Streaming algorithm

History

Though streaming algorithms had already been studied by Munro and Paterson as early as 1980, as well as Philippe Flajolet and G. Nigel Martin in 1982/83, the field of streaming algorithms was first formalized and popularized in a 1996 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.

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 a = ( a 1 , … , a n ) (initialized to the zero vector 0 ) that has updates presented to it in a stream. The goal of these algorithms is to compute functions of a using considerably less space than it would take to represent 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 ⟨ i , c ⟩ , 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 ⟨ i , c ⟩ , 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 ( ϵ , δ ) approximation meaning that the algorithm achieves an error of less than ϵ with probability 1 − δ .

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 as estimating the size of a join.

Frequency moments

The kth frequency moment of a set of frequencies a is defined as F k ( a ) = ∑ 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 ∞ 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.

Calculating Frequency Moments

A direct approach to find the frequency moments requires to maintain a register m_i for all distinct elements a_i ∈ (1,2,3,4,...,N) which requires at least memory of order Ω ( N ) . But we have space limitations and requires an algorithm that computes in much lower memory. This can be achieved by using approximations instead of exact values. An algorithm that computes an (ε,δ)approximation of F_k, where F'_k is the (ε,δ)- approximated value of F_k. Where ε is the approximation parameter and δ is the confidence parameter.

Calculating F₀ (Distinct Elements in a DataStream)

FM-Sketch Algorithm

Flajolet et al. in introduced probabilistic method of counting which was inspired from a paper by Robert Morris Counting large numbers of events in small registers. Morris in his paper says that if the requirement of accuracy is dropped, a counter n can be replaced by a counter log n which can be stored in log log n bits. Flajolet et al. in improved this method by using a hash function h which is assumed to uniformly distribute the element in the hash space (a binary string of length L).

h : [ m ] → [ 0 , 2 L − 1 ]

Let bit(y,k) represent the kth bit in binary representation of y

y = ∑ k ≥ 0 b i t ( y , k ) ∗ 2 k

Let ρ ( y ) represents the position of least significant 1-bit in the binary representation of y_i with a suitable convention for ρ ( 0 ) .

ρ ( y ) = { M i n ( b i t ( y , k ) ) if  y > 0 L if  y = 0

Let A be the sequence of data stream of length M whose cardinality need to be determined. Let BITMAP [0...L − 1] be the

hash space where the ρ(hashedvalues) are recorded. The below algorithm the determines approximate cardinality of A.

Procedure FM-Sketch: for i in 0 to L − 1 do BITMAP[i]:=0 end for for x in A: do Index:=ρ(hash(x)) i end if end for B:= Position of left most 0 bit of BITMAP[] return 2^B

If there are N distinct elements in a data stream.

For i ≫ log ⁡ ( N ) then BITMAP[i] is certainly 0

For i ≪ log ⁡ ( N ) then BITMAP[i] is certainly 1

For i ≈ log ⁡ ( N ) then BITMAP[i] is a fringes of 0's and 1's

K-Minimum Value Algorithm

The previous algorithm describes the first attempt to approximate F₀ in the data stream by Flajolet and Martin. Their algorithm picks a random hash function which they assume to uniformly distribute the hash values in hash space.

Bar-Yossef et al. in, introduces k-minimum value algorithm for determining number of distinct elements in data stream. They uses a similar hash function h which can be normalized to [0,1] as h : [ m ] → [ 0 , 1 ] . But they fixed a limit t to number of values in hash space. The value of t is assumed of the order O ( 1 ε 2 ) (i.e. less approximation-value ε requires more t). KMV algorithm keeps only t-smallest hash values in the hash space. After all the m values of stream are arrived, υ = M a x ( h ( a i ) ) is used to calculate F 0 ′ = t υ . That is, in a close-to uniform hash space, they expect at-least t elements to be less than O ( t F 0 ) .

Procedure 2 K-Minimum Value Initialize first t values of KMV for a in a1 to an do if h(a) < Max(KMV) then Remove Max(KMV) from KMV set Insert h(a) to KMV end if end for return t/Max(KMV)

Complexity analysis of KMV

KMV algorithm can be implemented in O ( ( 1 ε 2 ) ⋅ log ⁡ ( m ) ) memory bits space. Each hash value requires space of order O ( log ⁡ ( m ) ) memory bits. There are hash values of the order O ( 1 ε 2 ) . The access time can be reduced if we store the t hash values in a binary tree. Thus the time complexity will be reduced to O ( log ⁡ ( 1 ε ) ⋅ log ⁡ ( m ) ) .

Calculating F_k

Alon et al. in estimates F_k by defining random variables that can be computed within given space and time. The expected value of random variable gives the approximate value of F_k.

Let us assume length of sequence m is known in advance.

Construct a random variable X as follows:

Select a_p be a random member of sequence A with index at p, a p = l ∈ ( 1 , 2 , 3 , … , n )

Let r = | { q : q ≥ p , a p = l } | , represents the number of occurrences of l within the members of the sequence A following a_p.

Random variable X = m ( r k − ( r − 1 ) k ) .

Assume S₁ be of the order O ( n 1 − 1 / k / λ 2 ) and S₂ be of the order O ( log ⁡ ( 1 / ε ) ) . Algorithm takes S₂ random variable Y₁,Y₂,...,Y_S2 and outputs the median Y . Where Y_i is the average of X_ij where 1 ≤ j ≤ S₁.

Now calculate expectation of random variable E(X).

E ( X ) = ∑ i = 1 n ∑ i = 1 m i ( j k − ( j − 1 ) k ) = m m [ ( 1 k + ( 2 k − 1 k ) + … + ( m 1 k − ( m 1 − 1 ) k ) ) + ( 1 k + ( 2 k − 1 k ) + … + ( m 2 k − ( m 2 − 1 ) k ) ) + … + ( 1 k + ( 2 k − 1 k ) + … + ( m n k − ( m n − 1 ) k ) ) ] = ∑ i = 1 n m i k = F k

Complexity of F_k

From the algorithm to calculate F_k discussed above, we can see that each random variable X stores value of a_p and r. So, to compute X we need to maintain only log(n) bits for storing a_p and log(n) bits for storing r. Total number of random variable X will be the S 1 ∗ S 2 .

Hence the total space complexity the algorithm takes is of the order of O ( k log ⁡ 1 ε λ 2 n 1 − 1 k ( log ⁡ n + log ⁡ m ) )

Simpler approach to calculate F₂

The previous algorithm calculates F 2 in order of O ( n ( log ⁡ m + log ⁡ n ) ) memory bits. Alon et al. in simplified this algorithm using four-wise independent random variable with values mapped to { − 1 , 1 } .

This further reduces the complexity to calculate F 2 to O ( log ⁡ 1 ε λ 2 ( log ⁡ n + log ⁡ m ) )

Heavy hitters

Find the most frequent (popular) elements in a data stream. Some notable algorithms are:

Boyer–Moore majority vote algorithm

Karp-Papadimitriou-Shenker algorithm

Count-Min sketch

Sticky sampling

Lossy counting

Sample and Hold

Multi-stage Bloom filters

Count-sketch

Sketch-guided sampling

Event detection

Detecting events in data streams is often done using a heavy hitters algorithm as listed above: the most frequent items and their frequency are determined using one of these algorithms, then the largest increase over the previous time point is reported as trend. This approach can be refined by using exponentially weighted moving averages and variance for normalization.

Counting distinct elements

Counting the number of distinct elements in a stream (sometimes called the F₀ 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. It uses O(ε² + 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 a is defined as F k ( a ) = ∑ i = 1 n a i m log ⁡ a i m , where m = ∑ 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.

Feature hashing

Stochastic gradient descent

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.