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2  Introduction  Array Operations  Number of Elements in an array One-dimensional array Two dimensional array Multi-dimensional arrays Representation of Arrays in Memory One-dimensional array Two-dimensional arrays Three dimensional arrays N-dimensional array  Applications Sparse matrix Ordered lists ADT for arrays Outline

3  An array is an ADT whose objects are sequence of elements of the same type and the two operations performed on it are store and retrieve.  Engineering applications usually involve large chunk of data (of common type)  Arrays provide easy and efficient concept for data storage or management  Arrays are usually processed through loops (processing is very common)  Arrays are accessed by indicating an address or index

4  Arrays could be of one-dimension, two dimension, three-dimension or in general multi-dimension. ◦ A[1:5] ◦ A[1:5, 1:3] ◦ A[1:3,2:4,5:6,…,a:n]  N dimension array: ◦ A[a 0 :b 0,a 1 :b 1,…,a n-1 :b n-1 ]

5  An array when viewed as a data structure supports only two operations: ◦ storage of values (i.e.) writing into an array (STORE (a, i, e) ) ◦ r etrieval of values (i.e.) reading from an array ( RETRIEVE (a, i) )

6  Or (array’s size), is essential “memory locations”  A[l : u] where l is the lower bound and u is the upper bound of the index range, the number of elements is given by (u – l + 1).  The size of one dimension array:  Upper bound – lower bound + 1  A[l1 : u1, l2:u2] has a size of (u1-l1+1) (u2-l2+1) elements

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8  Calculate the size of the following arrays: ◦ A[1:10] ◦ A[-1:2] ◦ A[2:5,2:8] ◦ A[3:4,3:5,1:5]

9  The arrays are stored in memory in one of the two ways ◦ row major order or lexicographic order ◦ column major order.

10 One dimensional array

11  Given the address of the first element is (100), find ◦ In A[1:17] the address of A[7] ◦ In [-2:23] the address of A[16]  (106, 118)

12 Two dimensional array

13  Find : ◦ A[1:10,1:5] the address of A[8,3] ◦ A[-2:4,-6:10] the address of A[3,-5]  (257,306)

14 Three dimensional array

15  A[1:5,1:2,1:3] A[2,1,3]  A[-2:4,-6:10,1:3] A[-1,-4,2] ◦ (118,168)

16 N Dimensional arrays Let be an N-dimensional array. The address calculation for the retrieval of various elements are as given below:

17 Sparse Matrix  A matrix is a mathematical object which finds its applications in various scientific problems. A matrix is an arrangement of m X n elements arranged as m rows and n columns. The Sparse matrix is a matrix with zeros as the dominating elements. There is no precise definition for a sparse matrix. In other words, the “sparseness” is relatively defined  A matrix consumes a lot of space in memory. Thus, a 1000 X 1000 matrix needs 1 million storage locations in memory. Imagine the situation when the matrix is sparse! To store a handful of non-zero elements, voluminous memory is allotted and there by wasted!

18  To save valuable storage space, we resort to a 3-tuple representation viz., (i, j, value) to represent each non-zero element of the sparse matrix.  In other words, a sparse matrix A is represented by another matrix B[0:t, 1:3] with t+1 rows and 3 columns. Here t refers to the number of non-zero elements in the sparse matrix.  While rows 1 to t record the details pertaining to the non-zero elements as 3-tuples(that is 3 columns), the zeroth row viz. B[0,1], B[0,2] and B[0,3] record the number of non- zero elements of the original sparse matrix A.

19  One of the simplest and useful data objects in computer science is an ordered list or linear list.  An ordered list can be either empty or non empty. In the latter case, the elements of the list are known as atoms, chosen from a set D. The ordered lists provide a variety of operations such as retrieval, insertion, deletion, update etc.  The most common way to represent an ordered list is by using a one-dimensional array. Such a representation is termed sequential mapping through better forms of representation have been presented in the literature. Ordered lists

20  Given the following for the array RES. Find the address of RES[17]. ◦ Base address 520 ◦ Index of range 1:20 ◦ Array typeReal ◦ Size of memory location 4 bytes  For the following array B compute: ◦ The dimension of B ◦ The space occupied by B in the memory ◦ The address of B[7,2] ◦ Array B Column index 0:5 ◦ Base address 1003 ◦ Size of the memory location 4 bytes ◦ Row index 0:15

21 ADT for Arrays Data objects: A set of elements of the same type stored in a sequence Operations:  Store value VAL in the i th element of the array ARRAY ARRAY[i] = VAL  Retrieve the value in the i th element of array ARRAY as VAL = ARRAY[i]