices Implemented by Arrays Sparse Matrix 12.2 Generalized List 12.3 Storage management 12.4 Trie 12.5 Improved BST 2 Ming Zhang Data s and Algorithms

3 Basic Concepts Array is an ordered sequence with fixed number of elements and type. The size and type of static array must be specified at compile time Dynamic array is allocated memory at runtime 3 Ming Zhang Data s and Algorithms

4 Basic Concepts Multidimensional array is an extension of onedimensional array (vector). Vector of vectors make up an multidimensional array. Represented as ELEM A[c 1..d 1 ][c 2..d 2 ] [c n..d n ] c i and d i are upper and lower bounds of the indices in the i-th dimension. Thus, the total n number of elements is: i1 ( d i c i 1) 4 Ming Zhang Data s and Algorithms 4

5 of Array d1=3,d2=5 d2 d1 d1=3,d2=4, d3 d3=2 d1 d2 a[1][1] a[1][3][5] 2-dimensional array 3-dimensional array d1[0..2], d2[0..3], d3[0..1] are the three dimensions respectively 5 Ming Zhang Data s and Algorithms 5

6 Storage of Array Memory is one-dimensional, so arrays are stored linearly Stored row by row (row-major) Stored column by column (column-major) X= Ming Zhang Data s and Algorithms

7 Row-Major in Pascal a[1..k,1..m,1..n] a 111 a 112 a 113 a 11n a 122 a 123 a 12n a 121 a 1m1 a 1m2 a 1m3 a 1mn a 211 a 212 a 213 a 21n a 221 a 222 a 223 a 22n a 2m1 a 2m2 a 2m3 a 2mn a k11 a k12 a k13 a k1n a k21 a k22 a k23 a k2n a km1 a km2 a km3 a kmn 7 Ming Zhang Data s and Algorithms

Column-Major in FORTRAN a 111 a 211 a 311 a k11

a 12n a 22n a 32n a k2n a 1mn a 2mn a 3mn a kmn

8 Column-Major in FORTRAN a 111 a 211 a 311 a k11 a 121 a 221 a 321 a k21 a 1m1 a 2m1 a 3m1 a km1 a 112 a 212 a 312 a k12 a 122 a 222 a 322 a k22 a 1m2 a 2m2 a 3m2 a km2 a 11n a 21n a 31n a k1n a 12n a 22n a 32n a k2n a 1mn a 2mn a 3mn a kmn a[1..k, 1..m, 1..n] 8 Ming Zhang Data s and Algorithms

9 C++ multidimensional array ELEM A[d 1 ][ d 2 ] [d n ]; loc( A[ j, j,, j ]) loc( A[0,0,,0]) 1 2 d [ j d d j d d 1 2 n 2 3 j d j n n1 n n ] n1 loc( A[0,0,,0]) d [ j d j ] n n i k n i1 k i 1 9 Ming Zhang Data s and Algorithms 9

10 Special Matrices Implemented by Arrays Triangular matrix (upper/lower) Symmetric matrix Diagonal matrix Sparse matrix 10 Ming Zhang Data s and Algorithms 10

11 Lower Triangular Matrix One-dimensional array: list[0.. (n 2 +n)/2-1] The matrix element a i,j is stored in list[ (i 2 +i) /2 + j] (i>=j) Ming Zhang Data s and Algorithms 11

12 Symmetric Matrix Satisfies that a i,j = a j,i, 0 i, j < n The matrix on the right is a (symmetric) adjacent matrix for a undirected graph Store the lower triangle in a 1-dimensional array sa[0..n (n+1) /2-1] There is a one-to-one mapping between sa[k] and a i,j : k j( j 1) / 2 i, i j i( i 1) / 2 j, i j 12 Ming Zhang Data s and Algorithms

13 Diagonal Matrix Diagonal matrix: all non-zero elements are located at diagonal lines. Band matric: a[i][j] = 0 when i-j > 1 A band matrix with bandwidth 1 is shown as below a 0,0 a 0,1 a 1,0 a1,1 0 a 1,2 0 a n-1,n-2 a n-2,n-1 a n-1,n-1 13 Ming Zhang Data s and Algorithms 13

14 Sparse Matrix Few non-zero elemens, and these elements distribute unevenly A Ming Zhang Data s and Algorithms

15 Sparse Factor In a m n matrix, there are t non-zero elements, and the sparse factor is: When this value is lower than 0.05, the matrix could be considered a sparse matrix. 3-tuple (i, j, a ij ): commonly used for input/output i is the row number j is the column number a ij is the element value m t n 15 Ming Zhang Data s and Algorithms

16 Orthogonal Lists of a Sparse Matrix An orthogonal list consists of two sets of lists pointer sequense for rows and columns Each node has two pointers: one points to the successor on the same row; the other points to the successor on the same column head pointer for rows head pointer for columns Ming Zhang Data s and Algorithms 16

17 Classic Matrix Multiplication 17 Ming Zhang Data s and Algorithms 17

18 Time Cost of Classic Matrix Multiplication p=d1-c1+1,m=d3-c3+1,n=d2-c2+1; A is a p m matrix, B is a m n matrix, resulting in C, a p n matrix So the time cost of the classic matrix multiplication is O (p m n) for (i=c1; i<=d1; i++) for } (j=c2; j<=d2; j++){ sum = 0; for (k=c3; k<=d3; k++) sum = sum + A[i,k]*B[k,j]; C[i,j] = sum; 18 Ming Zhang Data s and Algorithms 18

19 head pointer for columns Sparse Matrix Multiplication = head pointer for rows finish Ming Zhang Data s and Algorithms 19

20 Time Cost of Sparse Matrix Multiplication A is a p m matrix, B is a m n matrix, resulting in C, a p n matrix. If the number of non-zero elements in a row of A is at most t a and the number of non-zero elements in a column of B is at most t b Overall running time is reduced to O ( (t a +t b ) p n) Time cost of classic matrix multiplication is O (p m n) 20 Ming Zhang Data s and Algorithms 20

21 Applications of Sparse Matrix polynomial of one indeterminate P n 2 ( x) a a x a x n 0 i0 a i 1 x i 2 a n x n 21 Ming Zhang Data s and Algorithms 21

22 Ming Zhang Data s and Algorithms Data s and Algorithms Thanks the National Elaborate Course (Only available for IPs in China) Ming Zhang, Tengjiao Wang and Haiyan Zhao Higher Education Press, (awarded as the "Eleventh Five-Year" national planning textbook)

Data Structures and Algorithms(2)

Ming Zhang Data Structures and Algorithms Data Structures and Algorithms(2) Instructor: Ming Zhang Textbook Authors: Ming Zhang, Tengjiao Wang and Haiyan Zhao Higher Education Press, 2008.6 (the "Eleventh