Advanced SearchTechniques for Large Scale Data Analytics Pavel Zezula and Jan Sedmidubsky Masaryk University http://disa.fi.muni.cz  In many data mining situations, we do not know the entire data set in advance  Stream Management is important when the input rate is controlled externally: ▪ Google queries ▪ Twitter or Facebook status updates  We can think of the data as infinite and non-stationary (the distribution changes over time) Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 2 3  Input elements enter at a rapid rate, at one or more input ports (i.e., streams) ▪ We call elements of the stream tuples  The system cannot store the entire stream accessibly  Q: How do you make critical calculations about the stream using a limited amount of (secondary) memory? Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 4 Processor Limited Working Storage . . . 1, 5, 2, 7, 0, 9, 3 . . . a, r, v, t, y, h, b . . . 0, 0, 1, 0, 1, 1, 0 time Streams Entering. Each is stream is composed of elements/tuples Ad-Hoc Queries Output Archival Storage Standing Queries  Types of queries one wants on answer on a data stream: ▪ Sampling data from a stream ▪ Construct a random sample ▪ Queries over sliding windows ▪ Number of items of type x in the last k elements of the stream ▪ Filtering a data stream ▪ Select elements with property x from the stream ▪ Counting distinct elements ▪ Number of distinct elements in the last k elements of the stream Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 5  Mining query streams ▪ Google wants to know what queries are more frequent today than yesterday  Mining click streams ▪ Yahoo wants to know which of its pages are getting an unusual number of hits in the past hour  Mining social network news feeds ▪ E.g., look for trending topics on Twitter, Facebook Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 6  Sensor Networks ▪ Many sensors feeding into a central controller  Telephone call records ▪ Data feeds into customer bills as well as settlements between telephone companies  IP packets monitored at a switch ▪ Gather information for optimal routing ▪ Detect denial-of-service attacks Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 7 As the stream grows the sample also gets bigger  Since we can not store the entire stream, one obvious approach is to store a sample  Two different problems: ▪ (1) Sample a fixed proportion of elements in the stream (say 1 in 10) ▪ (2) Maintain a random sample of fixed size over a potentially infinite stream ▪ At any “time” k we would like a random sample of s elements ▪ What is the property of the sample we want to maintain? For all time steps k, each of k elements seen so far has equal prob. of being sampled Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 9  Problem 1: Sampling fixed proportion  Scenario: Search engine query stream ▪ Stream of tuples: (user, query, time) ▪ Answer questions such as: How often did a user run the same query in a single days ▪ Have space to store 1/10th of query stream  Naïve solution: ▪ Generate a random integer in [0..9] for each query ▪ Store the query if the integer is 0, otherwise discard Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 10  Simple question: What fraction of queries by an average search engine user are duplicates? ▪ Suppose each user issues x queries once and d queries twice (total of x+2d queries) ▪ Correct answer: d/(x+d) ▪ Proposed solution: We keep 10% of the queries ▪ Sample will contain x/10 of the singleton queries and 2d/10 of the duplicate queries at least once ▪ But only d/100 pairs of duplicates ▪ d/100 = 1/10 ∙ 1/10 ∙ d ▪ Of d “duplicates” 18d/100 appear exactly once ▪ 18d/100 = ((1/10 ∙ 9/10)+(9/10 ∙ 1/10)) ∙ d ▪ So the sample-based answer is 𝑑 100 𝑥 10 + 𝑑 100 + 18𝑑 100 = 𝒅 𝟏𝟎𝒙+𝟏𝟗𝒅 Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 11 Solution:  Pick 1/10th of users and take all their searches in the sample  Use a hash function that hashes the user name or user id uniformly into 10 buckets Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 12  Stream of tuples with keys: ▪ Key is some subset of each tuple’s components ▪ e.g., tuple is (user, search, time); key is user ▪ Choice of key depends on application  To get a sample of a/b fraction of the stream: ▪ Hash each tuple’s key uniformly into b buckets ▪ Pick the tuple if its hash value is at most a Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 13 Hash table with b buckets, pick the tuple if its hash value is at most a. How to generate a 30% sample? Hash into b=10 buckets, take the tuple if it hashes to one of the first 3 buckets As the stream grows, the sample is of fixed size  Problem 2: Fixed-size sample  Suppose we need to maintain a random sample S of size exactly s tuples ▪ E.g., main memory size constraint  Why? Don’t know length of stream in advance  Suppose at time n we have seen n items ▪ Each item is in the sample S with equal prob. s/n Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 15 How to think about the problem: say s = 2 Stream: a x c y z k c d e g… At n= 5, each of the first 5 tuples is included in the sample S with equal prob. At n= 7, each of the first 7 tuples is included in the sample S with equal prob. Impractical solution would be to store all the n tuples seen so far and out of them pick s at random  Algorithm (a.k.a. Reservoir Sampling) ▪ Store all the first s elements of the stream to S ▪ Suppose we have seen n-1 elements, and now the nth element arrives (n > s) ▪ With probability s/n, keep the nth element, else discard it ▪ If we picked the nth element, then it replaces one of the s elements in the sample S, picked uniformly at random  Claim: This algorithm maintains a sample S with the desired property: ▪ After n elements, the sample contains each element seen so far with probability s/nPavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 16  We prove this by induction: ▪ Assume that after n elements, the sample contains each element seen so far with probability s/n ▪ We need to show that after seeing element n+1 the sample maintains the property ▪ Sample contains each element seen so far with probability s/(n+1)  Base case: ▪ After we see n=s elements the sample S has the desired property ▪ Each out of n=s elements is in the sample with probability s/s = 1 Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 17  Inductive hypothesis: After n elements, the sample S contains each element seen so far with prob. s/n  Now element n+1 arrives  Inductive step: For elements already in S, probability that the algorithm keeps it in S is:  So, at time n, tuples in S were there with prob. s/n  Time nn+1, tuple stayed in S with prob. n/(n+1)  So prob. tuple is in S at time n+1 = 𝒔 𝒏 ⋅ 𝒏 𝒏+𝟏 = 𝒔 𝒏+𝟏 Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 18 1 1 11 1                        n n s s n s n s Element n+1 discarded Element n+1 not discarded Element in the sample not picked  A useful model of stream processing is that queries are about a window of length N – the N most recent elements received  Interesting case: N is so large that the data cannot be stored in memory, or even on disk ▪ Or, there are so many streams that windows for all cannot be stored  Amazon example: ▪ For every product X we keep 0/1 stream of whether that product was sold in the n-th transaction ▪ We want answer queries, how many times have we sold X in the last k sales Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 20  Sliding window on a single stream: Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 21 q w e r t y u i o p a s d f g h j k l z x c v b n m q w e r t y u i o p a s d f g h j k l z x c v b n m q w e r t y u i o p a s d f g h j k l z x c v b n m q w e r t y u i o p a s d f g h j k l z x c v b n m Past Future N = 6 22  Problem: ▪ Given a stream of 0s and 1s ▪ Be prepared to answer queries of the form How many 1s are in the last k bits? where k ≤ N  Obvious solution: Store the most recent N bits ▪ When new bit comes in, discard the N+1st bit 0 1 0 0 1 1 0 1 1 1 0 1 0 1 0 1 1 0 1 1 0 1 1 0 Past Future Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) Suppose N=6  You can not get an exact answer without storing the entire window  Real Problem: What if we cannot afford to store N bits? ▪ E.g., we’re processing 1 billion streams and N = 1 billion  But we are happy with an approximate answer 23Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 0 1 0 0 1 1 0 1 1 1 0 1 0 1 0 1 1 0 1 1 0 1 1 0 Past Future  Q: How many 1s are in the last N bits?  A simple solution that does not really solve our problem: Uniformity assumption  Maintain 2 counters: ▪ S: number of 1s from the beginning of the stream ▪ Z: number of 0s from the beginning of the stream  How many 1s are in the last N bits? 𝑵 ∙ 𝑺 𝑺+𝒁  But, what if stream is non-uniform? ▪ What if distribution changes over time? Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 24 0 1 0 0 1 1 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 0 1 1 0 1 0 N Past Future  DGIM solution that does not assume uniformity  We store 𝑶(log 𝟐 𝑵) bits per stream  Solution gives approximate answer, never off by more than 50% ▪ Error factor can be reduced to any fraction > 0, with more complicated algorithm and proportionally more stored bits Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 25 [Datar, Gionis, Indyk, Motwani]  Solution that doesn’t (quite) work: ▪ Summarize exponentially increasing regions of the stream, looking backward ▪ Drop small regions if they begin at the same point as a larger region Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 26 0 1 0 0 1 1 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 0 1 1 0 1 0 N ? 01 12 23 4 106 We can reconstruct the count of the last N bits, except we are not sure how many of the last 6 1s are included in the N Window of width 16 has 6 1s  Stores only O(log2N ) bits ▪ 𝑶(log 𝑵) counts of log 𝟐 𝑵 bits each  Easy update as more bits enter  Error in count no greater than the number of 1s in the “unknown” area Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 27 28  As long as the 1s are fairly evenly distributed, the error due to the unknown region is small – no more than 50%  But it could be that all the 1s are in the unknown area at the end  In that case, the error is unbounded! Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 0 1 0 0 1 1 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 0 1 1 0 1 0 01 12 23 4 106 N ?  Idea: Instead of summarizing fixed-length blocks, summarize blocks with specific number of 1s: ▪ Let the block sizes (number of 1s) increase exponentially  When there are few 1s in the window, block sizes stay small, so errors are small 29Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 1001010110001011010101010101011010101010101110101010111010100010110010 N [Datar, Gionis, Indyk, Motwani] 30  Each bit in the stream has a timestamp, starting 1, 2, …  Record timestamps modulo N (the window size), so we can represent any relevant timestamp in 𝑶(𝒍𝒐𝒈 𝟐 𝑵) bits Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)  A bucket in the DGIM method is a record consisting of: ▪ (A) The timestamp of its end [O(log N) bits] ▪ (B) The number of 1s between its beginning and end [O(log log N) bits]  Constraint on buckets: Number of 1s must be a power of 2 ▪ That explains the O(log log N) in (B) above 31Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 1001010110001011010101010101011010101010101110101010111010100010110010 N  Either one or two buckets with the same power-of-2 number of 1s  Buckets do not overlap in timestamps  Buckets are sorted by size ▪ Earlier buckets are not smaller than later buckets  Buckets disappear when their end-time is > N time units in the past Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 32 Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 33 N 1 of size 2 2 of size 4 2 of size 8 At least 1 of size 16. Partially beyond window. 2 of size 1 1001010110001011010101010101011010101010101110101010111010100010110010 Three properties of buckets that are maintained: - Either one or two buckets with the same power-of-2 number of 1s - Buckets do not overlap in timestamps - Buckets are sorted by size  When a new bit comes in, drop the last (oldest) bucket if its end-time is prior to N time units before the current time  2 cases: Current bit is 0 or 1  If the current bit is 0: no other changes are needed 34Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)  If the current bit is 1: ▪ (1) Create a new bucket of size 1, for just this bit ▪ End timestamp = current time ▪ (2) If there are now three buckets of size 1, combine the oldest two into a bucket of size 2 ▪ (3) If there are now three buckets of size 2, combine the oldest two into a bucket of size 4 ▪ (4) And so on … 35Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 36 1001010110001011010101010101011010101010101110101010111010100010110010 0010101100010110101010101010110101010101011101010101110101000101100101 0010101100010110101010101010110101010101011101010101110101000101100101 0101100010110101010101010110101010101011101010101110101000101100101101 0101100010110101010101010110101010101011101010101110101000101100101101 0101100010110101010101010110101010101011101010101110101000101100101101 Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) Current state of the stream: Bit of value 1 arrives Two orange buckets get merged into a yellow bucket Next bit 1 arrives, new orange bucket is created, then 0 comes, then 1: Buckets get merged… State of the buckets after merging 37  To estimate the number of 1s in the most recent N bits: 1. Sum the sizes of all buckets but the last (note “size” means the number of 1s in the bucket) 2. Add half the size of the last bucket  Remember: We do not know how many 1s of the last bucket are still within the wanted window Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 38 N 1 of size 2 2 of size 4 2 of size 8 At least 1 of size 16. Partially beyond window. 2 of size 1 1001010110001011010101010101011010101010101110101010111010100010110010  Each element of data stream is a tuple  Given a list of keys S  Determine which tuples of stream are in S  Obvious solution: Hash table ▪ But suppose we do not have enough memory to store all of S in a hash table ▪ E.g., we might be processing millions of filters on the same stream 40Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)  Example: Email spam filtering ▪ We know 1 billion “good” email addresses ▪ If an email comes from one of these, it is NOT spam  Publish-subscribe systems ▪ You are collecting lots of messages (news articles) ▪ People express interest in certain sets of keywords ▪ Determine whether each message matches user’s interest 41Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)  Given a set of keys S that we want to filter  Create a bit array B of n bits, initially all 0s  Choose a hash function h with range [0,n)  Hash each member of s S to one of n buckets, and set that bit to 1, i.e., B[h(s)]=1  Hash each element a of the stream and output only those that hash to bit that was set to 1 ▪ Output a if B[h(a)] == 1 42Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)  Creates false positives but no false negatives ▪ If the item is in S we surely output it, if not we may still output it 43 Item 0010001011000 Output the item since it may be in S. Item hashes to a bucket that at least one of the items in S hashed to. Hash func h Drop the item. It hashes to a bucket set to 0 so it is surely not in S. Bit array B Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)  |S| = 1 billion email addresses |B|= 1GB = 8 billion bits  If the email address is in S, then it surely hashes to a bucket that has the big set to 1, so it always gets through (no false negatives)  Approximately 1/8 of the bits are set to 1, so about 1/8th of the addresses not in S get through to the output (false positives) ▪ Actually, less than 1/8th, because more than one address might hash to the same bit 44Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)  Consider: |S| = m, |B| = n  Use k independent hash functions h1 ,…, hk  Initialization: ▪ Set B to all 0s ▪ Hash each element s S using each hash function hi, set B[hi(s)] = 1 (for each i = 1,.., k)  Run-time: ▪ When a stream element with key x arrives ▪ If B[hi(x)] = 1 for all i = 1,..., k then declare that x is in S ▪ That is, x hashes to a bucket set to 1 for every hash function hi(x) ▪ Otherwise discard the element x Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 45 (note: we have a single array B!)  m = 1 billion, n = 8 billion ▪ k = 1: (1 – e-1/8) = 0.1175 ▪ k = 2: (1 – e-1/4)2 = 0.0493  What happens as we keep increasing k?  “Optimal” value of k: n/m ln(2) ▪ In our case: Optimal k = 8 ln(2) = 5.54 ≈ 6 ▪ Error at k = 6: (1 – e-1/6)2 = 0.0235 Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 46 0 2 4 6 8 10 12 14 16 18 20 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Number of hash functions, k Falsepositiveprob.  Bloom filters guarantee no false negatives, and use limited memory ▪ Great for pre-processing before more expensive checks  Suitable for hardware implementation ▪ Hash function computations can be parallelized  Is it better to have 1 big B or k small Bs? ▪ It is the same: (1 – e-km/n)k vs. (1 – e-m/(n/k))k ▪ But keeping 1 big B is simpler Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 47  Problem: ▪ Data stream consists of a universe of elements chosen from a set of size N ▪ Maintain a count of the number of distinct elements seen so far  Obvious approach: Maintain the set of elements seen so far ▪ That is, keep a hash table of all the distinct elements seen so far 49Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)  How many different words are found among the Web pages being crawled at a site? ▪ Unusually low or high numbers could indicate artificial pages (spam?)  How many different Web pages does each customer request in a week?  How many distinct products have we sold in the last week? 50Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)  Real problem: What if we do not have space to maintain the set of elements seen so far?  Estimate the count in an unbiased way  Accept that the count may have a little error, but limit the probability that the error is large 51Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)  Pick a hash function h that maps each of the N elements to at least log2 N bits  For each stream element a, let r(a) be the number of trailing 0s in h(a) ▪ r(a) = position of first 1 counting from the right ▪ E.g., say h(a) = 12, then 12 is 1100 in binary, so r(a) = 2  Record R = the maximum r(a) seen ▪ R = maxa r(a), over all the items a seen so far  Estimated number of distinct elements = 2R 52Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)  Very very rough and heuristic intuition why Flajolet-Martin works: ▪ h(a) hashes a with equal prob. to any of N values ▪ Then h(a) is a sequence of log2 N bits, where 2-r fraction of all as have a tail of r zeros ▪ About 50% of as hash to ***0 ▪ About 25% of as hash to **00 ▪ So, if we saw the longest tail of r=2 (i.e., item hash ending *100) then we have probably seen about 4 distinct items so far ▪ So, it takes to hash about 2r items before we see one with zero-suffix of length r Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 53