Sketching Sampled Data Streams

Transcription

1 Sketchng Sampled Data Streams Florn Rusu and Aln Dobra CISE Department Unversty of Florda March 31, 2009

2 Motvaton & Goal Motvaton Multcore processors How to use all the processng power? Parallel algorthms Sde tasks (analytcal, exploratory) Goal Analyze data at wre speed sngle pass, small memory Skew of a relaton read from the dsk Correlaton between flows passng through a hgh-speed router 2

3 Class of Queres Aggregates over Jons Equ-jon J between relatons F and G wth the jon constrant F.a = G.a Queres specfed by a jon and an aggregate: COUNT, SUM sze of jon (dot product), self-jon sze (second frequency moment) Example Stream F: a , frequency vector f: Stream G: a , frequency vector g: f g COUNT(F a G) = fg T = f g = [ ] 3 0 =

4 Man Idea: Basc Sketchng Technque [AGMS99] Summarze frequency table by projectng t onto a random vector sketch Use sketches to recover query result Random vectors: ξ = [ ξ 1...ξ n ] random vector of ±1 values, called ξ famly Sketches: Sketch of F, X F = fξ T Sketch of G, X G = gξ T X = X F X G estmates COUNT(F a G) snce E[X] = E[fξ T ξ g T ] = fe[ξ T ξ ]g T = fig T = fg T f E[ξ T ξ ] = I. dstnct elements of ξ must be par-wse ndependent, ξ 2 = 1, E[ξ ξ ] = 0 4

5 Basc Sketchng Technque Example: ξ = [ ξ 1 ξ 2 ξ 3 ] = [ ] Error of estmate X due to ts varance X F = fξ T = [ ] 1 +1 = 4 1 X G = gξ T = [ ] 1 +1 = 5 1 X = X F X G = ( 4)( 5) = Var[X] = f 2 j I g 2 j + ( ) 2 f g 2 f 2 g 2 Var(X) 2ff T gg T = 2 SJ(F) SJ(G) ξ s a famly of 4-wse ndependent random varables 5

6 Sketch Mantenance over Streams Choose seed s that generates ξ Stream F: a , frequency vector f: f X F = fξ T = f ξ = (ξ + + ξ ) = ξ t.a = ξ 1 + ξ 1 + ξ 2 + ξ 3 + ξ 1 + ξ 3 t F = h(s,1) + h(s,1) + h(s,2) + h(s,3) + h(s,1) + h(s,3) Stream G: a , frequency vector g: X G = gξ T = g ξ = (ξ + + ξ ) ξ t.a = ξ 3 + ξ 1 + ξ 3 + ξ 1 + ξ 1 = t G = h(s,3) + h(s,1) + h(s,3) + h(s,1) + h(s,1) g Counters X F and X G need only log space n the sze of the stream 6

7 !!,-,- () () % %$ Sketch Error Reducton Estmaton of COUNT(F a G) from sngle sketches of F and G s too nosy Soluton: Average 8Var(X) ε 2 E 2 [X] ndependent copes of X to reduce error to ε Compute medan of 2log1/δ such averages to ncrease confdence to 1 δ Stream F Sketches of F Seeds Stream G ξ ξ Sketches of G Independent copes of X Average Medan COUNT(F G)(1 ± ε) wth prob (1 δ) Memory requred ndependent of the sze of the stream 7

8 Speed-Up Methods Hashng Fast-AGMS sketches are faster and have better accuracy Pseudo-random number generatng schemes EH3 s as good as any 4-wse scheme + faster and denser 8

9 Fast-AGMS Sketches [CG05] Randomzaton Vector h of 2-unversal hash functons, h : I B Vector ξ of 4-wse ndependent ±1 random varables, ξ : I { 1,+1} Update Tme no. of rows m x x h (k) = x h (k) + w ξ (k) h ξ (k,w) 9

10 Sze of Jon Estmator Fast-AGMS Sketches [CG05] E [Z] = f ḡ = F G, Z ( f ḡ ± ε f 2 ḡ 2 ) wth probablty at least 1 δ f 2 = f f = f 2, ḡ 2 = ḡ ḡ = g 2 Sketch sze: B = n = O( 1 ε 2 ) and m = O(log 1 δ ) x(f) X j = x j (F) x j (G) h ξ x(g) Sum Z Y j = n =1 X j Medan 10

11 Update Tme Setup: sketch sze (row=1), Xeon 2.8 GHz, 512 KB cache, 4 GB man memory Tme / sketch update (ns) F-AGMS FC CM e+06 1e+07 1e+08 Bucket sze (log scale) 100 ns / sketch 10 sketches 1 µs 1 mllon ntegers / second 4 MB / second Desred rate 100 MB / second = 25X 11

12 Samplng Stream F: a , frequency vector f: Sample F : a 1 3 1, sampled frequency vector f : f f Stream G: a , frequency vector g: g Sample G : a 3 1, sampled frequency vector g : X = C f g T = C [2 0 1 ] 1 0 = C g

13 Samplng Sample at the tuple level Analyze n the frequency doman Random frequency vector (moment generatng functon) Bernoull Bnomal WR Multnomal WOR Multvarate hypergeometrc Var[X] = C 2 X = C f g E [X] = C E [ f ] E [ g ] E [ ( f f j] [ E g g ] j j I E [ f ] [ ] ) 2 E g 13

14 Sketches over Sampled Streams Stream F: a , frequency vector f: Sample F : a 1 3 1, sampled frequency vector f : f f Stream G: a , frequency vector g: g Sample G : a 3 1, sampled frequency vector g : ξ = [ ] [ ] ξ 1 ξ 2 ξ 3 = X F = f ξ T = [ ] 1 +1 = 3 1 X = C X FX G = C ( 3)( 2) = C g X G = g ξ T = [ ] 1 +1 =

15 Sketchng a Sample Buld the sketch over the non-materalzed sample [ [ Var[X] = C 2 E X = C f E [X] = C [ 2 E ] [ ] f 2 E g 2 j j I ] [ f 2 E ξ g jξ j j I E [ f ] [ ] E g g ( ] E [ f f [ j] E g g ] j j I E [ f ] [ ] ) 2 E g 15

16 Averagng Multple Sketches Sketches share the same sample correlaton Var [ 1 n n k=1 Var X k ] = C n ( [ 1 n E n k=1 X k ] = 1 n [ Var[Xk ] + (n 1) Cov k l [X k,x l ] ] E [ ( f f j] [ E g g ] j j I [ ] [ ] f 2 E g 2 j j I + E [ f j I E [ f ] [ ] ) 2 E g f j] [ [ E g g j] 2 E ] [ f 2 E g 2 ] )] Var sketch over samples = Var sketch + Var samplng + Var nteracton 16

17 p, q are samplng probabltes n F, G Bernoull Samplng Var [ 1 n n k=1 X k ] = 1 n f p p + 1 n j I g 2 j + [ 1 p p ( ) 2 f g 2 f 2 g 2 f g q q f g 2 j + 1 q j I, j q f 2 (1 p)(1 q) g + pq f g f 2 (1 p)(1 q) g j + j I, j pq ] f g j j I, j 17

18 Varance for Bernoull Samplng Term sgnfcance as a functon of the frequency dstrbuton 1 Interacton Samplng Sketch 1 Interacton Samplng Sketch Varance terms dstrbuton Varance terms dstrbuton Zpf coeffcent / p Zpf coeffcent / p Sze of Jon Self-Jon Sze 18

19 Error for Bernoull Samplng Settngs 100 mllon tuples F-AGMS sketches wth 5, 000 buckets 10 1 p= p= p= p=1.0 Relatve error (log scale) 1e-04 1e-05 1e ZIPF coeffcent 19

20 Error for WOR Samplng Settngs TPC-H scale 1, lnetem l orderkey = o orderkey orders 10 Relatve error (log scale) 1 1 Samplng rate (log scale) 20

21 Conclusons Sketches over sampled data Generc moment analyss Samplng n frequency doman Combned estmator Three types of samplng Bernoull Wth replacement Wthout replacement Expermental evaluaton 2 orders of magntude speed-up wthout sgnfcant error degradaton = Fast-AGMS sketches wth EH3 random varables over a sample 21

22 Questons 22