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Text of the page (random words):
variables and special notation for indexing array elements 1 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 3 special array types are often defined by the language s standard libraries dynamic lists are also more common and easier to implement dubious discuss than dynamic arrays array types are distinguished from record types mainly because they allow the element indices to be computed at run time as in the pascal assignment a i j a n i 2 j among other things this feature allows a single iterative statement to process arbitrarily many elements of an array variable in more theoretical contexts especially in type theory and in the description of abstract algorithms the terms array and array type sometimes refer to an abstract data type adt also called abstract array or may refer to an associative array a mathematical model with the basic operations and behavior of a typical array type in most languages basically a collection of elements that are selected by indices computed at run time depending on the language array types may overlap or be identified with other data types that describe aggregates of values such as lists and strings array types are often implemented by array data structures but sometimes by other means such as hash tables linked lists or search trees history edit heinz rutishauser s programming language superplan 1949 1951 included multi dimensional arrays however although rutishauser described how a compiler for his language should be built did not implement one assembly languages and low level languages like bcpl 4 generally have no syntactic support for arrays because of the importance of array structures for efficient computation the earliest high level programming languages including fortran 1957 cobol 1960 and algol 60 1960 provided support for multi dimensional arrays abstract arrays edit 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 5 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 implementations edit in order to effectively implement variables of such types as array structures with indexing done by pointer arithmetic many languages restrict the indices to integer data types 6 7 or other types that can be interpreted as integers such as bytes and enumerated types and require that all elements have the same data type and storage size most of those languages also restrict each index to a finite interval of integers that remains fixed throughout the lifetime of the array variable in some compiled languages in fact the index ranges may have to be known at compile time on the other hand some programming languages provide more liberal array types that allow indexing by arbitrary values such as floating point numbers strings objects references etc such index values cannot be restricted to an interval much less a fixed interval so these languages usually allow arbitrary new elements to be created at any time this choice precludes the implementation of array types as array data structures that is those languages use array like syntax to implement a more general associative array semantics and must therefore be implemented by a hash table or some other search data structure language support edit this section needs more citations please help improve this section by adding citations to reliable sources unsourced material may be challenged and removed january 2026 learn how and when to remove this message type edit languages have differing ways of defining an array type for example in c an array is actually a block of contiguous memory which is essentially treated as a pointer 8 in such cases array declarations can decay to pointers int a 10 array a of 10 ints int p a p points to the first element of a void foo int arr the parameter arr is an int arr is decayed to a pointer to its first element however in other languages such as java an array is an actual type for any type t it has a corresponding array type t which is an object with a length field 9 int a new int 5 declares an array a of 5 ints multi dimensional arrays edit a two dimensional array stored as a one dimensional array of one dimensional arrays rows the number of indices needed to specify an element is called the dimension dimensionality or rank of the array type a there are two common ways to support multi dimensional arrays via pointer chasing edit many languages support only one dimensional arrays in those languages a multi dimensional array is typically represented by an iliffe vector a one dimensional array of references to arrays of one dimension less a two dimensional array in particular would be implemented as a vector of pointers to its rows 10 thus an element in row i and column j of an array a would be accessed by double indexing a i j in typical notation this way of emulating multi dimensional arrays allows the creation of jagged arrays where each row may have a different size or in general where the valid range of each index depends on the values of all preceding indices this representation for multi dimensional arrays is quite prevalent in c and c software however c and c will use a linear indexing formula for multi dimensional arrays that are declared with compile time constant size e g by int a 10 20 or int a m n instead of the traditional int a 11 the c99 standard introduced variable length array types that let define array types with dimensions computed in run time the dynamic 4d array can be constructed using a pointer to 4d array e g int arr t u v w malloc sizeof arr the individual elements are accessed by first de referencing an array pointer followed by indexing e g arr i j k l alternatively n d arrays can be declared as pointers to its first element which is a n 1 dimensional array e g int arr u v w malloc t sizeof arr and accessed using more idiomatic syntax e g arr i j k l via computation edit some languages allow direct computation of element location languages with direct computation of element locations typically enclose the subscript list in a single pair of delimiters e g foo bar baz fortran pl i foo bar baz algol 60 pascal rather than placing each subscript in a pair of delimiters e g foo bar baz the order in which array elements are stored differs among languages e g fortran has row major arrays while pl i has column major arrays indexing notation edit most programming languages that support arrays support the store and select operations and have special syntax for indexing early languages used parentheses e g a i j as in fortran others choose square brackets e g a i j or a i j as in algol 60 and pascal to distinguish from the use of parentheses for function calls index types edit array data types are most often implemented as array structures with the indices restricted to integer or totally ordered values index ranges fixed at array creation time and multilinear element addressing this was the case in most third generation languages and is still the case of most systems programming languages such as ada c and c in some languages however array data types have the semantics of associative arrays with indices of arbitrary type and dynamic element creation this is the case in some scripting languages such as awk and lua and of some array types provided by standard c libraries bounds checking edit main article bounds checking some languages like pascal and modula perform bounds checking on every access raising an exception or aborting the program when any index is out of its valid range compilers may allow these checks to be turned off to trade safety for speed other languages like fortran and c trust the programmer and perform no checks good compilers may also analyze the program to determine the range of possible values that the index may have and this analysis may lead to bounds checking elimination index origin edit further information comparison of programming languages array some languages such as c provide only zero based array types for which the minimum valid value for any index is 0 12 this choice is convenient for array implementation and address computations with a language such as c a pointer to the interior of any array can be defined that will symbolically act as a pseudo array that accommodates negative indices this works only because c does not check an index against bounds when used other languages provide only one based array types where each index starts at 1 this is the traditional convention in mathematics for matrices and mathematical sequences a few languages such as pascal and lua support n based array types whose minimum legal indices are chosen by the programmer the relative merits of each choice have been the subject of heated debate zero based indexing can avoid off by one or fencepost errors 13 highest index edit the relation between numbers appearing in an array declaration and the index of that array s last element also varies by language in many languages such as c one should specify the number of elements contained in the array whereas in others such as pascal and visual basic net one should specify the numeric value of the index of the last element this distinction is not present in languages where the indices start at 1 such as lua array algebra edit some programming languages support array programming where operations and functions defined for certain data types are implicitly extended to arrays of elements of those types thus one can write a b to add corresponding elements of two arrays a and b usually these languages provide both the element by element multiplication and the standard matrix product of linear algebra and which of these is represented by the operator varies by language languages providing array programming capabilities have proliferated since the innovations in this area of apl these are core capabilities of domain specific languages such as gauss idl matlab and mathematica they are a core facility in newer languages such as julia and recent versions of fortran these capabilities are also provided via standard extension libraries for other general purpose programming languages such as the widely used numpy library for python string types and arrays edit many languages provide a built in string data type with specialized notation string literals to build values of that type in some languages such as c a string is just an array of characters or is handled in much the same way 14 other languages like pascal may provide vastly different operations for strings and arrays array index range queries edit some programming languages provide operations that return the size number of elements of a vector or more generally range of each index of an array in c and c arrays do not support the size function so programmers often have to declare separate variable to hold the size and pass it to procedures as a separate parameter elements of a newly created array may have undefined values as in c or may be defined to have a specific default value such as 0 or a null pointer as in java in c a std vector object supports the store select and append operations with the performance characteristics discussed above 15 vectors can be queried for their size and can be resized slower operations like inserting an element in the middle are also supported slicing edit an array slicing operation takes a subset of the elements of an array typed entity value or variable and then assembles them as another array typed entity possibly with other indices if array types are implemented as array structures many useful slicing operations such as selecting a sub array swapping indices or reversing the direction of the indices can be performed very efficiently by manipulating the dope vector of the structure the possible slicings depend on the implementation details for example fortran allows slicing off one column of a matrix variable but not a row and treat it as a vector on the other hand other slicing operations are possible when array types are implemented in other ways resizing edit some languages allow dynamic arrays also called resizable growable or extensible array variables whose index ranges may be expanded at any time after creation without changing the values of its current elements for one dimensional arrays this facility may be provided as an operation append a x that increases the size of the array a by one and then sets the value of the last element to x other array types such as pascal strings provide a concatenation operator which can be used together with slicing to achieve that effect and more in some languages assigning a value to an element of an array automatically extends the array if necessary to include that element in other array types a slice can be replaced by an array of different size with subsequent elements being renumbered accordingly as in python s list assignment a 5 5 10 20 30 that inserts three new elements 10 20 and 30 before element a 5 resizable arrays are conceptually similar to lists and the two concepts are synonymous in some languages an extensible array can be implemented as a fixed size array with a counter that records how many elements are actually in use the append operation merely increments the counter until the whole array is used when the append operation may be defined to fail this is an implementation of a dynamic array with a fixed capacity as in the string type of pascal alternatively the append operation may re allocate the underlying array with a larger size and copy the old elements to the new area see also edit array access analysis array database management system bounds checking elimination delimiter separated values index checking parallel array sparse array variable length array notes edit this nomenclature conflicts with the concept of dimension in linear algebra which expresses the shape of a matrix thus an array of numbers with 5 rows and 4 columns hence 20 elements is said to have dimension 2 in computing contexts bu...