1. By Graham Cormode and Marios Hadjieleftheriou Presented by Ankur Agrawal (09305002)

2.  Many data generation processes produce huge numbers of pieces of data, each of which is simple in isolation, but which taken together lead to a complex whole.  Examples ◦ Sequence of queries posed to an Internet search engine. ◦ Collection of transactions across all branches of a supermarket chain.  Data can arrive at enormous rates - hundreds of gigabytes per day or higher. Finding the Frequent Items in Streams of Data2

3.  Storing and indexing such large amount of data is costly.  Important to process the data as it happens.  Provide up to the minute analysis and statistics.  Algorithms take only a single pass over their input.  Compute various functions using resources that are sublinear in the size of input. Finding the Frequent Items in Streams of Data3

4. Problems range from simple to very complex.  Given a stream of transactions, finding the mean and standard deviation of the bill totals. ◦ Requires only few sufficient statistics to be stored.  Determining whether a search query has already appeared in the stream of queries. ◦ Requires a large amount of information to be stored.  Algorithms must ◦ Respond quickly to new information. ◦ Use very less amount of resources in comparison to total quantity of data. Finding the Frequent Items in Streams of Data4

5.  Given a stream of items, the problem is simply to find those items which occur most frequently.  Formalized as finding all items whose frequency exceeds a specified fraction of the total number of items.  Variations arise when the items are given weights, and further when these weights can also be negative. Finding the Frequent Items in Streams of Data5

6.  The problem is important both in itself and as a subroutine in more advanced computations.  For example, ◦ It can help in routing decisions, for in-network caching etc (if items represent packets on the Internet). ◦ Can help in finding popular terms if items represent queries made to an Internet search engine. ◦ Mining frequent itemsets inherently builds on this problem as a basic building block.  Algorithms for the problem have been applied by large corporations: AT&T and Google. Finding the Frequent Items in Streams of Data6

7.  Given a stream S of n items t 1 …t n, the exact ɸ-frequent items comprise the set {i | f i >ɸn}, where f i is the frequency of item i.  Solving the exact frequent items problem requires Ω(n) space.  Approximate version is defined based on tolerance for error parameterized by ɛ. Finding the Frequent Items in Streams of Data7

8.  Given a stream S of n items, the ɛ- approximate frequent items problem is to return a set of items F so that for all items i ε F, f i >(ɸ-ɛ)n, and there is no i∉F such that f i >ɸn. Finding the Frequent Items in Streams of Data8

9.  Given a stream S of n items, the frequency estimation problem is to process a stream so that, given any i, an f i * is returned satisfying f i ≤ f i * ≤ f i +ɛn. Finding the Frequent Items in Streams of Data9

10.  Two main classes of algorithms : ◦ Counter-based Algorithms ◦ Sketch Algorithms  Other Solutions : ◦ Quantiles : based on various notions of randomly sampling items from the input, and of summarizing the distribution of items. ◦ Less effective and have attracted less interest. Finding the Frequent Items in Streams of Data10

11.  Track a subset of items from the input and monitor their counts.  Decide for each new arrival whether to store or not.  Decide what count to associate with it.  Cannot handle negative weights. Finding the Frequent Items in Streams of Data11

12.  Problem posed by J. S. Moore in Journal of Algorithms, in 1981. Finding the Frequent Items in Streams of Data12

13.  Start with a counter set to zero. For each item: ◦ If counter is zero, store the item, set counter to 1. ◦ Else, if item is same as item stored, increment counter. ◦ Else, decrement counter.  If there is a majority item, it is the item stored.  Proof outline: ◦ Each decrement pairs up two different items and cancels them out. ◦ Since majority occurs > N/2 times, not all of its occurrences can be canceled out. Finding the Frequent Items in Streams of Data13

14.  First proposed by Misra and Gries in 1982.  Finds all items in a sequence whose frequency exceeds a 1/k fraction of the total count.  Stores k-1 (item, counter) pairs.  A generalization of the Majority algorithm. Finding the Frequent Items in Streams of Data14

15.  For each new item: ◦ Increment the counter if the item is already stored. ◦ If <k items stored, then store new item with counter set to 1. ◦ Otherwise decrement all the counters. ◦ If any counter equals 0, then delete the corresponding item. Finding the Frequent Items in Streams of Data15

16. 6 1 +1 3 120 2 Finding the Frequent Items in Streams of Data16

17. +1 3 12 1 2 Finding the Frequent Items in Streams of Data17 5 0 6 1

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19.  Time cost involves: ◦ O(1) dictionary operations per update. ◦ Cost of decrementing counts.  Can be performed in O(1) time.  Also solves the frequency estimation problem if executed with k=1/ɛ. Finding the Frequent Items in Streams of Data19

20.  Proposed by Manku and Motwani in 2002.  Stores tuples comprising ◦ An item ◦ A lower bound on its count ◦ A delta (△) value which records the difference between the upper bound and the lower bound. Finding the Frequent Items in Streams of Data20

21.  For processing i th item: ◦ Increment the lower bound if it is already stored. ◦ Else create a new tuple for the item with lower bound set to 1 and △ set to ⌊i/k⌋.  Periodically delete tuples with upper bound less than ⌊i/k⌋.  Uses O( k log(n/k) ) space in worst case.  Also solves frequency estimation problem with k=1/ɛ. Finding the Frequent Items in Streams of Data21

22. Finding the Frequent Items in Streams of Data22 * The CACM 2009 version of the paper has erroneously shown if block (line 8) outside for loop.

23.  Introduced by Metwally et al. in 2005.  Store k (item, count) pairs.  Initialize by first k distinct items and their exact counts.  If new item is not already stored, replace the item with least count and set the counter to 1 more than the least count.  Items which are stored by the algorithm early in the stream and are not removed have very accurate estimated counts. Finding the Frequent Items in Streams of Data23

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25.  The space required is O(k).  The time cost involves cost of: ◦ the dictionary operation of finding if an item is stored. ◦ finding and maintaining the item with minimum count.  Simple heap implementations can track the smallest count item in O(log k) time per update.  As in other counter based algorithms, it also solves frequency estimation problem with k=1/ɛ. Finding the Frequent Items in Streams of Data25

26.  The term “sketch” denotes ◦ a data structure. ◦ a linear projection of the input frequency vector.  Sketch is the product of frequency vector with a matrix.  Updates with negative values can easily be accommodated. ◦ Allows us to model the removal of items.  Primarily solve the frequency estimation problem.  Algorithms are randomized. ◦ Also take a failure probability δ so that the probability of failure is at most δ. Finding the Frequent Items in Streams of Data26

27.  Introduced by Charikar et al. in 2002.  Each update only affects a small subset of the sketch.  The sketch consists of a two-dimensional array C with d rows of w counters Each.  Uses two hash functions for each row: ◦ h j which maps input items onto [w]. ◦ g j which maps input items onto {−1, +1}.  Each input item i ◦ Add g j (i) to entry C[ j, h j (i)] in row j, for 1 ≤ j ≤ d. Finding the Frequent Items in Streams of Data27

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29. Finding the Frequent Items in Streams of Data29 For any row j, the value g j (i)*C[j,h j (i)] is an unbiased estimator for f i. The estimate f i * is the median of these estimates over the d rows.

30.  Setting d=log(4/δ) and w=O(1/ɛ 2 ) ensures that f i has error at most ɛn with probability of at least 1−δ. ◦ Requires the hash functions to be chosen randomly from a family of “fourwise independent” hash functions.  The total space used is.  The time per update is in worst case. Finding the Frequent Items in Streams of Data30

31.  Introduced by Cormode and Muthukrishnan in 2005.  Similarly uses a sketch consisting of 2-D array of d rows of w counters each.  Uses d hash functions h j, one for each row.  Each update is mapped onto d entries in the array, each of which is incremented.  Now frequency estimate for any item is f i * = min 1≤j≤d C[ j, h j (i)] Finding the Frequent Items in Streams of Data31

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33.  The estimate for an item in each row overestimates by less than n/w. ◦ Can be shown using Markov inequality.  Setting d=log(1/δ) and w=O(1/ɛ) ensures that f i has error at most ɛn with probability of at least 1−δ.  Similar to CountSketch with g j (i) always equal to 1.  The total space used is.  The time per update is in worst case. Finding the Frequent Items in Streams of Data33

34.  Sketches estimate the frequency of a single item. ◦ How to find frequent items without explicitly storing sketch for all items?  Divide-and-conquer approach limits search cost. ◦ Impose a binary tree over the domain ◦ Keep a sketch of each level of the tree ◦ Descend if a node is heavy, else stop  Correctness: all ancestors of a frequent item are also frequent.  Assumes that negative updates are possible but no negative frequencies are allowed.  Updates take \ hashing operations, and O(1) counter updates for each hash. Finding the Frequent Items in Streams of Data34

35.  Randomly divides the input into buckets.  Expect at most one frequent item in each bucket.  Within each bucket, the items are divided into subgroups so that the “weight” of each group indicates the identity of the frequent item.  Repeating this for all bit positions reveals the full identity of the item. Finding the Frequent Items in Streams of Data35

36.  Experiments performed using 10 million packets of HTTP traffic, representing 24 hours of traffic from a backbone router in a major network.  The frequency threshold ɸ varied from 0.0001 to 0.01 and the error guarantee ɛ set to ɸ.  Compare the efficiency of the algorithms with respect to: ◦ Update throughput, measured in number of updates per millisecond. ◦ Space consumed, measured in bytes. ◦ Recall, measured in the total number of true heavy hitters reported over the number of true heavy hitters given by an exact algorithm. ◦ Precision, measured in total number of true heavy hitters reported over the total number of answers reported. Finding the Frequent Items in Streams of Data36

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39.  Overall, the SpaceSaving algorithm appears conclusively better than other counter-based algorithms.  SSH yields very good estimates, typically achieving 100% recall and precision, consumes very small space, and is fairly fast to update.  SSL is the fastest algorithm with all the good characteristics of SSH but consumes twice as much space on average. Finding the Frequent Items in Streams of Data39

40.  No clear winner among the sketch algorithms.  CMH has small size and high update throughput, but is only accurate for highly skewed distributions.  CGT consumes a lot of space but it is the fastest sketch and is very accurate in all cases, with high precision and good frequency estimation accuracy.  CS has low space consumption and is very accurate in most cases, but has slow update rate and exhibits some random behavior. Finding the Frequent Items in Streams of Data40

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