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Data Stream Algorithms Intro, Sampling, EntropyPowerPoint Presentation

Data Stream Algorithms Intro, Sampling, Entropy

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Data Stream Algorithms Intro, Sampling, Entropy. Graham Cormode [email protected] Outline. Introduction to Data Streams Motivating examples and applications Data Streaming models Basic tail bounds Sampling from data streams Sampling to estimate entropy. Data is Massive.

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### Data Stream Algorithms Intro, Sampling, Entropy

Outline

### Data Stream Algorithms Frequency Moments

Frequency Moments

Frequency Moments

Frequency Moments

### Data Stream Algorithms Lower Bounds

### Data Stream Algorithms Extensions and Open Problems

Graham [email protected]

Outline

- Introduction to Data Streams
- Motivating examples and applications
- Data Streaming models
- Basic tail bounds

- Sampling from data streams
- Sampling to estimate entropy

Data Stream Algorithms

Data is Massive

- Data is growing faster than our ability to store or index it
- There are 3 Billion Telephone Calls in US each day, 30 Billion emails daily, 1 Billion SMS, IMs.
- Scientific data: NASA's observation satellites generate billions of readings each per day.
- IP Network Traffic: up to 1 Billion packets per hour per router. Each ISP has many (hundreds) routers!
- Whole genome sequences for many species now available: each megabytes to gigabytes in size

Data Stream Algorithms

Massive Data Analysis

Must analyze this massive data:

- Scientific research (monitor environment, species)
- System management (spot faults, drops, failures)
- Customer research (association rules, new offers)
- For revenue protection (phone fraud, service abuse)
Else, why even measure this data?

Data Stream Algorithms

Example: Network Data

- Networks are sources of massive data: the metadata per hour per router is gigabytes
- Fundamental problem of data stream analysis: Too much information to store or transmit
- So process data as it arrives: one pass, small space: the data stream approach.
- Approximate answers to many questions are OK, if there are guarantees of result quality

Data Stream Algorithms

Center (NOC)

SNMP/RMON,

NetFlow records

Source Destination DurationBytes Protocol

10.1.0.2 16.2.3.7 12 20K http

18.6.7.1 12.4.0.3 16 24K http

13.9.4.3 11.6.8.2 15 20K http

15.2.2.9 17.1.2.1 19 40K http

12.4.3.8 14.8.7.4 26 58K http

10.5.1.3 13.0.0.1 27 100K ftp

11.1.0.6 10.3.4.5 32 300K ftp

19.7.1.2 16.5.5.8 18 80K ftp

Peer

Converged IP/MPLS

Core

EnterpriseNetworks

PSTN

- Broadband Internet Access

DSL/Cable

Networks

- Voice over IP

- FR, ATM, IP VPN

- 24x7 IP packet/flow data-streams at network elements
- Truly massive streams arriving at rapid rates
- AT&T/Sprint collect ~1 Terabyte of NetFlow data each day

- Often shipped off-site to data warehouse for off-line analysis

Example NetFlow IP Session Data

Data Stream Algorithms

What are the top (most frequent) 1000 (source, dest) pairs seen over the last month?

DBMS

(Oracle, DB2)

How many distinct (source, dest) pairs have been seen by both R1 and R2 but not R3?

SELECT COUNT (R1.source, R2.dest)

FROM R1, R2

WHERE R1.dest = R2.source

Set-Expression Query

SQL Join Query

Network Monitoring Queries- Extra complexity comes from limited space and time
- Will introduce solutions for these and other problems

Off-line analysis – slow, expensive

Network Operations

Center (NOC)

R3

Peer

R1

R2

EnterpriseNetworks

PSTN

DSL/Cable

Networks

Data Stream Algorithms

Other Streaming Applications

- Sensor networks
- Monitor habitat and environmental parameters
- Track many objects, intrusions, trend analysis…

- Utility Companies
- Monitor power grid, customer usage patterns etc.
- Alerts and rapid response in case of problems

Data Stream Algorithms

Streams Defining Frequency Dbns.

- We will consider streams that define frequency distributions
- E.g. frequency of packets from source A to source B

- This simple setting captures many of the core algorithmic problems in data streaming
- How many distinct (non-zero) values seen?
- What is the entropy of the frequency distribution?
- What (and where) are the highest frequencies?

- More generally, can consider streams that define multi-dimensional distributions, graphs, geometric data etc.
- But even for frequency distributions, several models are relevant

Data Stream Algorithms

Data Stream Models

- We model data streams as sequences of simple tuples
- Complexity arises from massive length of streams
- Arrivals only streams:
- Example: (x, 3), (y, 2), (x, 2) encodesthe arrival of 3 copies of item x, 2 copies of y, then 2 copies of x.
- Could represent eg. packets on a network; power usage

- Arrivals and departures:
- Example: (x, 3), (y,2), (x, -2) encodes final state of (x, 1), (y, 2).
- Can represent fluctuating quantities, or measure differences between two distributions

x

y

x

y

Data Stream Algorithms

Approximation and Randomization

- Many things are hard to compute exactly over a stream
- Is the count of all items the same in two different streams?
- Requires linear space to compute exactly

- Approximation: find an answer correct within some factor
- Find an answer that is within 10% of correct result
- More generally, a (1) factor approximation

- Randomization: allow a small probability of failure
- Answer is correct, except with probability 1 in 10,000
- More generally, success probability (1-)

- Approximation and Randomization: (, )-approximations

Data Stream Algorithms

distribution

Tail probability

Markov:

Chebyshev:

Basic Tools: Tail Inequalities- General bounds on tail probability of a random variable (probability that a random variable deviates far from its expectation)
- Basic Inequalities: Let X be a random variable with expectation and variance Var[X]. Then, for any >0

Data Stream Algorithms

Tail Bounds

Markov Inequality:

For a random variable Y which takes only non-negative values.

Pr[Y k] E(Y)/k

(This will be < 1 only fork > E(Y))

Chebyshev’s Inequality:

For a random variable Y:

Pr[|Y-E(Y)| k] Var(Y)/k2

Proof:Set X = (Y – E(Y))2

- E(X) = E(Y2+E(Y)2–2YE(Y)) = E(Y2)+E(Y)2-2E(Y)2= Var(Y)
- So: Pr[|Y-E(Y)| k] = Pr[(Y – E(Y))2 k2].
- Using Markov: E(Y – E(Y))2/k2 = Var(Y)/k2

Data Stream Algorithms

Outline

- Introduction to Data Streams
- Motivating examples and applications
- Data Streaming models
- Basic tail bounds

- Sampling from data streams
- Sampling to estimate entropy

Data Stream Algorithms

Sampling From a Data Stream

- Fundamental prob: sample m items uniformly from stream
- Useful: approximate costly computation on small sample

- Challenge: don’t know how long stream is
- So when/how often to sample?

- Two solutions, apply to different situations:
- Reservoir sampling (dates from 1980s?)
- Min-wise sampling (dates from 1990s?)

Data Stream Algorithms

Reservoir Sampling

- Sample first m items
- Choose to sample the i’th item (i>m) with probability m/i
- If sampled, randomly replace a previously sampled item
- Optimization: when i gets large, compute which item will be sampled next, skip over intervening items. [Vitter 85]

Data Stream Algorithms

1 i i+1n-2n-1

…

i i+1 i+2 n-1 n

Reservoir Sampling - Analysis- Analyze simple case: sample size m = 1
- Probability i’th item is the sample from stream length n:
- Prob. i is sampled on arrival prob. i survives to end

= 1/n

- Case for m > 1 is similar, easy to show uniform probability
- Drawbacks of reservoir sampling: hard to parallelize

Data Stream Algorithms

Min-wise Sampling

- For each item, pick a random fraction between 0 and 1
- Store item(s) with the smallest random tag [Nath et al.’04]

0.391

0.908

0.291

0.555

0.619

0.273

- Each item has same chance of least tag, so uniform
- Can run on multiple streams separately, then merge

Data Stream Algorithms

Sampling Exercises

- What happens when each item in the stream also has a weight attached, and we want to sample based on these weights?
- Generalize the reservoir sampling algorithm to draw a single sample in the weighted case.
- Generalize reservoir sampling to sample multiple weighted items, and show an example where it fails to give a meaningful answer.
- Research problem: design new streaming algorithms for sampling in the weighted case, and analyze their properties.

Data Stream Algorithms

Outline

- Introduction to Data Streams
- Motivating examples and applications
- Data Streaming models
- Basic tail bounds

- Sampling from data streams
- Sampling to estimate entropy

Data Stream Algorithms

Application of Sampling: Entropy

- Given a long sequence of characters
S = <a1, a2, a3… am>each aj {1… n}

- Let fi = frequency of i in the sequence
- Compute the empirical entropy:
H(S) = - i fi/m log fi/m = - i pi log pi

- Example: S = < a, b, a, b, c, a, d, a>
- pa = 1/2, pb = 1/4, pc = 1/8, pd = 1/8
- H(S) = ½ + ¼ 2 + 1/8 3 + 1/8 3 = 7/4

- Entropy promoted for anomaly detection in networks

Data Stream Algorithms

Challenge

- Goal: approximate H(S) in space sublinear (poly-log) in m (stream length), n (alphabet size)
- (,) approx: answer is (1§)H(S) w/prob 1-

- Easy if we have O(n) space: compute each fi exactly
- More challenging if n is huge, m is huge, and we have only one pass over the input in order
- (The data stream model)

Data Stream Algorithms

Sampling Based Algorithm

- Simple estimator:
- Randomly sample a position j in the stream
- Count how many times aj appears subsequently = r
- Output X = -(r log (r/m) – (r-1) log((r-1)/m))

- Claim: Estimator is unbiased –E[X] = H(S)
- Proof: prob of picking j = 1/m, sum telescopes correctly

- Variance of estimate is not too large – Var[X] = O(log2 m)
- Observe that |X| ≤ log m
- Var[X] = E[(X – E[X])2] < (max(X) – min(X))2 = O(log2 m)

Data Stream Algorithms

Analysis of Basic Estimator

- A general technique in data streams:
- Repeat in parallel an unbiased estimator with bounded variance, take average of estimates to improve variance
- Var[ 1/k (Y1 + Y2 + ... Yk) ] = 1/k Var[Y]
- By Chebyshev, need k repetitions to be Var[X]/2E2[X]
- For entropy, this means space k = O(log2m/2H2(S))

- Problem for entropy: when H(S) is very small?
- Space needed for an accurate approx goes as 1/H2!

Data Stream Algorithms

Low Entropy

- But... what does a low entropy stream look like?
- aaaaaaaaaaaaaaaaaaaaaaaaaaaaaabaaaaa

- Very boring most of the time, we are only rarely surprised
- Can there be two frequent items?
- aabababababababaababababbababababababa
- No! That’s high entropy (¼ 1 bit / character)

- Only way to get H(S) =o(1) is to have only one character with pi close to 1

Data Stream Algorithms

Removing the frequent character

- Write entropy as
- -pa log pa + (1-pa) H(S’)
- Where S’ = stream S with all ‘a’s removed

- Can show:
- Doesn’t matter if H(S’) is small: as pa is large, additive error on H(S’) ensures relative error on (1-pa)H(S’)
- Relative error (1-pa) on pa gives relative error on pa log pa
- Summing both (positive) terms gives relative error overall

Data Stream Algorithms

Finding the frequency character

- Ejecting a is easy if we know in advance what it is
- Can then compute pa exactly

- Can find online deterministically
- Assume pa > 2/3 (if not, H(S) > 0.9, and original alg works)
- Run a ‘heavy hitters’ algorithm on the stream (see later)
- Modify analysis, find a and pa§ (1-pa)

- But... how to also compute H(S’) simultaneously if we don’t know a from the start... do we need two passes?

Data Stream Algorithms

Always have a back up plan...

- Idea: keep two samples to build our estimator
- If at the end one of our samples is ‘a’, use the other
- How to do this and ensure uniform sampling?

- Pick first sample with ‘min-wise sampling’:
- At end of the stream, if the sampled character = ‘a’, we want to sample from the stream ignoring all ‘a’s
- This is just “the character achieving the smallest label distinct from the one that achieves the smallest label”
- Can track information to do this in a single pass, constant space

Data Stream Algorithms

B

B

B

min tag

min tag amongst remaining tokens

second smallest tag, but we don’t want this; same token as min tag!

Sampling Two TokensStream:

C

A

A

B

B

A

B

D

C

A

B

A

Tags:

0.408

0.815

0.217

0.191

0.770

0.082

0.366

0.228

0.549

0.173

0.627

0.202

Repeats:

A

A

A

- Assign tags, choose first token as before
- Delete all occurrences of first token
- Choose token with min remaining tag; count repeats
- Implementation: keep track of two triples
- (min tag, corresponding token, number of repeats)

Data Stream Algorithms

Putting it all together

- Can combine all these pieces
- Build an estimator based on tracking this information, deciding whether there is a frequent character or not
- A more involved Chernoff bounds argument improves number of repetitions of estimator from O(-2Var[X]/E2[X]) to O(-2Range[X]/E[X]) = O(-2 log m)
- In O(-2 log m log 1/) space (words) we can compute an (,) approximation to H(S) in a single pass

Data Stream Algorithms

Entropy Exercises

- As a subroutine, we need to find an element that occurs more than 2/3 of the time and estimate its weight
- How can we find a frequently occurring item?
- How can we estimate its weight p with (1-p) error?
- Our algorithm uses O(-2 log m log 1/) space, could this be improved or is it optimal (lower bounds)?
- Our algorithm updates each sampled pair for every update, how quickly can we implement it?
- (Research problem) What if there are multiple distributed streams and we want to compute the entropy of their union?

Data Stream Algorithms

- Introduction to Data Streams
- Motivating examples and applications
- Data Streaming models
- Basic tail bounds

- Sampling from data streams
- Sampling to estimate entropy

Data Stream Algorithms

Graham [email protected]

Frequency Moments

- Introduction to Frequency Moments and Sketches
- Count-Min sketch for Fand frequent items
- AMS Sketch for F2
- Estimating F0
- Extensions:
- Higher frequency moments
- Combined frequency moments

Data Stream Algorithms

Last Time

- Introduced data streams and data stream models
- Focus on a stream defining a frequency distribution

- Sampling to draw a uniform sample from the stream
- Entropy estimation: based on sampling

Data Stream Algorithms

This Time: Frequency Moments

- Given a stream of updates, let fi be the number of times that item i is seen in the stream
- Define Fk of the stream as i (fi)k – the k’th Frequency Moment
- “Space Complexity of the Frequency Moments” by Alon, Matias, Szegedy in STOC 1996 studied this problem
- Awarded Godel prize in 2005
- Set the pattern for much streaming algorithms to follow
- Frequency moments are at the core of many streaming problems

Data Stream Algorithms

Frequency Moments

- F0 : count 1 if fi 0 – number of distinct items
- F1 : length of stream, easy
- F2 : sum the squares of the frequencies – self join size
- Fk : related to statistical moments of the distribution
- F : (really lim k Fk1/k) dominated by the largest fk, finds the largest frequency
- Different techniques needed for each one.
- Mostly sketch techniques, which compute a certain kind of random linear projection of the stream

Data Stream Algorithms

Sketches

- Not every problem can be solved with sampling
- Example: counting how many distinct items in the stream
- If a large fraction of items aren’t sampled, don’t know if they are all same or all different

- Other techniques take advantage that the algorithm can “see” all the data even if it can’t “remember” it all
- (To me) a sketch is a linear transform of the input
- Model stream as defining a vector, sketch is result of multiplying stream vector by an (implicit) matrix

linear projection

Data Stream Algorithms

Trivial Example of a Sketch

1 0 1 1 1 0 1 0 1 …

- Test if two (asynchronous) binary streams are equal d= (x,y) = 0 iff x=y, 1 otherwise
- To test in small space: pick a random hash function h
- Test h(x)=h(y) : small chance of false positive, no chance of false negative.
- Compute h(x), h(y) incrementally as new bits arrive (e.g. h(x) = xiti mod p for random prime p, and t < p)
- Exercise: extend to real valued vectors in update model

1 0 1 1 0 0 1 0 1 …

Data Stream Algorithms

Frequency Moments

- Introduction to Frequency Moments and Sketches
- Count-Min sketch for Fand frequent items
- AMS Sketch for F2
- Estimating F0
- Extensions:
- Higher frequency moments
- Combined frequency moments

Data Stream Algorithms

Count-Min Sketch

- Simple sketch idea, can be used for as the basis of many different stream mining tasks.
- Model input stream as a vector x of dimension U
- Creates a small summary as an array of w d in size
- Use d hash function to map vector entries to [1..w]
- Works on arrivals only and arrivals & departures streams

W

Array: CM[i,j]

d

Data Stream Algorithms

h1(j)

+c

+c

hd(j)

+c

Count-Min Sketch Structure- Each entry in vector x is mapped to one bucket per row.
- Merge two sketches by entry-wise summation
- Estimate x[j] by taking mink CM[k,hk(j)]
- Guarantees error less than eF1 in size O(1/e log 1/d)
- Probability of more error is less than 1-d

j,+c

d=log 1/

w = 2/

[C, Muthukrishnan ’04]

Data Stream Algorithms

Approximation of Point Queries

Approximate point query x’[j] = mink CM[k,hk(j)]

- Analysis: In k'th row, CM[k,hk(j)] = x[j] + Xk,j
- Xk,j = S x[i] | hk(i) = hk(j)
- E(Xk,j) = Si j x[i]*Pr[hk(i)=hk(j)] Pr[hk(i)=hk(k)] * Si x[i] = e F1/2 by pairwise independence of h
- Pr[Xk,j eF1] = Pr[Xk,j 2E(Xk,j)] 1/2 by Markov inequality

- So, Pr[x’[j] x[j] + eF1] = Pr[ k. Xk,j > eF1] 1/2log 1/d= d
- Final result: with certainty x[j] x’[j] and with probability at least1-d, x’[j] < x[j] + e F1

Data Stream Algorithms

Applications of Count-Min to F

- Count-Min sketch lets us estimate fi for any i (up to F1)
- F asks to find maxi fi
- Slow way: test every i after creating sketch
- Faster way: test every i after it is seen in the stream, and remember largest estimated value
- Alternate way:
- keep a binary tree over the domain of input items, where each node corresponds to a subset
- keep sketches of all nodes at same level
- descend tree to find large frequencies, discarding branches with low frequency

Data Stream Algorithms

Count-Min Exercises

- The median of a distribution is the item so that the sum of the frequencies of lexicographically smaller items is ½ F1. Use CM sketch to find the (approximate) median.
- Assume the input frequencies follow the Zipf distribution so that the i’th largest frequency is (i-z) for z>1. Show that CM sketch only needs to be size -1/z to give same guarantee
- Suppose we have arrival and departure streams where the frequencies of items are allowed to be negative. Extend CM sketch analysis to estimate these frequencies (note, Markov argument no longer works)
- How to find the large absolute frequencies when some are negative? Or in the difference of two streams?

Data Stream Algorithms

Frequency Moments

- Introduction to Frequency Moments and Sketches
- Count-Min sketch for Fand frequent items
- AMS Sketch for F2
- Estimating F0
- Extensions:
- Higher frequency moments
- Combined frequency moments

Data Stream Algorithms

F2 estimation

- AMS sketch (for Alon-Matias-Szegedy) proposed in 1996
- Allows estimation of F2 (second frequency moment)
- Used at the heart of many streaming and non-streaming mining applications: achieves dimensionality reduction

- Here, describe AMS sketch by generalizing CM sketch.
- Uses extra hash functions g1...glog 1/d{1...U} {+1,-1}
- Now, given update (j,+c), set CM[k,hk(i)] += c*gk(j)

linear projection

AMS sketch

Data Stream Algorithms

+c*g1(j)

h1(j)

+c*g2(j)

+c*g3(j)

hd(j)

+c*g4(j)

F2 analysis- Estimate F2 = mediankåi CM[k,i]2
- Each row’s result is åi g(i)2x[i]2 + åh(i)=h(j) 2 g(i) g(j) x[i] x[j]
- But g(i)2 = -12 = +12 = 1, andåi x[i]2 = F2
- g(i)g(j) has 1/2 chance of +1 or –1 : expectation is 0 …

j,+c

d=8log 1/

w = 4/2

Data Stream Algorithms

F2 Variance

- Expectation of row estimate Rk = åi CM[k,i]2is exactly F2
- Variance of row k, Var[Rk], is an expectation:
- Var[Rk] = E[ (buckets b (CM[k,b])2 – F2)2 ]
- Good exercise in algebra: expand this sum and simplify
- Many terms are zero in expectation because of terms like g(a)g(b)g(c)g(d) (degree at most 4)
- Requires that hash function g is four-wise independent: it behaves uniformly over subsets of size four or smaller
- Such hash functions are easy to construct

Data Stream Algorithms

F2 Variance

- Terms with odd powers of g(a) are zero in expectation
- g(a)g(b)g2(c), g(a)g(b)g(c)g(d), g(a)g3(b)

- Leaves Var[Rk] i g4(i) x[i]4 + 2 j i g2(i) g2(j) x[i]2 x[j]2 + 4 h(i)=h(j) g2(i) g2(j) x[i]2 x[j]2 - (x[i]4 +j i 2x[i]2 x[j]2) F22/w
- Row variance can finally be bounded by F22/w
- Chebyshev for w=4/2gives probability ¼ of failure
- How to amplify this to small probability of failure?

Data Stream Algorithms

Tail Inequalities for Sums

- We derive stronger bounds on tail probabilities for the sum of independent Bernoulli trials via theChernoff Bound:
- Let X1, ..., Xm be independent Bernoulli trials s.t. Pr[Xi=1] = p (Pr[Xi=0] = 1-p).
- Let X = i=1m Xi ,and = mp be the expectation of X.
- Then, for any >0,

Data Stream Algorithms

Applying Chernoff Bound

- Each row gives an estimate that is within relative error with probability p > ¾
- Take d repetitions and find the median. Why the median?
- Because bad estimates are either too small or too large
- Good estimates form a contiguous group “in the middle”
- At least d/2 estimates must be bad for median to be bad

- Apply Chernoff bound to d independent estimates, p=3/4
- Pr[ More than d/2 bad estimates ] < 2exp(d/8)
- So we set d = (ln ) to give probability of failure

- Same outline used many times in data streams

Data Stream Algorithms

Aside on Independence

- Full independence is expensive in a streaming setting
- If hash functions are fully independent over n items, then we need (n) space to store their description
- Pairwise and four-wise independent hash functions can be described in a constant number of words

- The F2 algorithm uses a careful mix of limited and full independence
- Each hash function is four-wise independent over all n items
- Each repetition is fully independent of all others – but there are only O(log 1/) repetitions.

Data Stream Algorithms

AMS Sketch Exercises

- Let x and y be binary streams of length n. The Hamming distance H(x,y) = |{i | x[i] y[i]}|Show how to use AMS sketches to approximate H(x,y)
- Extend for strings drawn from an arbitrary alphabet
- The inner product of two strings x, y is x y = i=1n x[i]*y[i]Use AMS sketches to estimate x y
- Hint: try computing the inner product of the sketches.Show the estimator is unbiased (correct in expectation)
- What form does the error in the approximation take?
- Use Count-Min Sketches for the same problem and compare the errors.
- Is it possible to build a (1) approximation of x y?

Data Stream Algorithms

- Introduction to Frequency Moments and Sketches
- Count-Min sketch for Fand frequent items
- AMS Sketch for F2
- Estimating F0
- Extensions:
- Higher frequency moments
- Combined frequency moments

Data Stream Algorithms

F0 Estimation

- F0 is the number of distinct items in the stream
- a fundamental quantity with many applications

- Early algorithms by Flajolet and Martin [1983] gave nice hashing-based solution
- analysis assumed fully independent hash functions

- Will describe a generalized version of the FM algorithm due to Bar-Yossef et. al with only pairwise indendence

Data Stream Algorithms

vt

F0 Algorithm- Let m be the domain of stream elements
- Each item in stream is from [1…m]

- Pick a random hash function h: [m] [m3]
- With probability at least 1-1/m, no collisions under h

- For each stream item i, compute h(i), and track the t distinct items achieving the smallest values of h(i)
- Note: if same i is seen many times, h(i) is same
- Let vt =t’th smallest value of h(i) seen.

- If F0 < t, give exact answer, else estimate F’0 = tm3/vt
- vt/m3 fraction of hash domain occupied by t smallest

0m3

m3

Data Stream Algorithms

Analysis of F0 algorithm

- Suppose F’0 = tm3/vt > (1+) F0[estimate is too high]
- So for stream = set S 2[m], we have
- |{ s S | h(s) < tm3/(1+)F0 }| > t
- Because < 1, we have tm3/(1+)F0 (1-/2)tm3/F0
- Pr[ h(s) < (1-/2)tm3/F0] 1/m3 * (1-/2)tm3/F0 = (1-/2)t/F0
- (this analysis outline hides some rounding issues)

0m3

vt

tm3/(1+)F0

m3

Data Stream Algorithms

Chebyshev Analysis

- Let Y be number of items hashing to under tm3/(1+)F0
- E[Y] = F0 * Pr[ h(s) < tm3/(1+)F0] = (1-/2)t
- For each item i, variance of the event = p(1-p) < p
- Var[Y] = s S Var[ h(s) < tm3/(1+)F0] < (1-/2)t
- We sum variances because of pairwise independence

- Now apply Chebyshev:
- Pr[ Y > t ] Pr[|Y – E[Y]| > t/2] 4Var[Y]/2t2< 4t/(2t2)
- Set t=20/2 to make this Prob 1/5

Data Stream Algorithms

Completing the analysis

- We have shownPr[ F’0 > (1+) F0 ] < 1/5
- Can show Pr[ F’0 < (1-) F0 ] < 1/5 similarly
- too few items hash below a certain value

- So Pr[ (1-) F0 F’0 (1+)F0] > 3/5 [Good estimate]
- Amplify this probability: repeat O(log 1/) times in parallel with different choices of hash function h
- Take the median of the estimates, analysis as before

Data Stream Algorithms

F0 Issues

- Space cost:
- Store t hash values, so O(1/2 log m) bits
- Can improve to O(1/2 + log m) with additional tricks

- Time cost:
- Find if hash value h(i) < vt
- Update vt and list of t smallest if h(i) not already present
- Total time O(log 1/ + log m) worst case

Data Stream Algorithms

Range Efficiency

- Sometimes input is specified as a stream of ranges [a,b]
- [a,b] means insert all items (a, a+1, a+2 … b)
- Trivial solution: just insert each item in the range

- Range efficient F0[Pavan, Tirthapura 05]
- Start with an alg for F0 based on pairwise hash functions
- Key problem: track which items hash into a certain range
- Dives into hash fns to divide and conquer for ranges

- Range efficient F2[Calderbank et al. 05, Rusu,Dobra 06]
- Start with sketches for F2 which sum hash values
- Design new hash functions so that range sums are fast

Data Stream Algorithms

F0 Exercises

- Suppose the stream consists of a sequence of insertions and deletions. Design an algorithm to approximate F0 of the current set.
- What happens when some frequencies are negative?

- Give an algorithm to find F0 of the most recent W arrivals
- Use F0 algorithms to approximate Max-dominance: given a stream of pairs (i,x(i)), approximate i max(i, x(i)) x(i)

Data Stream Algorithms

- Introduction to Frequency Moments and Sketches
- Count-Min sketch for Fand frequent items
- AMS Sketch for F2
- Estimating F0
- Extensions:
- Higher frequency moments
- Combined frequency moments

Data Stream Algorithms

Higher Frequency Moments

- Fk for k>2. Use sampling trick as with Entropy [Alon et al 96]:
- Uniformly pick an item from the stream length 1…n
- Set r = how many times that item appears subsequently
- Set estimate F’k = n(rk – (r-1)k)

- E[F’k]=1/n*n*[ f1k - (f1-1)k + (f1-1)k - (f1-2)k + … + 1k-0k]+… = f1k + f2k + … = Fk
- Var[F’k]1/n*n2*[(f1k-(f1-1)k)2 + …]
- Use various bounds to bound the variance by k m1-1/k Fk2
- Repeat k m1-1/k times in parallel to reduce variance

- Total space needed is O(k m1-1/k) machine words

Data Stream Algorithms

Improvements

- [Coppersmith and Kumar ‘04]: Generalize the F2 approach
- E.g. For F3, set p=1/m, and hash items onto {1-1/p, -1/p} with probability {1/p, 1-1/p} respectively.
- Compute cube of sum of the hash values of the stream
- Correct in expectation, bound variance O(mF32)

- [Indyk, Woodruff ‘05, Bhuvangiri et al. ‘06]:Optimal solutions by extracting different frequencies
- Use hashing to sample subsets of items and fi’s
- Combine these to build the correct estimator
- Cost is O(m1-2/kpoly-log(m,n,1/)) space

Data Stream Algorithms

Combined Frequency Moments

Consider network traffic data: defines a communication graph

eg edge: (source, destination)

or edge: (source:port, dest:port)

Defines a (directed) multigraph

We are interested in the underlying (support) graph on n nodes

- Want to focus on number of distinct communication pairs, not size of communication
- So want to compute moments of F0 values...

Data Stream Algorithms

Multigraph Problems

- Let G[i,j] = 1 if (i,j) appears in stream: edge from i to j. Total of m distinct edges
- Let di = Sj=1n G[i,j]: degree of node i
- Find aggregates of di’s:
- Estimate heavy di’s (people who talk to many)
- Estimate frequency moments:number of distinct di values, sum of squares
- Range sums of di’s (subnet traffic)

Data Stream Algorithms

F (F0) using CM-FM

- Find i’s such that di > fåi diFinds the people that talk to many others
- Count-Min sketch only uses additions, so can apply:

Data Stream Algorithms

Accuracy for F(F0)

- Focus on point query accuracy: estimate di.
- Can prove estimate has only small bias in expectation
- Analysis is similar to original CM sketch analysis, but now have to take account of F0 estimation of counts

- Gives an bound of O(1/3 poly-log(n)) space:
- The product of the size of the sketches

- Remains to fully understand other combinations of frequency moments, eg. F2(F0), F2(F2) etc.

Data Stream Algorithms

Exercises / Problems

- (Research problem) What can be computed for other combinations of frequency moments, e.g. F2 of F2 values, etc.?
- The F2 algorithm uses the fact that +1/-1 values square to preserve F2 but are 0 in expectation. Why won’t it work to estimate F4 with h {-1, +1, -i, +i}?
- (Research problem) Read, understand and simplify analysis for optimal Fk estimation algorithms
- Take the sampling Fk algorithm and combine it with F0 estimators to approximate Fk of node degrees
- Why can’t we use the sketch approach for F2 of node degrees? Show there the analysis breaks down

Data Stream Algorithms

- Introduction to Frequency Moments and Sketches
- Count-Min sketch for Fand frequent items
- AMS Sketch for F2
- Estimating F0
- Extensions:
- Higher frequency moments
- Combined frequency moments

Data Stream Algorithms

Graham [email protected]

Bob

Streaming Lower Bounds- Lower bounds for data streams
- Communication complexity bounds
- Simple reductions
- Hardness of Gap-Hamming problem
- Reductions to Gap-Hamming

1 0 1 1 1 0 1 0 1 …

Data Stream Algorithms

This Time: Lower Bounds

- So far, have seen many examples of things we can do with a streaming algorithm
- What about things we can’t do?
- What’s the best we could achieve for things we can do?
- Will show some simple lower bounds for data streams based on communication complexity

Data Stream Algorithms

Bob

Streaming As Communication- Imagine Alice processing a stream
- Then take the whole working memory, and send to Bob
- Bob continues processing the remainder of the stream

1 0 1 1 1 0 1 0 1 …

Data Stream Algorithms

Streaming As Communication

- Suppose Alice’s part of the stream corresponds to string x, and Bob’s part corresponds to string y...
- ...and that computing the function on the stream corresponds to computing f(x,y)...
- ...then if f(x,y) has communication complexity (g(n)), then the streaming computation has a space lower bound of (g(n))
- Proof by contradiction: If there was an algorithm with better space usage, we could run it on x, then send the memory contents as a message, and hence solve the communication problem

Data Stream Algorithms

Deterministic Equality Testing

1 0 1 1 1 0 1 0 1 …

- Alice has string x, Bob has string y, want to test if x=y
- Consider a deterministic (one-round, one-way) protocol that sends a message of length m < n
- There are 2m possible messages, so some strings must generate the same message: this would cause error
- So a deterministic message (sketch) must be (n) bits
- In contrast, we saw a randomized sketch of size O(log n)

1 0 1 1 0 0 1 0 1 …

Data Stream Algorithms

Hard Communication Problems

- INDEX: x is a binary string of length ny is an index in [n]Goal: output x[y]Result: (one-way) (randomized) communication complexity of INDEX is (n) bits
- DISJOINTNESS: x and y are both length n binary strings Goal: Output 1 if i: x[i]=y[i]=1, else 0Result: (multi-round) (randomized) communication complexity of DISJOINTNESS is (n) bits

Data Stream Algorithms

Simple Reduction to Disjointness

x: 1 0 1 1 0 1

1, 3, 4, 6

- F: output the highest frequency in a stream
- Input: the two strings x and y from disjointness
- Stream: if x[i]=1, then put i in stream; then same for y
- Analysis: if F=2, then intersection; if F1, then disjoint.
- Conclusion: Giving exact answer to F requires (N) bits
- Even approximating up to 50% error is hard
- Even with randomization: DISJ bound allows randomness

y: 0 0 0 1 1 0

4, 5

Data Stream Algorithms

Simple Reduction to Index

x: 1 0 1 1 0 1

1, 3, 4, 6

- F0: output the number of items in the stream
- Input: the strings x and index y from INDEX
- Stream: if x[i]=1, put i in stream; then put y in stream
- Analysis: if (1-)F’0(xy)>(1+)F’0(x) then x[y]=1, else it is 0
- Conclusion: Approximating F0 for <1/N requires (N) bits
- Implies that space to approximate must be (1/)
- Bound allows randomization

y: 5

5

Data Stream Algorithms

Hardness Reduction Exercises

Use reductions to DISJ orINDEX to show the hardness of:

- Frequent items: find all items in the stream whose frequency > N, for some .
- Sliding window: given a stream of binary (0/1) values, compute the sum of the last N values
- Can this be approximated instead?

- Min-dominance: given a stream of pairs (i,x(i)), approximate i min(i, x(i)) x(i)
- Rank sum: Given a stream of (x,y) pairs and query (p,q) specified after stream, approximate |{(x,y)| x<p, y<q}|

Data Stream Algorithms

Bob

Streaming Lower Bounds- Lower bounds for data streams
- Communication complexity bounds
- Simple reductions
- Hardness ofGap-Hammingproblem
- Reductions to Gap-Hamming

1 0 1 1 1 0 1 0 1 …

Data Stream Algorithms

Gap Hamming

GAP-HAMM communication problem:

- Alice holds x {0,1}N, Bob holds y {0,1}N
- Promise: H(x,y) is either N/2 - pN or N/2 + pN
- Which is the case?
- Model: one message from Alice to Bob
Requires (N) bits of one-way randomized communication

[Indyk, Woodruff’03, Woodruff’04, Jayram, Kumar, Sivakumar ’07]

Data Stream Algorithms

Hardness of Gap Hamming

- Reduction to an instance ofINDEX
- Map string x to u by 1 +1, 0 -1 (i.e. u[i] = 2x[i] -1 )
- Assume both Alice and Bob have access to public random strings rj, where each bit of rjis iid {-1, +1}
- Assume w.l.o.g. that length of string n is odd (important!)
- Alice computes aj = sign(rj u)
- Bob computes bj = sign(rj[y])
- Repeat N times with different random strings, and consider the Hamming distance of a1... aN with b1 ... bN

Data Stream Algorithms

Probability of a Hamming Error

- Consider the pair aj= sign(rj u), bj = sign(rj[y])
- Let w = i y u[i] rj[i]
- w is a sum of (n-1) values distributed iid uniform {-1,+1}

- Case 1: w 0. So |w| 2, since (n-1) is even
- so sign(aj) = sign(w), independent of x[y]
- Then Pr[aj bj] = Pr[sign(w) sign(rj[y]) = ½

- Case 2: w = 0. So aj = sign(rju) = sign(w + u[y]rj[y]) = sign(u[y]rj[y])
- Then Pr[aj bj] = Pr[sign(u[y]rj[y]) = sign(rj[y])]
- This probability is 1 is u[y]=+1, 0 if u[y]=-1
- Completely biased by the answer to INDEX

Data Stream Algorithms

Finishing the Reduction

- So what is Pr[w=0]?
- w is sum of (n-1) iid uniform {-1,+1} values
- Textbook: Pr[w=0] =c/n, for some constant c

- Do some probability manipulation:
- Pr[aj = bj] = ½ + c/2n if x[y]=1
- Pr[aj = bj] = ½ - c/2n if x[y]=0

- Amplify this bias by making strings of length N=4n/c2
- Apply Chernoff bound on N instances
- With prob>2/3, either H(a,b)>N/2 + N or H(a,b)<N/2 - N

- If we could solve GAP-HAMMING, could solve INDEX
- Therefore, need(N) = (n)bits forGAP-HAMMING

Data Stream Algorithms

Bob

Streaming Lower Bounds- Lower bounds for data streams
- Communication complexity bounds
- Simple reductions
- Hardness of Gap-Hamming problem
- Reductions toGap-Hamming

1 0 1 1 1 0 1 0 1 …

Data Stream Algorithms

(2,1)

(3,0)

(4,0)

(5,1)

(6,1)

(1,1)

(2,1)

(3,0)

(4,0)

(5,1)

(6,0)

Lower Bound for EntropyAlice: x {0,1}N, Bob: y {0,1}N

Entropy estimation algorithm A

- Alice runs A on enc(x) =(1,x1), (2,x2), …, (N,xN)
- Alice sends over memory contents to Bob
- Bob continues A on enc(y) =(1,y1), (2,y2), …, (N,yN)

Alice

0

1

0

0

1

1

Bob

1

1

0

0

1

0

Data Stream Algorithms

(2,1)

(3,0)

(4,0)

(5,1)

(6,1)

(1,1)

(2,1)

(3,0)

(4,0)

(5,1)

(6,0)

Lower Bound for Entropy- Observe: there are
- 2H(x,y) tokens with frequency 1 each
- N-H(x,y) tokens with frequency 2 each

- So, H(S) = log N + H(x,y)/N
- Thus size of Alice’s memory contents = (N). Set = 1/(p(N) log N) to show bound of (/log 1/)-2)

Alice

0

1

0

0

1

1

Bob

1

1

0

0

1

0

Data Stream Algorithms

Lower Bound for F0

- Same encoding works for F0 (Distinct Elements)
- 2H(x,y) tokens with frequency 1 each
- N-H(x,y) tokens with frequency 2 each

- F0(S) = N + H(x,y)
- Either H(x,y)>N/2 + N or H(x,y)<N/2 - N
- If we could approximate F0with < 1/N, could separate
- But space bound = (N) = (-2) bits

- Dependence onfor F0 is tight
- Similar arguments show (-2) bounds for Fk,
- Proof assumes k (and hence 2k) are constants

Data Stream Algorithms

Lower Bounds Exercises

- Formally argue the space lower bound for F2 via Gap-Hamming
- Argue space lower bounds for Fk via Gap-Hamming
- (Research problem) Extend lower bounds for the case when the order of the stream is random or near-random
- (Research problem) Kumar conjectures the multi-round communication complexity of Gap-Hamming is (n) – this would give lower bounds for multi-pass streaming

Data Stream Algorithms

Bob

Streaming Lower Bounds- Lower bounds for data streams
- Communication complexity bounds
- Simple reductions
- Hardness of Gap-Hamming problem
- Reductions to Gap-Hamming

1 0 1 1 1 0 1 0 1 …

Data Stream Algorithms

Graham [email protected]

This Time: Extensions

- Have given “the basics” of streaming: streams of items, frequency moments, upper and lower bounds
- Many variations with many open problems
- Streams representing different combinatorial objects
- Streams that are distributed, correlated, uncertain
- Systems for processing streams
- Different models of streams

- See also “Open problems in Data Streams” [McGregor ’07]
- Result of a workshop held at IIT Kanpur in Dec 2006

Data Stream Algorithms

Deterministic Streaming Algorithms

- Focus so far has been on randomized algorithms
- Many important problems can be solved deterministically!
- Finding frequent items/ heavy hitters
- Finding quantiles of a distribution

- For many problems, lower bounds show randomization is necessary for sublinear space:
- Anything involving equality testing as a special case
- Frequency moments

- When they are possible, deterministic algorithms are often faster and use less space: more practical to implement

Data Stream Algorithms

Clustering On Data Streams

- Goal: output k cluster centers at end - any point can be classified using these centers.
- Use divide and conquer approach [Guha et al. ’00]:
- Buffer as many points as possible, then cluster them
- Cluster the clusters
- Cluster the cluster clusters, etc...
- Each level of clustering gives up extra factors in quality

Input:

Output:

Data Stream Algorithms

Geometric Streaming

- Stream specifies a sequence of d-dimensional points
- Answer various geometric problems such as:
- Convex hull
- Minimum spanning tree weight
- Facility location
- Minimum enclosing ball

- Gridding approach reduces to Fk or related problems [Indyk ’03]
- Core-set: keep a carefully chosen small subset of points and evaluate on them [Har-Peled 02, Chan’06]
- Simple example: For minimum enclosing ball, keep extremal points in evenly-space directions

Data Stream Algorithms

Sliding Window Computations

- In a sliding window, we only consider the last W items
- W still very large, so want poly-log(W) solutions

- Exponential Histograms [Datar et al.02]and Waves [Gibbons Tirthapura’02]
- Deterministic structure tracks counts in a window
- Based on doubling bucket sizes to give relative error
- Same structure + sketches solves for aggregates

- Asynchronous streams: items not in timestamp order
- Relative error counts possible [Busch, Tirthapura ’07]
- Extend concept to other aggregates [C. et al. ’08]

Data Stream Algorithms

Time Decay

- Assign a weight to each item as a function of its age
- E.g. Exponential decay or polynomial decay
- Implies “weighted” versions of problems

- Cohen and Strauss [2003]:
- Can reduce sum and counts to multiple instances of sliding window queries

- C., Korn and Tirthapura [2008]:
- Same observations applies to othercomputations (quantiles, frequent items)

age

Data Stream Algorithms

Multi-Pass Algorithms

- Some situations allow multiple passes of the stream
- E.g. scanning over slow storage (tape): random access not possible, but can scan multiple times

- Earliest work in streaming [Munro, Paterson ’78] studied the pass/space tradeoff for finding medians
- Lower bounds can follow from multi-round communication complexity bounds

1 0 1 1 1 0 1 0 1 …

Data Stream Algorithms

Other Massive Data Models

- Massive Unordered Data (MUD) model [Feldman et al. ‘08]
- Abstracts computations in MapReduce/Hadoop settings
- Can provably simulate deterministic streaming algs
- What about randomized computations, multiple passes?

Data Stream Algorithms

items sorted by frequency

log frequency

log rank

Skewed Streams- In practice, not all frequency distributions are worst case
- Few items are frequent, then a long tail of infrequent items
- Such skew is prevalent in network data, word frequency, paper citations, city sizes, etc.
- “Zipfian” distribution with skew z > 0 (z = [1..2] typical)

- Analyze algorithms under assumption of skewed data
- Improved F2 space cost = O(e-2/(1+z) log 1/d), provided z>1

Data Stream Algorithms

5

1

4

2

Graph Streaming- Stream specifies a massive graph edge by edge
- Most natural problems have (|V|) space lower bounds
- Semi-streaming model: allow (|V|) but o(|E|) spaceTherefore also o(|V|2) space also

- Allow one (or few) passes to approximate:
- Minimum Spanning Tree Weight
- Graph Distances (based on spanners)
- Maximum weight matching
- Counting Triangles

(4,5) (2,3) (1,3) (3,5) (1,2) (2,4) (1,5) (3,4)

Data Stream Algorithms

( )( )

A C U R

Matrix Streaming- Stream specifies a massive n nmatrix
- Either by giving entries in some order, or updates to entries

- In one (or few) passes, find:
- CUR Decomposition
- Page Rank Vector
- Approximate Matrix product
- Singular Vector Decomposition

- Current methods take small constant number of passes, sample constant number of rows and columns by weight
- Sketching methods don’t seem so useful here

O(1) Columns

O(1) Rows

Carefully chosen U

Data Stream Algorithms

3

4

2

3

4

1

2

Permutation Streaming- Stream presents a permutation of items
- Abstraction of several settings, more of theoretic interest

- Approximate number of inversions in the stream
- Locations where i >j but i appears before j in stream
- Can be reduced to a variation of quantiles [Gupta, Zane’03]

- Find length of longest increasing subsequence
- Reduce (up to factor 2) to simpler function [Ergun, Jowhari ’08]
- Approximate this using a different variation of quantiles
- Deterministic lower bound (N1/2), randomized bound open

Data Stream Algorithms

Random Order Streaming

- Lower bounds are sometimes based on carefully creating adversarial orders of streams
- Random order streams: order is uniformly permuted
- Can sometimes give much better upper bounds– prefix of stream gives a good sample of dbn. to come
- Lower bounds in random order give stronger evidence of “robust” hardness, e.g. [Chakrabarti et al. ’08]
- Hardness via communication complexity of random partitions
- GAP-HAMMING still has linear lower bound
- t2-party DISJOINTNESS has (n/t) lower bound

Data Stream Algorithms

Probabilistic Streams

- Example: S = (x, ½, y, 1/3, y, ¼)
- Encodes 6 “possible worlds”:

- Instead of exact values, stream of discrete distributions
- Specify exponentially many possible worlds

- Adds complexity to previously studied problems
- Sum and Count are easy (by linearity of expectation)
- Avg=Sum/Count is hard! –because of ratio [McGregor et al. ’07]

- Linearity of expectation, summation of variance
- Allows estimation of Fk over streams [C, Garofalakis ’07]

Data Stream Algorithms

Distributed Streams

- Motivated by Sensor Networks – large wireless nets
- Communication drains battery: compute more, send less

- Key problem: design stream summary data structures that can be combined to summarize the union of streams
- Most sketches (AMS, Count-Min, F0) naturally distribute
- Similar results needed for other problems

http://www.intel.com/research/exploratory/motes.htm

base station

(root, coordinator…)

Data Stream Algorithms

Track Q(S1,…,Sm)

local stream(s) seen at each site

m sites

S1

Sm

Continuous Distributed Model- Goal:Continuously track (global) query over streams at the coordinator while bounding the communication
- Large-scale network-event monitoring, real-time anomaly/ DDoS attack detection, power grid monitoring, …

- Results known for quantiles, Fk, clustering...
- Cost not much higher than one time computation [C et al. 08]

Data Stream Algorithms

Extensions for P2P Networks

- Much work focused on specifics of sensor and wired nets
- P2P and Grid computing present alternate models
- Structure of multi-hop overlay networks
- “Controlled failure” model: nodes explicitly leave and join

- Allows us to think beyond model of “highly resource constrained” sensors.
- Implementations such as OpenDHT over PlanetLab [Rhea et al.’05]

Data Stream Algorithms

Authenticated Stream Aggregation

- Wide-area query processing
- Possible malicious aggregators
- Can suppress or add spurious information

- Authenticate query results at the querier?
- Perhaps, to within some approximation error

- Initial steps in [Garofalakis et al.’06],
- Sliding window: [Hadjieleftheriou et al. ’07]

Data Stream Algorithms

Data Stream Algorithms

- Slides are on the web on my website
- Long list of references also on the web
- http://dimacs.rutgers.edu/~graham

Data Stream Algorithms

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