VIEWS: 2 PAGES: 114 POSTED ON: 3/27/2013
Data Stream Algorithms Intro, Sampling, Entropy Graham Cormode graham@research.att.com Outline Introduction to Data Streams – Motivating examples and applications – Data Streaming models – Basic tail bounds Sampling from data streams Sampling to estimate entropy 2 Data Stream Algorithms - Introduction 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 3 Data Stream Algorithms - Introduction 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? 4 Data Stream Algorithms - Introduction 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 5 Data Stream Algorithms - Introduction IP Network Monitoring Application Source Destination Duration Bytes Protocol 10.1.0.2 16.2.3.7 12 20K http Network Operations 18.6.7.1 12.4.0.3 16 24K http SNMP/RMON, 13.9.4.3 11.6.8.2 15 20K http Center (NOC) NetFlow records 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 Peer 11.1.0.6 10.3.4.5 32 300K ftp Converged IP/MPLS Core 19.7.1.2 16.5.5.8 18 80K ftp Example NetFlow IP Session Data Enterprise PSTN Networks • FR, ATM, IP VPN DSL/Cable • Broadband • Voice over IP Networks Internet Access 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 6 Data Stream Algorithms - Introduction Packet-Level Data Streams Single 2Gb/sec link; say avg packet size is 50bytes Number of packets/sec = 5 million Time per packet = 0.2 microsec If we only capture header information per packet: src/dest IP, time, no. of bytes, etc. – at least 10bytes. – Space per second is 50Mb – Space per day is 4.5Tb per link – ISPs typically have hundreds of links! Analyzing packet content streams – order(s) of magnitude harder 7 Data Stream Algorithms - Introduction Network Monitoring Queries Back-end Data Warehouse DBMS (Oracle, DB2) What are the top (most frequent) 1000 (source, dest) Off-line analysis – pairs seen over the last month? slow, expensive Network Operations How many distinct (source, dest) pairs have Center (NOC) been seen by both R1 and R2 but not R3? Peer R3 Set-Expression Query R1 SELECT COUNT (R1.source, R2.dest) R2 FROM R1, R2 WHERE R1.dest = R2.source Enterprise PSTN SQL Join Query Networks DSL/Cable Networks Extra complexity comes from limited space and time Will introduce solutions for these and other problems 8 Data Stream Algorithms - Introduction Streaming Data Questions Network managers ask questions requiring us to analyze the data: – How many distinct addresses seen on the network? – Which destinations or groups use most bandwidth? – Find hosts with similar usage patterns? Extra complexity comes from limited space and time Will introduce solutions for these and other problems 9 Data Stream Algorithms - Introduction 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 10 Data Stream Algorithms - Introduction 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 11 Data Stream Algorithms - Introduction 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) encodes x the arrival of 3 copies of item x, y 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 x final state of (x, 1), (y, 2). y – Can represent fluctuating quantities, or measure differences between two distributions 12 Data Stream Algorithms - Introduction 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 13 Data Stream Algorithms - Introduction Basic Tools: Tail Inequalities General bounds on tail probability of a random variable (probability that a random variable deviates far from its expectation) Probability distribution Tail probability Basic Inequalities: Let X be a random variable with expectation and variance Var[X]. Then, for any >0 Markov: Chebyshev: 1 Var[X] Pr(X (1 ε)μ) Pr(| X μ | με) 2 2 1 ε με 14 Data Stream Algorithms - Introduction 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 for k > 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 15 Data Stream Algorithms - Introduction Outline Introduction to Data Streams – Motivating examples and applications – Data Streaming models – Basic tail bounds Sampling from data streams Sampling to estimate entropy 16 Data Stream Algorithms - Introduction 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?) 17 Data Stream Algorithms - Introduction 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] 18 Data Stream Algorithms - Introduction 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 i i+1 … n-2 n-1 i i+1 i+2 n-1 n = 1/n Case for m > 1 is similar, easy to show uniform probability Drawbacks of reservoir sampling: hard to parallelize 19 Data Stream Algorithms - Introduction 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 20 Data Stream Algorithms - Introduction Sampling Exercises What happens when each item in the stream also has a weight attached, and we want to sample based on these weights? 1. Generalize the reservoir sampling algorithm to draw a single sample in the weighted case. 2. Generalize reservoir sampling to sample multiple weighted items, and show an example where it fails to give a meaningful answer. 3. Research problem: design new streaming algorithms for sampling in the weighted case, and analyze their properties. 21 Data Stream Algorithms - Introduction Outline Introduction to Data Streams – Motivating examples and applications – Data Streaming models – Basic tail bounds Sampling from data streams Sampling to estimate entropy 22 Data Stream Algorithms - Introduction 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 23 Data Stream Algorithms - Introduction 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) 24 Data Stream Algorithms - Introduction 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) 25 Data Stream Algorithms - Introduction 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] Var[X] Pr(| X μ | με) 2 2 με – By Chebyshev, need k repetitions to be Var[X]/2E2[X] – For entropy, this means space O(log2m/2H2(S)) Problem for entropy: when H(S) is very small? – Space needed for an accurate approx goes as 1/H2! 26 Data Stream Algorithms - Introduction 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 27 Data Stream Algorithms - Introduction 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 28 Data Stream Algorithms - Introduction 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? 29 Data Stream Algorithms - Introduction 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 30 Data Stream Algorithms - Introduction Sampling Two Tokens Stream: C A A B B A B D C A B A 0.408 0.815 0.217 0.191 0.770 0.082 0.366 0.228 0.549 0.173 0.627 0.202 Tags: Repeats: B B A B A B A min min tag amongst tag remaining tokens second smallest Assign tags, choose first token as before tag, but we don’t want this; same Delete all occurrences of first token token as min tag! Choose token with min remaining tag; count repeats Implementation: keep track of two triples (min tag, corresponding token, number of repeats) 31 Data Stream Algorithms - Introduction 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 space O(-2 log m log 1/) space we can compute an (,) approximation to H(S) in a single pass 32 Data Stream Algorithms - Introduction Entropy Exercises As a subroutine, we need to find an element that occurs more than 2/3 of the time and estimate its weight 1. How can we find a frequently occurring item? 2. How can we estimate its weight p with (1-p) error? 3. Our algorithm uses O(-2 log m log 1/) space, could this be improved or is it optimal (lower bounds)? 4. Our algorithm updates each sampled pair for every update, how quickly can we implement it? 5. (Research problem) What if there are multiple distributed streams and we want to compute the entropy of their union? 33 Data Stream Algorithms - Introduction Outline Introduction to Data Streams – Motivating examples and applications – Data Streaming models – Basic tail bounds Sampling from data streams Sampling to estimate entropy 34 Data Stream Algorithms - Introduction Data Stream Algorithms Frequency Moments Graham Cormode graham@research.att.com Frequency Moments Introduction to Frequency Moments and Sketches Count-Min sketch for F and frequent items AMS Sketch for F2 Estimating F0 Extensions: – Higher frequency moments – Combined frequency moments 36 Data Stream Algorithms - Introduction 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 37 Data Stream Algorithms - Introduction 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 38 Data Stream Algorithms - Introduction 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 : 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 39 Data Stream Algorithms - Introduction 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 40 Data Stream Algorithms - Introduction Trivial Example of a Sketch 1 0 1 1 1 0 1 0 1 … 1 0 1 1 0 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 (Karp-Rabin: h(x) = xi2i mod p for random prime p) – Exercise: extend to real valued vectors in update model 41 Data Stream Algorithms - Introduction Frequency Moments Introduction to Frequency Moments and Sketches Count-Min sketch for F and frequent items AMS Sketch for F2 Estimating F0 Extensions: – Higher frequency moments – Combined frequency moments 42 Data Stream Algorithms - Introduction 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 43 Data Stream Algorithms - Introduction CM Sketch Structure +c h1(j) d=log 1/ j,+c +c +c hd(j) +c w = 2/ 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 F1 in size O(1/ log 1/) – Probability of more error is less than 1- 44 [C, Muthukrishnan ’04] Data Stream Algorithms - Introduction Approximation Approximate 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] = F1/2 by pairwise independence of h – Pr[Xk,j F1] = Pr[Xk,j 2E(Xk,j)] 1/2 by Markov inequality So, Pr[x’[j] x[j] + F1] = Pr[ k. Xk,j> F1] 1/2log 1/ = Final result: with certainty x[j] x’[j] and with probability at least 1-, x’[j]< x[j] + F1 45 Data Stream Algorithms - Introduction Applications of CM to F CM sketch lets us estimate fi for any i 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 46 Data Stream Algorithms - Introduction Count-Min Exercises 1. 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. 2. 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 3. 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) 4. How to find the large absolute frequencies when some are negative? 47 Data Stream Algorithms - Introduction Frequency Moments Introduction to Frequency Moments and Sketches Count-Min sketch for F and frequent items AMS Sketch for F2 Estimating F0 Extensions: – Higher frequency moments – Combined frequency moments 48 Data Stream Algorithms - Introduction 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/ {1...U} {+1,-1} Now, given update (j,+c), set CM[k,hk(i)] += c*gk(j) linear projection AMS sketch 49 Data Stream Algorithms - Introduction F2 analysis +c*g1(j) h1(j) d=8log 1/ j,+c +c*g2(j) +c*g3(j) hd(j) +c*g4(j) w = 4/2 Estimate F2 = mediank i CM[k,i]2 Each row’s result is i g(i)2xi2 + h(i)=h(j) 2 g(i) g(j) fi fj But g(i)2 = -12 = +12 = 1, and i xi2 = F2 g(i)g(j) has 1/2 chance of +1 or –1 : expectation is 0 … 50 Data Stream Algorithms - Introduction F2 Variance Expectation of row estimate is exactly F2 Variance of row k is an expectation: – Vark = E[ buckets b (CM[k,b])4 – F22 ] – 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 Row variance can finally be bounded by F22/w – Chebyshev for w=4/2 gives probability ¼ of failure – How to amplify this to small probability of failure? 51 Data Stream Algorithms - Introduction Tail Inequalities for Sums We derive stronger bounds on tail probabilities for the sum of independent Bernoulli trials via the Chernoff 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, με2 Pr(| X μ | με) 2exp 2 52 Data Stream Algorithms - Introduction 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 53 Data Stream Algorithms - Introduction 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. 54 Data Stream Algorithms - Introduction AMS Sketch Exercises 1. 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) 2. Extend for strings drawn from an arbitrary alphabet 3. 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? 55 Data Stream Algorithms - Introduction Frequency Moments Introduction to Frequency Moments and Sketches Count-Min sketch for F and frequent items AMS Sketch for F2 Estimating F0 Extensions: – Higher frequency moments – Combined frequency moments 56 Data Stream Algorithms - Introduction 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 57 Data Stream Algorithms - Introduction 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 0m3 vt m3 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 58 Data Stream Algorithms - Introduction Analysis of F0 algorithm Suppose F’0 = tm3/vt > (1+) F0 [estimate is too high] 0m3 vt tm3/(1+)F0 m3 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) 59 Data Stream Algorithms - Introduction 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 4/(2t2) – Set t=6/2 to make this Prob 1/9 60 Data Stream Algorithms - Introduction Completing the analysis We have shown Pr[ F’0 > (1+) F0 ] < 1/9 Can show Pr[ F’0 < (1-) F0 ] < 1/9 similarly – too few items hash below a certain value So Pr[ (1-) F0 F’0 (1+)F0] > 7/9 [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 61 Data Stream Algorithms - Introduction 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 62 Data Stream Algorithms - Introduction 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 63 Data Stream Algorithms - Introduction 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 j max(i, x(i)) x(i) 64 Data Stream Algorithms - Introduction Frequency Moments Introduction to Frequency Moments and Sketches Count-Min sketch for F and frequent items AMS Sketch for F2 Estimating F0 Extensions: – Higher frequency moments – Combined frequency moments 65 Data Stream Algorithms - Introduction 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 66 Data Stream Algorithms - Introduction 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/k poly-log(m,n,1/)) space 67 Data Stream Algorithms - Introduction 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... 68 Data Stream Algorithms - Introduction 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) 69 Data Stream Algorithms - Introduction F (F0) using CM-FM Find i’s such that di > f i di Finds the people that talk to many others Count-Min sketch only uses additions, so can apply: 70 Data Stream Algorithms - Introduction 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. 71 Data Stream Algorithms - Introduction Exercises / Problems 1. (Research problem) Read, understand and simplify analysis for optimal Fk estimation algorithms 2. Take the sampling Fk algorithm and combine it with F0 estimators to approximate Fk of node degrees 3. Why can’t we use the sketch approach for F2 of node degrees? Show there the analysis breaks down 4. (Research problem) What can be computed for other combinations of frequency moments, e.g. F2 of F2 values, etc.? 72 Data Stream Algorithms - Introduction Frequency Moments Introduction to Frequency Moments and Sketches Count-Min sketch for F and frequent items AMS Sketch for F2 Estimating F0 Extensions: – Higher frequency moments – Combined frequency moments 73 Data Stream Algorithms - Introduction Data Stream Algorithms Lower Bounds Graham Cormode graham@research.att.com Streaming Lower Bounds Lower bounds for data streams – Communication complexity bounds – Simple reductions – Hardness of Gap-Hamming problem – Reductions to Gap-Hamming Alice 1 0 1 1 1 0 1 0 1 … Bob 75 Data Stream Algorithms - Introduction 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 76 Data Stream Algorithms - Introduction Streaming As Communication Alice 1 0 1 1 1 0 1 0 1 … Bob Imagine Alice processing a stream Then take the whole working memory, and send to Bob Bob continues processing the remainder of the stream 77 Data Stream Algorithms - Introduction 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 78 Data Stream Algorithms - Introduction Deterministic Equality Testing 1 0 1 1 1 0 1 0 1 … 1 0 1 1 0 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) 79 Data Stream Algorithms - Introduction Hard Communication Problems INDEX: x is a binary string of length n y 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 0 Result: (multi-round) (randomized) communication complexity of DISJOINTNESS is (n) bits 80 Data Stream Algorithms - Introduction Simple Reduction to Disjointness x: 1 0 1 1 0 1 1, 3, 4, 6 y: 0 0 0 1 1 0 4, 5 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 81 Data Stream Algorithms - Introduction Simple Reduction to Index x: 1 0 1 1 0 1 1, 3, 4, 6 y: 5 5 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’(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 82 Data Stream Algorithms - Introduction Hardness Reduction Exercises Use reductions to DISJ or INDEX to show the hardness of: Frequent items: find all items in the stream whose frequency > fN, for some f. 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 j 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}| 83 Data Stream Algorithms - Introduction Streaming Lower Bounds Lower bounds for data streams – Communication complexity bounds – Simple reductions – Hardness of Gap-Hamming problem – Reductions to Gap-Hamming Alice 1 0 1 1 1 0 1 0 1 … Bob 84 Data Stream Algorithms - Introduction 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] 85 Data Stream Algorithms - Introduction Hardness of Gap Hamming Reduction from an instance of INDEX 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 rj is 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 86 Data Stream Algorithms - Introduction 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 87 Data Stream Algorithms - Introduction 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 for GAP-HAMMING 88 Data Stream Algorithms - Introduction Streaming Lower Bounds Lower bounds for data streams – Communication complexity bounds – Simple reductions – Hardness of Gap-Hamming problem – Reductions to Gap-Hamming Alice 1 0 1 1 1 0 1 0 1 … Bob 89 Data Stream Algorithms - Introduction Lower Bound for Entropy Alice: 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) 0 1 0 0 1 1 Alice (1,0) (2,1) (3,0) (4,0) (5,1) (6,1) (1,1) (2,1) (3,0) (4,0) (5,1) (6,0) Bob 1 1 0 0 1 0 90 Data Stream Algorithms - Introduction 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) 0 1 0 0 1 1 Alice (1,0) (2,1) (3,0) (4,0) (5,1) (6,1) (1,1) (2,1) (3,0) (4,0) (5,1) (6,0) Bob 1 1 0 0 1 0 91 Data Stream Algorithms - Introduction 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 F0 with < 1/N, could separate – But space bound = (N) = (-2) bits Dependence on for F0 is tight Similar arguments show (-2) bounds for Fk, – Proof assumes k (and hence 2k) are constants 92 Data Stream Algorithms - Introduction Lower Bounds Exercises 1. Formally argue the space lower bound for F2 via Gap- Hamming 2. Argue space lower bounds for Fk via Gap-Hamming 3. (Research problem) Extend lower bounds for the case when the order of the stream is random or near-random 4. (Research problem) Kumar conjectures the multi-round communication complexity of Gap-Hamming is (n) – this would give lower bounds for multi-pass streaming 93 Data Stream Algorithms - Introduction Streaming Lower Bounds Lower bounds for data streams – Communication complexity bounds – Simple reductions – Hardness of Gap-Hamming problem – Reductions to Gap-Hamming Alice 1 0 1 1 1 0 1 0 1 … Bob 94 Data Stream Algorithms - Introduction Data Stream Algorithms Extensions and Open Problems Graham Cormode graham@research.att.com 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 96 Data Stream Algorithms - Introduction 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 97 Data Stream Algorithms - Introduction 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: 98 Data Stream Algorithms - Introduction 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 99 Data Stream Algorithms - Introduction 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] 100 Data Stream Algorithms - Introduction 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]: age – Can reduce sum and counts to multiple instances of sliding window queries C., Korn and Tirthapura [2008]: – Same observations applies to other computations (quantiles, frequent items) 101 Data Stream Algorithms - Introduction 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 … 102 Data Stream Algorithms - Introduction 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? 103 Data Stream Algorithms - Introduction Skewed Streams log frequency frequency items sorted by frequency log rank 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(-2/(1+z) log 1/), provided z>1 104 Data Stream Algorithms - Introduction Graph Streaming 3 1 5 (4,5) (2,3) (1,3) (3,5) (1,2) (2,4) (1,5) (3,4) 4 2 Stream specifies a massive graph edge by edge – Most natural problems have (|V|) space lower bounds – Semi-streaming model: allow (|V|) but o(|E|) space Therefore 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 105 Data Stream Algorithms - Introduction Matrix Streaming Stream specifies a massive n n matrix – Either by giving entries in some order, or updates to entries O(1) Columns O(1) Rows In one (or few) passes, find: – CUR Decomposition – Page Rank Vector ( )=( A C )( )( U R Carefully ) – Approximate Matrix product chosen U – 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 106 Data Stream Algorithms - Introduction Permutation Streaming 1 3 4 2 3 4 1 2 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 107 Data Stream Algorithms - Introduction 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] GAP-HAMMING still has linear lower bound t2-party DISJOINTNESS has (n/t) lower bound 108 Data Stream Algorithms - Introduction Probabilistic Streams Example: S = (x, ½, y, 1/3, y, ¼) Encodes 6 “possible worlds”: G f x y x,y y,y x,y,y Pr[G] ¼ ¼ 5/24 5/24 1/24 1/24 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] 109 Data Stream Algorithms - Introduction Distributed Streams http://www.intel.com/research/exploratory/motes.htm 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 base station (root, coordinator…) 110 Data Stream Algorithms - Introduction Continuous Distributed Model Coordinator Track Q(S1,…,Sm) local stream(s) seen at each m sites site S1 Sm 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] 111 Data Stream Algorithms - Introduction 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] 112 Data Stream Algorithms - Introduction 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] 113 Data Stream Algorithms - Introduction Data Stream Algorithms Slides are on the web on my website Long list of references also on the web http://dimacs.rutgers.edu/~graham 114 Data Stream Algorithms - Introduction