Data Structures
Internal
Overview
A data structure is an arrangement of data in computer's memory or external storage, designed to facilitate a specific way to access or modify the data, quickly and usefully. Data structures include arrays, linked lists, stacks, queues, binary trees, search trees, heaps, hash tables, bloom filters, union-find, etc.
Data structures are crucial ingredients in the design of fast algorithms. A data structure is responsible for organizing data in a way that supports fast queries. Different data structures support different types of operations and are suited for different kinds of tasks.
Data Structure Characteristics
A data structure can be described based on several characteristics:
- space requirements.
- what operations the data structure supports.
- what is the running time of those operations.
In general, fewer operations a data structure supports, the faster the operations and smallest the space overhead required by the data structure will be.
Data Structures as Sets
All data structures used to represent data in the memory of a computer are essentially sets. Unlike a mathematical set, the sets manipulated by computers are finite, and their composition change in time, so they are called dynamic sets. We know from mathematics that the elements of a set are distinct. Data structure sets also consist of distinct elements, if we conceptualize the fact that the elements are maintained in distinct memory locations. In a typical implementation of a dynamic set, each element is represented by an object (memory location) whose attributes can be examined and manipulated if we have a pointer or a reference to the objects. Different objects have different references.
Some dynamic sets assume that one of the element's attributes is an identifying key. If the keys are all different, we can think of the dynamic set as being a set of key values. Some dynamic sets assume that the keys are drawn from a totally ordered set, such as the real numbers, or the set of all words under the usual alphabetic ordering. A total ordering allows defining a minimum element of the set, or to speak of the next element larger than the given element.
Augmenting a Data Structure
Augmenting a data structure means modifying the data structure to maintain additional information about the data structure itself. An example is maintaining the number of nodes contained by the subtree of a certain node in a binary search tree, which is useful to implement SELECT() and RANK() on binary search trees, or the "color" of a red-black tree. Care should be taken to update the augmented state in case of any operation that modifies the data structure.
Dictionary
A dynamic set that allows insertion of new elements, deletion of existing elements and membership testing (or search) for a specific element is called a dictionary. Insertion and deletion modify the set, and they are called modifying operations. Search does not modify the set and it belongs to the query category of operations. Depending on the contexts, the concept of dictionary may imply an underlying total ordered set and the associated queries on totally ordered sets.
Modifying Dictionary Operations
Assuming that X is a pointer, or a reference, not a key, a dictionary supports the following modifying operations:
INSERT(X)
Add the element pointed by X to the set.
DELETE(X)
Remove the element pointed by X.
Queries
SEARCH(K)
A query that given a key value K, returns a pointer to an element with the given key, or NULL if there is no such element. Different data structures implement search in different ways, resulting in different running times relative to the size of the set. For example, a static sorted array or a binary search tree implement the search operation in O(log n). Also see:
Queries on Totally Ordered Sets
A totally ordered set, also known as a linearly ordered set, is formally defined here.
We usually measure the time taken the execute an operation in terms of the size of the set n. Asymptotic notation is used to describe the behavior of these operations, as a function of the set size.
If the set is totally ordered, the following operations may be exposed:
SELECT(ith order statistics)
Return the ith order statistics in the set. The ith order statistics of a set of n numbers is the ith smallest number in the set.
For more details about the selection problem see:
MINIMUM()
Return the pointer to the element with the smallest key. MINIMUM() is the particular case of the 0th order statistic in a sorted set, returned by SELECT().
MAXIMUM()
Returns the pointer to the element with the largest key. MAXIMUM() is the particular case of the n-1th order statistic in a sorted set, returned by SELECT().
PREDECESSOR(X)
A query that, given an element X whose key is from a totally ordered set, returns a pointer to the next smaller element, or NULL if X is the minimum element.
SUCCESSOR(X)
A query that, given an element X whose key is from a totally ordered set, returns a pointer to the next larger element, or NULL if X is the maximum element.
RANK(X)
How many keys stored in the data structure are less or equal than the given value.
Arrays
An array is a fundamental data structure in computer science. Arrays are how data is physically stored in memory. An array consists in a contiguous set of memory locations. The elements are maintained in linear order, which is determined by the array index. Given the pointer to the first element of the array, accessing any element is a Θ(1) operation that consists in adding the index multiplied with the element size to the first element pointer to obtain the pointer of the desired element.
In Java, fast copies for primitive arrays with System.arraycopy(source, sourcePosition, destination, destinationPosition, count).