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Approximate Frequency Counts over Data Streams. Gurmeet Singh Manku (Standford) Rajeev Motwani (Standford). Presented by Michal Spivak November, 2003. Stream. The Problem …. Identify all elements whose current frequency exceeds support threshold s = 0.1%. Stream. Related problem ….

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### Approximate Frequency Counts over Data Streams

Gurmeet Singh Manku (Standford) Rajeev Motwani (Standford)

Presented by Michal Spivak

November, 2003

The Problem…

Identify all elements whose current frequency exceeds support threshold s = 0.1%.

Related problem…

Identify all subsets of items whose current frequency exceeds s=0.1%

Present an algorithm for computing frequency counts exceeding a user-specified threshold over data streams with the following advantages:

• Simple

• Low memory footprint

• Output is approximate but guaranteed not to exceed a user specified error parameter.

• Can be deployed for streams of singleton items and handle streams of variable sized sets of items.

• Introduction

• Frequency counting applications

• Problem definition

• Algorithm for Frequent Items

• Algorithm for Frequent Sets of Items

• Experimental results

• Summary

### Introduction

• Iceberg Query Perform an aggregate function over an attribute and eliminate those below some threshold.

• Association RulesRequire computation of frequent itemsets.

• Iceberg DatacubesGroup by’s of a CUBE operator whose aggregate frequency exceeds threshold

• Traffic measurement Require identification of flows that exceed a certain fraction of total traffic

What’s out there today…

• Algorithms that compute exact results

• Attempt to minimize number of data passes (best algorithms take two passes).

• Only one pass is allowed.

• Results are expected to be available with short response time.

• Fail to provide any a-priori guarantee on the quality of their output.

Why Streams?Streams vs. Stored data

• Volume of a stream over its lifetime can be huge

• Queries for streams require timely answers, response times need to be small

As a result it is not possible to store the stream as an entirety.

### Frequency counting applications

• Iceberg Query Perform an aggregate function over an attribute and eliminate those below some threshold.

• Association RulesRequire computation of frequent itemsets.

• Iceberg DatacubesGroup by’s of a CUBE operator whose aggregate frequency exceeds threshold

• Traffic measurement Require identification of flows that exceed a certain fraction of total traffic

Iceberg QueriesIdentify aggregates that exceed a user-specified threshold r

One of the published algorithms to compute iceberg queries efficiently uses repeated hashing over multiple passes.*

Basic Idea:

• In the first pass a set of counters is maintained

• Each incoming item is hashed to one of the counters which is incremented

• These counters are then compressed to a bitmap, with a 1 denoting large counter value

• In the second pass exact frequencies are maintained for only those elements that hash to a counter whose bitmap value is 1

This algorithm is difficult to adapt for streams because it requires two passes

* M. FANG, N. SHIVAKUMAR, H. GARCIA-MOLINA,R. MOTWANI, AND J. ULLMAN. Computing iceberg

queries efficiently. In Proc. of 24th Intl. Conf. on Very Large Data Bases, pages 299–310, 1998.

Definitions

• Transaction – subset of items drawn from I, the universe of all Items.

• Itemset X I has support s if X occurs as a subset at least a fraction - s of all transactions

• Associations rules over a set of transactions are of the form X=>Y, where X and Y are subsets of I such that X∩Y = 0 and XUY has support exceeding a user specified threshold s.

• Confidence of a rule X => Y is the value support(XUY) / support(X)

U|

• For support = 50%, confidence = 50%, we have the following rules

• 1 => 3 with 50% support and 66% confidence

• 3 => 1 with 50% support and 100% confidence

For support = 50%, confidence = 50%

• For the rule 1 => 3:

• Support = Support({1, 3}) = 50%

• Confidence = Support({1,3})/Support({1}) = 66%

Toivonen’s algorithm

• Based on sampling of the data stream.

• Basically, in the first pass, frequencies are computed for samples of the stream, and in the second pass these the validity of these items is determined.Can be adapted for data stream

• Problems:- false negatives occur because the error in frequency counts is two sided- for small values of e, the number of samples is enormous ~ 1/e (100 million samples)

• Flow – sequence of transport layer packets that share the same source+destination addresses

• Estan and Verghese proposed an algorithm for this identifying flows that exceed a certain threshold.The algorithm is a combination of repeated hashing and sampling, similar to those for iceberg queries.

• Algorithm presented in this paper is directly applicable to the problem of network flow identification. It beats the algorithm in terms of space and requirements.

### Problem definition

• Algorithm accepts two user-specified parameters- support threshold s E (0,1)- error parameter εE (0,1)- ε << s

• N – length of stream (i.e no. of tuples seen so far)

• Itemset – set of items

• Denote item(set) to be item or itemset

• At any point of time, the algorithm can be asked to produce a list of item(set)s along with their estimated frequency.

• All item(set)s whose true frequency exceeds sN are output. There are no false negatives.

• No item(set)s whose true frequency is less than (s- ε(N is output.

• Estimated frequencies are less than true frequencies by at mostεN

• S = 0.1%

• ε as a rule of thumb, should be set to one-tenth or one-twentieth of s. ε = 0.01%

• As per property 1, ALL elements with frequency exceeding 0.1% will be output.

• As per property 2, NO element with frequency below 0.09% will be output

• Elements between 0.09% and 0.1% may or may not be output. Those that “make their way” are false positives

• As per property 3, all individual frequencies are less than their true frequencies by at most 0.01%

• An algorithm maintains an ε-deficient synopsis if its output satisifies the aforementioned properties

• Goal:to devise algorithms to support ε-deficient synopsis using as little main memory as possible

### The Algorithms for frequent Items

Sticky Sampling

Lossy Counting

Sticky Sampling Algorithm

34

15

30

28

31

41

23

35

19

 Create counters by sampling

• Data structure S - set of entries of the form (e,f)

• f – estimates the frequency of an element e.

• r – sampling rate. Sampling an element with rate = r means we select the element with probablity = 1/r

• Initially – S is empty, r = 1.

• For each incoming element eif (e exists in S) increment corresponding felse { sample element with rate r if (sampled) add entry (e,1) to S else ignore }

• Let t = 1/ ε log(s-1-1) ( = probability of failure)

• First 2t elements are sampled at rate=1

• The next 2t elements at rate=2

• The next 4t elements at rate=4

and so on…

Whenever the sampling rate r changes: for each entry (e,f) in S repeat { toss an unbiased coin if (toss is not successful) diminsh f by one if (f == 0) { delete entry from S break }} until toss is successful

• The number of unsuccessful coin tosses folows a geometric distribution.

• Effectively, after each rate change S is transformed to exactly the state it would have been in, if the new rate had been used from the beginning.

• When a user requests a list of items with threshold s, the output are those entries in S where f ≥ (s –ε)N

Sticky Sampling computes an ε-deficient synopsis with probability at least 1 -  using at most 2/ ε log(s-1-1) expected number of entries.

• First 2t elements find their way into S

• When r ≥ 2 N = rt + rt` ( t`E [1,t) ) => 1/r ≥ t/N

• Error in frequency corresponds to a sequence of unsuccessful coin tosses during the first few occurrences of e.the probability that this length exceeds εN is at most (1 – 1/r)εN < (1 – t/N)-εN < e-εt

• Number of elements with f > s is no more than 1/s => the probability that the estimate for any of them is deficient by εN is at most e-εt/s

Theorem 1 – proof cont…

• Probability of failure should be at most . This yieldse-εt/s < 

• t ≥ 1/ ε log(s-1-1)

since the space requirements are 2t, the space bound follows…

• The algorithm name is called sticky sampling because S sweeps over the stream like a magnet, attracting all elements which already have an entry in S

• The space complexity is independent of N

• The idea of maintaining samples was first presented by Gibbons and Matias who used it to solve the top-k problem.

• This algorithm is different in that the sampling rate r increases logarithmically to produce ALL items with frequency > s, not just the top k

bucket 3

bucket 2

Lossy Counting

Divide the stream into buckets

Keep exact counters for items in the buckets

Prune entrys at bucket boundaries

• A deterministic algorithm that computes frequency counts over a stream of singleitem transactions, satisfying the guarantees outlined in Section 3 using at most 1/εlog(εN) space where N denotes the current length of the stream.

• The user specifies two parameters:- support s- error ε

• The incoming stream is conceptually divided into buckets of width w = ceil(1/e)

• Buckets are labeled with bucket ids, starting from 1

• Denote the current bucket id by bcurrent whose value is ceil(N/w)

• Denote feto be the true frequency of an element e in the stream seen so far

• Data stucture D is a set of entries of the form (e,f,D)

• Initially D is empty

• Receive element eif (e exists in D) increment its frequency (f) by 1else create a new entry (e, 1, bcurrent– 1)

• If it bucket boundary prune D by the following the rule:(e,f,D) is deleted if f + D≤ bcurrent

• When the user requests a list of items with threshold s, output those entries in D where f ≥ (s –ε)N

• For an entry (e,f,D) f represents the exact frequency count for e ever since it was inserted into D.

• The value D is the maximum number of times e could have occurred in the first bcurrent– 1 buckets ( this value is exactly bcurrent– 1)

• Once a value is inserted into D its value D is unchanged

Counts

+

First Bucket

Lossy counting in action

D is Empty

At window boundary, remove entries that for themf+D ≤bcurrent

Counts

+

Next Bucket

Lossy counting in action cont

At window boundary, remove entries that for themf+D ≤bcurrent

• Whenver deletions occur, bcurrent≤eN

Proof: N = bcurrentw + neN = bcurrent + eneN ≥ bcurrent

• Whenever an entry (e,f,D) gets deleted fe≤ bcurrent

Proof by induction

• Base case: bcurrent = 1

(e,f,D) is deleted only if f = 1 Thus fe ≤ bcurrent (fe = f)

• Induction step: - Consider (e,f,D) that gets deleted for some bcurrent > 1. - This entry was inserted when bucket D+1 was being processed. - It was deleted at late as the time as bucket D became full. - By induction the true frequency for e was no more than D. - f is the true frequency of e since it was inserted.- fe ≤ f+D combined with the deletion rule f+D ≤ bcurrent => fe ≤ bcurrent

• If e does not appear in D, then fe≤eN

Proof: If the lemma is true for an element e whenever it gets deleted, it is true for all other N also.From lemmas 1, 2 we infer that fe ≤ eN whenever it gets deleted.

• If (e,f,D) E D, then f ≤ fe≤ f + eN

Proof:

If D=0, f=fe.

Otherwisee was possibly deleted in the first D buckets.From lemma 2 fe ≤ f+DD ≤ bcurrent– 1 ≤ eN

Conclusionf ≤ fe≤ f + eN

• Lemma 3 shows that all elements whose true frequency exceed eN have entries in D

• Lemma 4 shows that the estimated frequency of all such elements are accurate to within eN

=> D correctly maintains an e-deficient synopsis

• Lossy counting computes an e-deficient synopsis using at most 1/elog(eN) entries

• Let B = bcurrent

• di– denote the number of entries in D whose bucket id is B - i + 1 (iE[1,B])

• e corresponding to di must occur at least i times in buckets B-i+1 through B, otherwise it would have been deleted

• We get the following constraint:(1) Sidi ≤ jw for j = 1,2,…B. i = 1..j

Theorem 2 – proof

• The following inequality can be proved by induction:Sdi ≤ Sw/ifor j = 1,2,…B i = 1..j

• |D| = Sdi for i = 1..B

• From the above inequality |D| ≤ Sw/i ≤ 1/elogB = 1/elog(eN)

z

Sticky Sampling vs. Lossy counting

Log10 of N (stream length)

Support s = 1%

Error ε = 0.1%

N (stream length)

Sticky Sampling vs.Lossy counting cont…

Kinks in the curve for sticky sampling correspond to re-sampling

Kinks in the curve for lossy counting correspond to bucket boundaries

Sticky Sampling vs. Lossy counting cont…

SS – Sticky Sampling LC – Lossy Counting

Zipf – zipfian distribution Uniq – stream with no duplicates

Sticky Sampling vs. Lossy summary

• Lossy counting is superior by a large factor

• Sticky sampling performs worse because of its tendency to remember every unique element that gets sampled

• Lossy counting is good at pruning low frequency elements quickly

• Toivonen – sampling algorithm for association rules.

• Sticky sampling beats the approach by roughly a factor of 1/e

• KPS02 – In the first path the algorithm maintains 1/e elements with their frequencies. If a counter exists for an element it is increased, if there is a free counter it is inserted, otherwise all existing counters are reduced by one

• Can be used to maintain e-deficient synopsis with exactly 1/e space

• If the input stream is ZipfianLossy Counting takes less than 1/e spacefor e=0.01% roughly 2000 entries ~ 20% 1/e

### Frequent Sets of Items

From theory to Practice

Frequent Sets of Items

Identify all subsets of items whose

current frequency exceeds s = 0.1%.

• Input: stream of transactions, each transaction is a set of items from I

• N: length of the stream

• User specifies two parameters: support s, error e

• Challenge: - handling variable sized transactions- avoiding explicit enumeration of all subsets of any transaction

• Data structure D – set of entries of the form (set, f, D)

• Transactions are divided into buckets

• w = ceil(1/e) – no. of transactions in each bucket

• bcurrent– current bucket id

• Transactions are not processed one by one. Main memory is filled with as many transactions as possible. Processing is done on a batch of transactions.B– no. of buckets in main memory in the current batch being processed.

• UPDATE_SET: for each entry (f,set,D) E D, update f by counting occurrences of set in the current batch. If the updated entry satisfies f+D ≤ bcurrent, we delete this entry

• NEW_SET: if a set set has frequency f ≥ B in the current batch and set does not occur in D, create a new entry (set,f,bcurrent–B)

• If fset ≥ eN it has an entry in D

• If (set,f,D)ED then the true frequency of fset satisfies the inequality f≤ fset ≤ f+D

• When a user requests a list of items with threshold s, output those entries in D wheref ≥ (s-e)N

• B needs to be a large number. Any subset of I that occurs B+1 times or more contributes to D.

SUBSET-GEN

BUFFER

Three modules

maintains the data structureD

implement UPDATE_SET, NEW_SET

operates on the current batch of transactions

repeatedly reads in a batch of transactionsinto available main memory

bucket 1 bucket 2 bucket 3 bucket 4 bucket 5 bucket 6

Module 1 - Buffer

• Read a batch of transactions

• Transactions are laid out one after the other in a big array

• A bitmap is used to remember transaction boundaries

• After reading in a batch, BUFFER sorts each transaction by its item-id’s

50

40

31

29

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30

50 40 30 31 29

45 32 42

Sets with frequency counts

Module 2 - TRIE

Module 2 – TRIE cont…

• Nodes are labeled {item-id, f, D, level}

• Children of any node are ordered by their item-id’s

• Root nodes are also ordered by their item-id’s

• A node represents an itemset consisting of item-id’s in that node and all its ancestors

• TRIE is maintained as an array of entries of the form {item-id, f, D, level} (pre-order of the trees). Equivalent to a lexicographic ordering of subsets it encodes.

• No pointers, level’s compactly encode the underlying tree structure.

2 2 1

2 1

3

1

1

Frequency counts

of subsets

in lexicographic order

BUFFER

Module 3 - SetGen

SetGen uses the following pruning rule:if a subset S does not make its way into TRIE after application of both UPDATE_SET and NEW_SET, then no supersets of S should be considered

2 2 1

2 1

3

1

1 SUBSET-GEN

BUFFER

TRIE

new TRIE

Overall Algorithm

• If item-id’s are successive integers from 1 thru |I|, and I is small enough (less than 1 million) Maintain exact frequency counts for singleton sets. Prune away those item-id’s whose frequency is less than eN and then sort the transactions

• If |I| = 105, array size = 0.4 MB

• Take advantage of the fact that the sets produced by SetGen are lexicographic.

• Maintain TRIE as a set of fairly large-sized chunks of memory instead of one huge array

• Instead of modifying the original TRIE, create a new TRIE.

• Chunks from the old TRIE are freed as soon as they are not required.

• By the time SetGen finishes, the chunks of the original TRIE have been discarder.

• Employs a priority queue called Heap

• Initially contains pointers to smallest item-id’s of all transactions in buffer

• Duplicate members are maintained together and constitute a single item in the Heap. Chain all these pointers together.

• Derive the space from BUFFER. Change item-id’s with pointers.

Efficient ImplementationsSetGen cont…

2

1

Efficient ImplementationsSetGen cont…

• Repeatedly process the smallest item-id in Heap to generate singleton sets.

• If the singleton belongs to TRIE after UPDATE_SET and NEW_SET try to generate the next set by extending the current singleton set.

• This is done by invoking SetGen recursively with a new Heap created out of successors of the pointers to item-id’s just processed and removed.

• When the recursive call returns, the smallest entry in Heap is removed and all successors of the currently smallest item-id are added.

Efficient ImplementationsSetGen cont…

2

1

2

3

• Buffer scans the incoming stream by memory mapping the input file.

• Use standard qsort to sort transactions

• Threading SetGen and Buffer does not help because SetGen is significantly slower.

• The rate at which tries are scanned is much smaller than the rate at which sequiential disk I/O can be done

• Possible to maintain TRIE on disk without loss in performance

System issues and optimizationsTRIE on disk advantages

• The size of TRIE is not limited by main memory – this algorithm can function with a low amount of main memory.

• Since most available main memory can be devoted to BUFFER, this algorithm can handle smaller values of e than other algorithms can handle.

• No candidate generation phase.

• Compact disk-based tries is novel

• Able to compute frequent itemsets under low memory conditions.

• Able to handle smaller values of support threshold than previously possible.

### Experimental results

N = 1Million Avg Tran Size = 10 Input Size = 49MB

IBM synthetic dataset T15.I6.1000K

N = 1Million Avg Tran Size = 15 Input Size = 69MB

Frequent word pairs in 100K web documents

N = 100K Avg Tran Size = 134 Input Size = 54MB

Frequent word pairs in 806K Reuters newsreports

N = 806K Avg Tran Size = 61 Input Size = 210MB

Experimental Results

• Support threshold s

• Number of transactions N

• Size of BUFFER B

• Total time taken t

set e = 0.1s

B = 4 MB

B = 4 MB

B = 16 MB

Time taken in seconds

B = 16 MB

B = 28 MB

B = 28 MB

B = 40 MB

B = 40 MB

Support threshold s

Support threshold s

IBM test dataset T10.I4.1000K.

Reuters 806k docs.

Varying buffer sizes and support s

Decreasing s leads to increases in running time

S = 0.004

S = 0.008

S = 0.001

S = 0.012

S = 0.002

S = 0.016

S = 0.004

S = 0.020

S = 0.008

BUFFER size in MB

BUFFER size in MB

IBM test dataset T10.I4.1000K.

Reuters 806k docs.

Varying support s and buffer size B

Time taken in seconds

Kinks occur due to TRIE optimization

on last batch

S = 0.002

S = 0.001

S = 0.004

S = 0.002

S = 0.004

Time taken in seconds

Length of stream in Thousands

Length of stream in Thousands

IBM test dataset T10.I4.1000K.

Reuters 806k docs.

Varying length N and support s

Running time is linear proportional to the length of the stream

The curve flattens in the end as processing the last batch is faster

Dataset: IBM T10.I4.1000K with 1M transactions, average size 10

Query: Identify all word pairs in 100K web documents

which co-occur in at least 0.5% of the documents.

[FSGM+98] multiple pass algorithm:

7000 seconds with 30 MB memory

Our single-pass algorithm:

4500 seconds with 26 MB memory

### Summary

A Novel Algorithm for computing approximate

frequency counts over Data Streams

• Require provably small main memory footprints

• Each of the motivating examples can now be solved over streaming data

• Handle smaller values of support threshold than previously possible

• Remains practical in environments with moderate main memory

• Give an Apriori error guarantee

• Work for variable sized transactions.

• Optimized implementation for frequent itemsets

• For the datasets tested, the algorithm runs in one pass and produces exact results, beating previous algorithms in terms of time.

### Questions?

More questions/comments can be sent to [email protected]