Streaming algorithm

In computer science, streaming algorithms are algorithms for
processing data streams in which the input is presented as a sequence of
items and can be examined in only a few passes (typically just one). These
algorithms have limited memory available to them (much less than the input
size) and also limited processing time per item.
These constraints may mean that an algorithm produces an approximate answer based on a summary or "sketch" of the data stream in memory. 
==History==
Though streaming algorithms had already been studied by Munro and
Paterson as well as Flajolet and
Martin, the field of streaming
algorithms was first formalized and popularized in a paper by Noga Alon,
Yossi Matias, and Mario Szegedy. 
For this paper, the authors later won the Gödel Prize in 2005 "for their foundational
contribution to streaming algorithms." There has since been a large body
of work centered around data streaming algorithms that spans a diverse
spectrum of computer science fields such as theory, databases, networking,
and natural language processing.
==Models==
In the data stream model, some or all of the input data that are
to be operated on are not available for random access from disk or
memory, but rather arrive as one or more continuous data streams. 
Streams can be denoted as an ordered sequence of points (or "updates") that must be accessed in order and can be read only once or a small number of times.
Much of the streaming literature is concerned with computing statistics on
frequency distributions that are too large to be stored. For this class of
problems, there is a vector \mathbf{a} = (a_1, \dots, a_n)
(initialized to the zero vector \mathbf{0}) that has updates
presented to it in a stream. The goal of these algorithms is to compute
functions of \mathbf{a} using considerably less space than it
would take to represent \mathbf{a} precisely. There are two
common models for updating such streams, called the "cash register" and
"turnstile" models.
In the cash register model each update is of the form \langle i,
c\rangle, so that a_i is incremented by some positive
integer c. A notable special case is when c = 1
(only unit insertions are permitted).
In the turnstile model each update is of the form \langle i,
c\rangle, so that a_i is incremented by some (possible
negative) integer c. In the "strict turnstile" model, no
a_i at any time may be less than zero.
Several papers also consider the "sliding window" model. In this model,
the function of interest is computing over a fixed-size window in the
stream. As the stream progresses, items from the end of the window are
removed from consideration while new items from the stream take their
place.
Besides the above frequency-based problems, some other types of problems
have also been studied. Many graph problems are solved in the setting
where the adjacency matrix or the adjacency list of the graph is streamed in
some unknown order. There are also some problems that are very dependent
on the order of the stream (i.e., asymmetric functions), such as counting
the number of inversions in a stream and finding the longest increasing
subsequence.
== Evaluation ==
The performance of an algorithm that operates on data streams is measured by three basic factors:
* The number of passes the algorithm must make over the stream.
* The available memory.
* The running time of the algorithm.
These algorithms have many similarities with online algorithms since they both require decisions to be made before all data are available, but they are not identical. Data stream algorithms only have limited memory available but they may be able to defer action until a group of points arrive, while online algorithms are required to take action as soon as each point arrives.
If the algorithm is an approximation algorithm then the accuracy of the answer is another key factor. The accuracy is often stated as an (\epsilon,\delta) approximation meaning that the algorithm achieves an error of less than \epsilon with probability 1-\delta.
==Applications==
Streaming algorithms have several applications in networking such as
monitoring network links for elephant flows, counting the number of
distinct flows, estimating the distribution of flow sizes, and so
on. They also have applications in
databases, such estimating the size of a join.
==Some streaming problems==
===Frequency moments===
The kth frequency moment of a set of frequencies
\mathbf{a} is defined as F_k(\mathbf{a}) = \sum_{i=1}^n
a_i^k.
The first moment F_1 is simply the sum of the frequencies
(i.e., the total count). The second moment F_2 is useful for
computing statistical properties of the data, such as the Gini coefficient
of variation. F_{\infty} is defined as the frequency of the
most frequent item(s).
The seminal paper of Alon, Matias, and Szegedy dealt with the
problem of estimating the frequency moments.
===Heavy hitters===
Find all elements i whose frequency a_i > T, say. Some
notable algorithms are:
* Karp-Papadimitriou-Shenker algorithm
* Count-min sketch
* Sticky sampling
* Lossy counting
* Sample and Hold
* Multi-stage Bloom filters
* Count-sketch
* Sketch-guided sampling
===Counting distinct elements===
Counting the number of distinct elements in a stream (sometimes called the
F_0 moment) is another problem that has been well studied.
The first algorithm for it was proposed by Flajolet and Martin, and the
currently best known algorithm is due to Bar-Yossef et al.
===Entropy===
The (empirical) entropy of a set of frequencies \mathbf{a} is
defined as F_k(\mathbf{a}) = \sum_{i=1}^n
\frac{a_i}{m}\log{a_i/m}, where m = \sum_{i=1}^n a_i.
Estimation of this quantity in a stream has been done by:
* McGregor et al.
* Do Ba et al.
* Lall et al.
* Chakraborty et al.
==Lower bounds==
Lower bounds have been computed for many of the data streaming problems
that have been studied. By far, the most common technique for computing
these lower bounds has been using communication complexity.
== See also ==
* Data stream mining
* Stream processing
* Data stream clustering
==Notes==
==References==
* .
** First published as: .
* .
* . From SIGMETRICS 2007.
* 
* 
* .
==External links==
* Princeton Lecture Notes
* Streaming Algorithms for Geometric Problems, by Piotr Indyk, MIT
* Dagstuhl Workshop on Sublinear Algorithms
* IIT Kanpur Workshop on Data Streaming
* List of open problems in streaming (compiled by Andrew McGregor) from discussion at the IITK Workshop on Algorithms for Data Streams, 2006.
* StreamIt - programming language and compilation infrastructure by MIT CSAIL
* IBM Spade - Stream Processing Application Declarative Engine
* IBM InfoSphere Streams
;Tutorials and surveys
* Data Stream Algorithms and Applications by S. Muthu Muthukrishnan
* Stanford STREAM project survey
* Network Applications of Bloom filters, by Broder and Mitzenmacher
* Xu's SIGMETRICS 2007 tutorial
* Lecture notes from Data Streams course at Barbados in 2009, by Andrew McGregor and S. Muthu Muthukrishnan
;Courses
* MIT
* Rice
* Rutgers
* University at Buffalo
Category:Algorithms
fr:Algorithme de fouille de flots de données