1. Mining Top-K Multidimensional Gradients Department of Informatics School of Engineering University of Minho PORTUGAL Ronnie Alves, Orlando Belo and Joel Ribeiro 9th International Conference on Data Warehousing and Knowledge Discovery (DaWaK 2007) 3-7 September 2007, Regensburg, Germany

4. Gradients (A=a1, B=b1, C=c1) (A=a1, B=b1, C=c1, D=d1) (A=a1, B=b1) (A=a1, B=b1, C=c2) roll-up(C) drill-down(D=d1) mutate(C=c2) cubegrade operations Even when considering only iceberg cells , It may still generate a very large number of pairs . > Mining gradients with constraints: a) significance , b) probe and c) gradient > LiveSet-Driven strategy Constrained Gradients Mining Top-K Multidimensional Gradients Dong et al TKDM’02, vol.16 *Introduction

7. Gradient Regions *Top-K Gradients Mining Top-K Multidimensional Gradients countXY( ) sumXY( ) avgXY() convex non-convex gradient region (GR) > Avg() is an algebraic function and It also has an unpredictable spreading factor regarding its distribution value > There are also sets of GRs to looking for Different shapes of aggregating functions

9. Definitions *Top-K Gradients Mining Top-K Multidimensional Gradients Base Table closed cell maximal cell maximal probe cell matchable cells A cell cg is said to be gradient cell of a probe cell cp , when they are matchable cells and their delta change, given by Δg(cg, cp)  (g(cg, cp) ≥  ) is true, where  is a constant value and g is a gradient function .

12. Cubing *Top-K Gradients Mining Top-K Multidimensional Gradients X,Y,Z: Selecting dimensions Value list Inverted index Spreading factors C i ={x1,y3,*}={4} Cuboid cell {1,4} {4} U Count (Ci)=1 Intersect tids aggregating function > Assembling high-dimensional cubes from low-dimensional ones > Follows Frag-Cubing ideas Li et al VLDB’04

13. *Top-K Gradients Set Enumeration Tree Mining Top-K Multidimensional Gradients Gradient Region Top-K sets Min_sf>0.25, valid GR > Lattice is formed by projecting GR[x1] >> GR[y2] >> GR[z2] > Find local gradients Agg_value Probe cells 1

14. *Top-K Gradients Mining Top-K Multidimensional Gradients 2 Projecting probe cells GR[x1] >> GR[y3] Top-K sets Matchable links Bin x1 = [1,4] Min_avg>2.7, valid Top-KGR

15. *Top-K Gradients Mining Top-K Multidimensional Gradients 3 Projecting probe cells GR[x1] >> GR[z1] Top-3 = {i, L, j} {x1,y2,*} -> {x1,y3,*} {x1,*,z3} -> {x1,*,z1} {x1,*,*} -> {x1,y3,*} Top-K sets Matchable links That’s it!!

17. Min_ sf pruning effects *Evaluation Study Mining Top-K Multidimensional Gradients Datasets Running time(s) Min_sf() D2 D1

18. Min_avg pruning effects *Evaluation Study Mining Top-K Multidimensional Gradients Datasets D2 D1 Running time(s) Min_avg()

19. K effects *Evaluation Study Mining Top-K Multidimensional Gradients Running time(s) K-cells D2

20. General pruning effects *Evaluation Study Mining Top-K Multidimensional Gradients D1 & D2 Valid cells Min_sf() K=5, avg>200 1.3M cells 420K cells 200s 170s 1Gb Ram 1M GRs

22. Mining Top-K Multidimensional Gradients QUESTIONS??? Department of Informatics School of Engineering University of Minho PORTUGAL Ronnie Alves, Orlando Belo and Joel Ribeiro 9th International Conference on Data Warehousing and Knowledge Discovery (DaWaK 2007) 3-7 September 2007, Regensburg, Germany Web : http://alfa.di.uminho.pt/~ronnie/