1. Space-Efficient Online Computation of Quantile Summaries SIGMOD 01 Michael Greenwald & Sanjeev Khanna Presented by ellery

2. Outline Introduction The summary data structure Operation and algorithm Tree representation Analysis and experimental result Conclusion

3. Introduction Space-efficient computation of quantile summaries of very large data sets in a single pass. Quantile queries: Given a quantile, , return the value whose rank is  N 

4. t0t0 t1t1 t2t2 t3t3 t4t4 t5t5 t6t6 t7t7 t8t8 t9t9 t 10 t 11 t 12 t 13 t 14 t 15 121011101 119678 4523 sorting 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 10, 11, 11, 11, 12 N = 16 0.5 quantile returns element ranked 8 ( 0.5*16) which is 8 0.75 quantile returns element ranked 12 (0.75*16) which is 10

5. Requirements Explicit & tunable a priori guarantees on the precision of the approximation As small a memory footprint as possible Online: Single pass over the data Data Independent Performance: guarantees should be unaffected by arrival order, distribution of values, or cardinality of observations. Data Independent Setup: no a priori knowledge required about data set (size, range, distribution, order).

6. ε- approximate A quantile summary for a data sequence is ε- approximate if, for any given rank r, it returns a value whose rank r’ is guaranteed to be within the interval [r -εN, r + εN ] Example : A data stream with 100 elements, 0.5 – quantile with ε= 0.1 returns a value v. The true rank of v is within [40,60]

7. The Summary Data Structure Let r min (v) and r max (v) denote the lower and upper bounds on the rank of v Each tuple ti = (v i, g i,Δ i )

8. Example .01, N=1750 192 {15,2} 201 {28,7} 204 {10,1} [501,503] [529,536] [539,540]

9. Query Sketch S is ε- approximate, That is for each ψ (0,1], there is a (v i, r min ( v i ), r max ( v i ) ) in S such that v i is our answer for ψ-quantile

10. Corollary If at any time n, the summary S(n) satisfies the property that then we can answer any ψ-quantile query to within an εn precision.

11. Overview of Summary Data Structure Quantile  =.29? Compute r and choose best v i 192 [501,503] {15,2} 201 [529,536] {28,7} .01, N=1800 204 [539,540] { 10,1 }  =.29 r =  N = 522

12. Overview of Summary Data Structure If (r max (v i+1 ) - r min (v i )) ≦ 2  N, then  - approximate summary. Our goal: always maintain this property. Tuple formulation of this rule: g i +  I ≦ 2  N 192 [ 501,503 ] {15,2} 201 [529,536] {28,7} .01, N=1800 204 [539,540] {10,1} 2  N=36

13. Overview of Summary Data Structure Goal: always maintain  -approximate summary (r max (v i+1 ) - r min (v i )) = (g i +  I ) ≦ 2  N Insert new observations into summary 192 [501,503] {15,2} 201 [529,536] {28,7} .01, N=1800 204 [539,540] {10,1} 197 2  N=36

14. Overview of Summary Data Structure 192 [501,503] {15,2} 201 [529,536] {28,7} .01, N=1800 204 [539,540] {10,1} 197 [502,536] 2  N=36 Goal: always maintain  -approximate summary (r max (v i+1 ) - r min (v i )) = (g i +  I ) ≦ 2  N Insert new observations into summary

15. Overview of Summary Data Structure Goal: always maintain  -approximate summary (r max (v i+1 ) - r min (v i )) = (g i +  I ) ≦ 2  N Insert new observations into summary Insert tuple before the ith tuple. g new = 1;  new = g i +  I - 1; 192 [501,503] {15,2} 201 [530,537] {28,7} .01, N=1801 204 [540,541] {10,1} 197 [502,536] 2  N=36.02 {1,34}

16. Overview of Summary Data Structure Goal: always maintain  -approximate summary (r max (v i+1 ) - r min (v i )) = (g i +  I ) ≦ 2  N Insert new observations into summary Delete all “superfluous” entries. 192 [501,503] {15,2} 201 [530,537] {28,7} .01, N=1801 204 [540,541] {10,1} 197 [502,536] 2  N=36.02 {1,34}

17. Overview of Summary Data Structure Goal: always maintain  -approximate summary (r max (v i+1 ) - r min (v i )) = (g i +  I ) ≦ 2  N Insert new observations into summary Delete all “superfluous” entries. 192 [501,503] {15,2} 201 [530,537] { 28,7 } .01, N=1801 204 [540,541] {10,1} 2  N=36.02 {1,34}

18. Overview of Summary Data Structure Goal: always maintain  -approximate summary (r max (v i+1 ) - r min (v i )) = (g i +  I ) ≦ 2  N Insert new observations into summary Delete all “superfluous” entries. g i = g i + g i-1 192 [501,503] {15,2} 201 [530,537] {29,7} .01, N=1801 204 [540,541] {10,1} 2  N=36.02

19. Overview of Summary Data Structure Insert: g new = 1;  new = g i +  I - 1; Delete: g i = g i + g i-1 192 [501,503] {15,2} 201 [530,537] {29,7} .01, N=1801 204 [540,541] {10,1} 2  N=36.02

20. Terminology Full tuple: A tuple is full if g i +  I = 2  N Full tuple pair: A pair of tuples is full if deleting the left-hand tuple would overfill the right one Capacity: number of observations that can be counted by g i before the tuple becomes full. (= 2  N -  I ) General strategy will be to delete tuples with small capacity and preserve tuples with large capacity.

21. Operations Insert(v) ： Find the smallest i, such that, and insert Delete(v i ) ： to delete from S, replace and by the new tuple Compress() ： from right to left, merge all mergeable pair.

22. GK Algorithm To add the n+1st observation, v, to summary S(n) yes no COMPRESS()INSERT

23. Tree Representation .001, N=7,000 2  N=14  -range CapacityBand 0-78-153 8-114-72 12-132-31 1410 000000111111112223333 Group tuples with similar capacities into bands First (least index) node to the right with higher capacity band becomes parent.

24. Tree Representation .001, N=7,000 2  N=14  -range CapacityBand 0-78-153 8-114-72 12-132-31 1410 Group tuples with similar capacities into bands First (least index) node to the right with higher capacity band becomes parent. 00000011111111222 3333

25. Tree Representation .001, N=7,000 2  N=14  -range CapacityBand 0-78-153 8-114-72 12-132-31 1410 Group tuples with similar capacities into bands First (least index) node to the right with higher capacity band becomes parent. 00000011111111 222 333 3

26. Tree Representation .001, N=7,000 2  N=14  -range CapacityBand 0-78-153 8-114-72 12-132-31 1410 Group tuples with similar capacities into bands First (least index) node to the right with higher capacity band becomes parent. 000000 1 111 1 1 11 2 22 333 3 R

27. Operation (compress) General strategy: delete tuples with small capacity and preserve tuples with large capacity. 1) Deletion cannot leave descendants unmerged --- it must delete entire subtrees 2) Deletion can only merge a tuple with small capacity into a tuple with similar or larger capacity. 3) Deletion cannot create an over-full tuple (i.e with g+  > floor(2  N))

28. Analysis Theorem At any time n, the total number of tuples stored in S(n) is at most

29. Experimental Result Measurement: |S| Observed  (vs. desired  ) : max, avg, and for 16 representative quantiles Optimal max observed  Compared 3 algorithms MRL Preallocated (1/3 number of stored observations as MRL) Adaptive: allocate a new quantile only when observed error is about to exceed desired 

30. Conclusion Better worst-case behavior than previous algorithms It does not require a priori knowledge of the parameter N

31. Any Question ?