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In computer science, an array type is a data type that is meant to describe a collection of elements (values or variables), each selected by one or more indices (identifying keys) that can be computed at run time by the program. Such a collection is usually called an array variable, array value, or simply array. By analogy with the mathematical concepts of vector and matrix, array types with one and two indices are often called vector type and matrix type, respectively.

Language support for array types may include certain built-in array data types, some syntactic constructions (array type constructors) that the programmer may use to define such types and declare array variables, and special notation for indexing array elements.For example, in the Pascal programming language, the declaration type MyTable = array [1..4,1..2] of integer, defines a new array data type called MyTable. The declaration var A: MyTable then defines a variable A of that type, which is an aggregate of eight elements, each being an integer variable identified by two indices. In the Pascal program, those elements are denoted A[1,1], A[1,2], A[2,1],… A[4,2].Special array types are often defined by the language's standard libraries.


An array data structure can be mathematically modeled as an abstract data structure (an abstract array) with two operations

    get(A, I): the data stored in the element of the array A whose indices are the integer tuple I.
    set(A,I,V): the array that results by setting the value of that element to V.

These operations are required to satisfy the axioms

    get(set(A,I, V), I) = V
    get(set(A,I, V), J) = get(A, J) if I ≠ J

for any array state A, any value V, and any tuples I, J for which the operations are defined.

The first axiom means that each element behaves like a variable. The second axiom means that elements with distinct indices behave as disjoint variables, so that storing a value in one element does not affect the value of any other element.

These axioms do not place any constraints on the set of valid index tuples I, therefore this abstract model can be used for triangular matrices and other oddly-shaped arrays.



Multidimensional arrays

For a two-dimensional array, the element with indices i,j would have address B + c · i + d · j, where the coefficients c and d are the row and column address increments, respectively.

More generally, in a k-dimensional array, the address of an element with indices i1, i2, …, ik is

    B + c1 · i1 + c2 · i2 + … + ck · ik.

For example: int a[3][2];

This means that array a has 3 rows and 2 columns, and the array is of integer type. Here we can store 6 elements they are stored linearly but starting from first row linear then continuing with second row. The above array will be stored as a11, a12, a13, a21, a22, a23.

This formula requires only k multiplications and k−1 additions, for any array that can fit in memory. Moreover, if any coefficient is a fixed power of 2, the multiplication can be replaced by bit shifting.

The coefficients ck must be chosen so that every valid index tuple maps to the address of a distinct element.

If the minimum legal value for every index is 0, then B is the address of the element whose indices are all zero. As in the one-dimensional case, the element indices may be changed by changing the base address B. Thus, if a two-dimensional array has rows and columns indexed from 1 to 10 and 1 to 20, respectively, then replacing B by B + c1 - − 3 c1 will cause them to be renumbered from 0 through 9 and 4 through 23, respectively. Taking advantage of this feature, some languages (like FORTRAN 77) specify that array indices begin at 1, as in mathematical tradition; while other languages (like Fortran 90, Pascal and Algol) let the user choose the minimum value for each index.

Reference: Wikipedia