# Streaming algorithm

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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[1] as well as Flajolet and Martin,[2] the field of streaming algorithms was first formalized and popularized in a paper by Noga Alon, Yossi Matias, and Mario Szegedy.[3] 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.

Semi-streaming algorithms were introduced in 2005 as an extension of streaming algorithms that allows for a constant or logarithmic number of passes over the dataset [1].

## 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.[4]

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 (possibly 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.[5] They also have applications in databases, such as estimating the size of a join[citation needed].

## Some streaming problems

### Frequency moments

The $k$th 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. In 2010, D. Kane, J. Nelson and D. Woodruff found an asymptotically optimal algorithm for this problem.[6] It uses O(ε^2 + log d) space, with O(1) worst-case update and reporting times, as well as universal hash functions and a r-wise independent hash family where r = Ω(log(1/ε)/ log log(1/ε)) ..

### Entropy

The (empirical) entropy of a set of frequencies $\mathbf{a}$ is defined as $F_k(\mathbf{a}) = \sum_{i=1}^n \frac{a_i}{m}\log{\frac{a_i}{m}}$, 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.
• Chakrabarti et al.

### Online learning

Learn a model (e.g. a classifier) by a single pass over a training set.

## 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.