Efficient Computation of Frequent and Top- k Elements in Data Streams

1. Efficient Computation of Frequent and Top-k Elements in Data Streams

2. Motivation • Motivated by Internet advertising commissioners • Before rendering an advertisement for user, query clicks stream for advertisements to display. • If the user's profile is not a frequent “clicker”, then s/he will probably not click any displayed advertisement. • Show Pay-Per-Impression advertisements. • If the user's profile is a frequent “clicker”, then s/he may click a displayed advertisement. • Show Pay-Per-Click advertisements. • Retrieve top advertisements to choose what to display.

3. Problem Definition • Given alphabet A, stream S of size N, a frequent element, E, is an element whose frequency, F, exceeds a user specified support, φN • Top-k elements are the k elements with highest frequency • Both problems: • Very related, though, no integrated solution has been proposed • Exact solution is O(min(N,A)) space • approximate variations

4. φN (φ - ) N Practical Frequent Elements • -Deficient Frequent Elements [Manku ‘02]: • All frequent elements output should have F > (φ - )N, where  is the user-defined error.

5. F4 (1 - ) F4 Practical Top-k • FindApproxTop(S, k, ) [Charikar ‘02]: • Retrieve a list of k elements such that every element, Ei, in the list has Fi > (1 - ) Fk, where Ek is the kthranked element.

6. The Space-Saving Algorithm • Space-Saving is counter-based • Monitor only m elements • Only over-estimation errors • Frequency estimation is more accurate for significant elements • Keep track of max. possible errors

7. Space-Saving By Example A B B A C A B B D D B E C Space-Saving Algorithm • For every element in the stream S • If a monitored element is observed • Increment its Count • If a non-monitored element is observed, • Replace the element with minimum hits, min • Increment the minimum Count to min + 1 • maximum possible over-estimation is error Space-Saving Algorithm • For every element in the stream S • If a monitored element is observed • Increment its Count • If a non-monitored element is observed, • Replace the element with minimum hits, min • Increment the minimum Count to min + 1 • maximum possible over-estimation is error Space-Saving Algorithm • For every element in the stream S • If a monitored element is observed • Increment its Count • If a non-monitored element is observed, • Replace the element with minimum hits, min • Increment the minimum Count to min + 1 • maximum possible over-estimation is error Space-Saving Algorithm • For every element in the stream S • If a monitored element is observed • Increment its Count • If a non-monitored element is observed, • Replace the element with minimum hits, min • Increment the minimum Count to min + 1 • maximum possible over-estimation is error Space-Saving Algorithm • For every element in the stream S • If a monitored element is observed • Increment its Count • If a non-monitored element is observed, • Replace the element with minimum hits, min • Increment the minimum Count to min + 1 • maximum possible over-estimation is error

8. Space-Saving Observations S = ABBACABBDDBEC N = 13 • Observations: • The summation of the Counts is N • Minimum number of hits, min ≤ N/m • In this example, min = 4 • The minimum number of hits, min, is an upper bound on the error of any element

9. Space-Saving Proved Properties S = ABBACABBDDBEC N = 13 S = ABBACABBDDBEC N = 13 • If Element E has frequency F > min, then E must be in Stream-Summary. F(B) = F1 = 5, min = 4. The Count at position i in Stream-Summary is no less than Fi, the frequency of the ith ranked element. F(A) = F2 = 3, Count2 = 4.

10. Space-Saving Data Structure • We need a data structure that • Increments counters in constant time • Keeps elements sorted by their counters • We propose the Stream-Summary structure, similar to the data structure in [Demaine ’02]

11. Frequent Elements Queries • Traverse Stream-Summary, and report all elements that satisfy the user support • Any element whose guaranteed hits = (Count – error) > φN is guaranteed to be a frequent element

12. Frequent Elements Example • For N = 73,m = 8,φ = 0.15: • Frequent Elements should have support of 11 hits. • Candidate Frequent Elements are B, D, and G. • Guaranteed Frequent Elements are B, and D, since their guaranteed hits > 11.

13. Top-k Elements Queries • Traverse the Stream-Summary, and report top-k elements. • From Property 2, we assert: • Guaranteed top-k elements: • Any element whose guaranteed hits = (Count – error) ≥ Countk+1, is guaranteed to be in the top-k. • Guaranteed top-k’ (where k’≈k): • The top-k’ elements reported are guaranteed to be the correct top-k’ iff for every element in the top-k’, guaranteed hits = (Count – error) ≥ Countk’+1.

14. Top-k Elements Example • For k = 3,m = 8: • B, D, and G are the top-3candidates. • B, and D are guaranteed to be in the top-3. • B , D, G and A are guaranteed to be the top-4. Here k’ = 4. • B , and D are guaranteed to be the top-2. Another k’ = 2.

15. Frequent Elements Precision

16. Frequent Elements Run Time

17. Frequent Elements Space Used

18. Max freq. element in stream • Can we promise to find it with less than m buckets?