computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...

, an online algorithm is one that can process its input piece-by-piece in a serial fashion, i.e., in the order that the input is fed to the

algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...

, without having the entire input available from the start. In contrast, an offline algorithm is given the whole problem data from the beginning and is required to output an answer which solves the problem at hand. In

operations research Operations research ( en-GB, operational research) (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a discipline that deals with the development and application of analytical methods to improve deci ...

, the area in which online algorithms are developed is called

online optimization Online optimization is a field of optimization theory, more popular in computer science and operations research, that deals with optimization problems having no or incomplete knowledge of the future (online). These kind of problems are denoted as o ...

. As an example, consider the

sorting algorithms In computer science, a sorting algorithm is an algorithm that puts elements of a list into an order. The most frequently used orders are numerical order and lexicographical order, and either ascending or descending. Efficient sorting is important ...

selection sort and insertion sort: selection sort repeatedly selects the minimum element from the unsorted remainder and places it at the front, which requires access to the entire input; it is thus an offline algorithm. On the other hand, insertion sort considers one input element per iteration and produces a partial solution without considering future elements. Thus insertion sort is an online algorithm. Note that the final result of an insertion sort is optimum, i.e., a correctly sorted list. For many problems, online algorithms cannot match the performance of offline algorithms. If the ratio between the performance of an online algorithm and an optimal offline algorithm is bounded, the online algorithm is called competitive. Not every ''offline algorithm'' has an efficient ''online'' counterpart.

Definition

Because it does not know the whole input, an online algorithm is forced to make decisions that may later turn out not to be optimal, and the study of online algorithms has focused on the quality of decision-making that is possible in this setting. Competitive analysis formalizes this idea by comparing the relative performance of an online and offline algorithm for the same problem instance. Specifically, the competitive ratio of an algorithm, is defined as the worst-case ratio of its cost divided by the optimal cost, over all possible inputs. The competitive ratio of an online problem is the best competitive ratio achieved by an online algorithm. Intuitively, the competitive ratio of an algorithm gives a measure on the quality of solutions produced by this algorithm, while the competitive ratio of a problem shows the importance of knowing the future for this problem.

Other interpretations

For other points of view on ''online inputs to algorithms'', see * streaming algorithm: focusing on the amount of memory needed to accurately represent past inputs; *

dynamic algorithm Dynamic problems in computational complexity theory are problems stated in terms of the changing input data. In the most general form a problem in this category is usually stated as follows: * Given a class of input objects, find efficient algorit ...

: focusing on the time complexity of maintaining solutions to problems with online inputs.

Examples

Some ''online algorithms'': * Insertion sort * Perceptron * Reservoir sampling * Greedy algorithm * Adversary model *

Metrical task systems Task systems are mathematical objects used to model the set of possible configuration of online algorithms. They were introduced by Borodin, Linial and Saks (1992) to model a variety of online problems. A task system determines a set of states and ...

* Odds algorithm * Page replacement algorithm * Algorithms for calculating variance * Ukkonen's algorithm

Online problems

A problem exemplifying the concepts of online algorithms is the Canadian Traveller Problem. The goal of this problem is to minimize the cost of reaching a target in a weighted graph where some of the edges are unreliable and may have been removed from the graph. However, that an edge has been removed (''failed'') is only revealed to ''the traveller'' when she/he reaches one of the edge's endpoints. The worst case for this problem is simply that all of the unreliable edges fail and the problem reduces to the usual

Shortest Path Problem In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. The problem of finding the shortest path between tw ...

. An alternative analysis of the problem can be made with the help of competitive analysis. For this method of analysis, the offline algorithm knows in advance which edges will fail and the goal is to minimize the ratio between the online and offline algorithms' performance. This problem is PSPACE-complete. There are many formal problems that offer more than one ''online algorithm'' as solution: * ''k''-server problem * Job shop scheduling problem *

List update problem The List Update or the List Access problem is a simple model used in the study of competitive analysis of online algorithms. Given a set of items in a list where the cost of accessing an item is proportional to its distance from the head of the lis ...

Bandit problem In probability theory and machine learning, the multi-armed bandit problem (sometimes called the ''K''- or ''N''-armed bandit problem) is a problem in which a fixed limited set of resources must be allocated between competing (alternative) choices ...

Secretary problem The secretary problem demonstrates a scenario involving optimal stopping theory For French translation, secover storyin the July issue of ''Pour la Science'' (2009). that is studied extensively in the fields of applied probability, statistics, an ...

Search games A search game is a two-person zero-sum game which takes place in a set called the search space. The searcher can choose any continuous trajectory subject to a maximal velocity constraint. It is always assumed that neither the searcher nor the hider ...

Ski rental problem In computer science, the ski rental problem is a name given to a class of problems in which there is a choice between continuing to pay a repeating cost or paying a one-time cost which eliminates or reduces the repeating cost. The problem Many o ...

Linear search problem In computational complexity theory, the linear search problem is an optimal search problem introduced by Richard E. Bellman and independently considered by Anatole Beck. The problem "An immobile hider is located on the real line according to a k ...

* Portfolio selection problem

References

*{{cite book , author-link = Allan Borodin , author = Borodin, A. , author2=El-Yaniv, R. , url = https://www.cs.technion.ac.il/~rani/book.html , title = Online Computation and Competitive Analysis , publisher = Cambridge University Press , year = 1998 , isbn = 0-521-56392-5

External links