1. 1 Pushing Aggregate Constraints by Divide-and-Approximate Ke Wang, Yuelong Jiang, Jeffrey Xu Yu, Guozhu Dong and Jiawei Han

2. 2 No Easy to Push Constraints The exists a gap between the interesting criterion and the techniques used in mining patterns from a large amount of data Anti-monotonicity is too loose as a pruning strategy. Anti-monotonicity is too restricted as an interesting criterion. Should we design new algorithms to mine those patterns that can only be found using anti-monotonicity? Mining patterns with “general” constraints

3. 3 Iceberg-Cube Mining A iceberg-cube mining query select A, B, C, count(*) from R cube by A, B, C having count(*) >= 2 Count(*) >= 2 is an anti- monotone constraint. ABC M 123 100 124 200 224 300 ABCCount(*) 1-- 2 -2- 2 --4 2 12- 2 -24 2

4. 4 Iceberg-Cube Mining Another query select A, B, C, sum(M) from R cube by A, B, C having sum(M) >= 150 sum(M) >= 150 is an anti-monotone constraint, when all values in M are positive. sum(M) >= 150 is not an anti- monotone constraint, when some values in M are negative. ABC M 123 100 134 20 234 130 ABC M 123 -100 134 200 234 -300 R2 R1

5. 5 The Main Idea Study Iceberg-Cube Mining Consider f(v) θ σ f is a function with SQL-like aggregates and arithmetic operators (+, -, *, /); v is a variable; σ is a constant, and θ is either ≤ or ≥. Can we push the constraints into iceberg-cube mining that are not anti-monotone or monotone? If so, what is pushing method that is not specific to a particular constraint? Divide-Approximate: find a “stronger approximate” for the constraint in a subspace.

6. 6 Some Definitions A relation with many dimensions Di and one or more measures Mi. A cell is, di…dk, from Di, …, Dk. Use c as a cell variable Use di…dk for a cell value (representative) SAT(d1…dk) (or SAT(c)) contains all tuples that contains all values in d1…dk (or c). C’ is a super-cell of c, or c is a sub-cell of c’, if c’ contains all the values in c. Let C be a constraint (f(v) θ σ). CUBE(C) denotes the set of cells that satisfy C. A constraint C is weaker than C’ if CUBE(C’) ⊆ CUBE(C) ABC M 123 -100 134 200 234 -300

7. 7 An Example Iceberg-Cube Mining select A, B, C, sum(M) from R cube by A, B, C having sum(M) >= 150 sum(c) >= 150 is neither anti-monotone nor monotone. Let the space be S = {ABC, AB, AC, BC, A, B, C} Let sum(c) = psum(c) – nsum(c) >= 150. psum(c) is the profit, and nsum(c) is the cost. Push an anti-monotone approximator Use psum(c) >= 150, and ignore nsum(c). If nsum(c) is large, there are have many false positive. Use a min nsum in S: psum(c) – nsum min (ABC) >= 150. nsum min (ABC) is the minimum nsum in S. Use a min nsum in a subspace of S (a stronger constraint) ABC M 123 -100 134 200 234 -300

8. 8 The Search Strategy (using a lexicographic tree) A node represents a group-by BUC (BottomUpCube): Partition the database in the depth-first order of the lexicographic tree. 0 A B CD E AB ACAD AE DE ACE ADE ABC BD BCBC BCD CE BEBE CD ACD ACDE CDE BDE BCDE ABCD ABCDE ABDE BCE ABCE ABE AB D ABCDE M 12345 100 12345 12345-150 12345-100 12346 50 12356 40 12456 400

9. 9 Another Example Iceberg-Cube Mining select A, B, C, D, E, sum(M) from R cube by A, B, C having sum(M) >= 200 At node ABCDE, sum(12345) = psum(12345) – nsum(12345) = 200 – 250 = -50. (fails). Backtracking to ABC, psum(123) – nsum min (12345) = 290 - 100 = 190. (fails) Then, at node ABCE, p[1235], must fail. Therefore, all tuples, t[1235], can be pruned. ABCDE M 12345 100 12345 12345-150 12345-100 12346 50 12356 40 12456 400

10. 10 Find a cell p at u0 fails C, and then extract an anti-monotone approximator Cp. Consider an ancestor uk of u0, where u0 is the left-most leaf in tree(uk). p[u] denote p projected onto u (a cell of u). tree(uk, p) = {p[u] | u is a node in tree(uk)}. p is the max cell in tree(uk, p) and p[uk] is the min cell. In tree(uk, p). If p[uk] fails Cp, all cells in tree(uk, p) fails. Note: tree(uk, p) ≠ tree(uk, p’) if p’ ≠ p. 0 A BC D E AB AC ADAE DE ACE ADE ABC BDBC BCD CECE BE CD ACD ACDE CDEBDE BCDE ABCD ABCDEABDE BCE ABCE ABE ABD uk u0 Tree(uk) A node in tree(uk) is group-by attributes A cell in tree(uk, p) is group-by values u0’ uk’

11. 11 On the backtracking from u0 to uk Check if u0 is on the left-most path in tree(uk) Check if p[uk] can use the same anti-monotone approximator as p[u0] Check if p[uk] fails Cp. If all conditions are met, then For every unexplored child ui of uk, we prune all the tuples that match p on tail(ui), because such tuples generate only cells in tree(uk, p), which fail Cp. tail(u): the set of all dimensions appearing in tree(u). 0 A BCDE AB AC ADAE DE ACE ADE ABC BDBC BCD CE BE CD ACD ACDE CDEBDE BCDE ABCD ABCDEABDE BCE ABCE ABE ABD uk u0 Tree(uk) The Pruning

12. 12 Suppose that a cell p[ABCDE] fails. On the backtracking from ABCDE to ABC, If conditions are met (p[ABC] fails) Prune tuples such that t[ABCE] = p[ABCE] On the backtracking from ABC to AB, If conditions are met (p[AB] fails) Prune tuples such that t[ABDE] = p[ABDE] from tree (ABD) Prune tuples such that t[ABE] = p[ABE] from tree(ABE) 0 A BCDE AB AC ADAE DE ACE ADE ABC BDBC BCD CE BE CD ACD ACDE CDEBDE BCDE ABCD ABCDEABDE BCE ABCE ABE ABD uk u0 ui ui’ uk’ Given a leaf node u0 and a cell p at u0. Let the leftmost path uk…u0 in tree(uk), k >= 0. p is a pruning anchor wrt (uk,u0). Tree(uk, p) the pruning scope.

13. 13 The D&A Algorithm Modify BUC. Push up a pruning anchor p along the leftmost path from u0 to uk. Partition the prunning anchors pushed up to the current node, in addition to partitioning the tuples ABCDE M 12345 100 12345 12345-150 12345-100 12346 50 12356 40 12456 400

14. 14 With Min-Support Suppose cell abcd is frequent, but cell abcde is infrequent. (Shoud stop at abcd) If cell abcd is anchored at node A, cannot prune ae, abe, ace, ade in tree(A, abcd). 0 A B CD E AB AC AD AEDE ACE ADE ABC BD BCBC BCD CE BEBE CD ACD ACDE CDE BDE BCDE ABCD ABCDE ABDE BCE ABCE ABE AB D ABCDE M 12345 100 12345 10 12348 -50 12348-100 12356 -50 12356 40 12357 10 Min-sup = 3 sum(M) >= 100

15. 15 Rollback tree RBtree(AD), RBtree(AC), RBtree(ABD), RBtree(D), RBtree(C), and RBtree(B) do not have E. If abcd is anchored at the root, we can prune tuples from RBtree(D), RBtree(C), and RBtree(B). 0 AE D C B AB ACAD AE CB AECAED ABC ED EBEB EBC DC ECEC DB ADC AECD DBC ED C BBCD ABCD ABCDE ABED EBD ABCE ABE AB D ABCDE M 12345 100 12345 10 12348 -50 12348-100 12356 -50 12356 40 12357 10 Min-sup = 3 sum(M) >= 100

16. 16 Constraint/Function Monotonicity A constraint C is a-monotone if whenever a cell is not in CUBE(C), neither is any super-cell. A constraint C is m-monotone if whenever a cell is in CUBE(C), so its every super-cell. A function x(y) is a-monotone wrt y if x decreases as y grows (for cell-valued y) or increases (for real-valued y). A function x(y) is m-monotone wrt y if x increases as y grows (for cell-valued y) or increases (for real-values y). An example: sum(v) = psum(v) – nsum(v) sum(v) is m-monotone wrt psum(v) sum(v) is a-monotone wrt nsum(v)

17. 17 Constraint/Function Monotonicity Let a denote m, and m denote a. Let τ denote either a or m. Example: psum(v) ≥ σ is a-monotone, then psum(v) ≤ σ is m-monotone If psum(c1) ≥ σ is not held, then psum(c2) ≥ σ is not true, where c2 is a super cell of c1. (say c1 is a cell of ABC, and c2 is a cell of ABCD) f(v) ≥ σ is τ-monotone if and only if f(v) is τ-monotone wrt v. f(v) ≤ σ is τ-monotone if and only if f(v) is τ-monotone wrt v. An example: sum(v) = psum(v) – nsum(v) ≥ σ. sum(v) ≥ σ is m-monotone with psum(v), because sum(v) is m- monotone wrt psum(v). sum(v) ≥ σ is a-monotone with nsum(v), because sum(v) is a- monotone wrt nsum(v).

18. 18 Find Approximators Consider f(v) ≥ σ. Divide f(v) ≥ σ into two groups. A+: As cell v grows (becomes a super cell), f monotonically increases. A-: As cell grows (becomes a super cell), f monotonically decreases. Consider sum(v) = psum(v) – nsum(v) ≥ σ. A+ = {nsum(v)} A- = {psum(v)} f(A+; A-/c min ) ≥ σ and f(A+/c min ; A-) ≤ σ are m-monotone approximators in a subspace Si, where c min is a min cell instantiation in Si. f(A+/c max ; A-) ≥ σ and f(A+; A-/c max ) ≤ σ are a-monotone approximators in a subspace Si, where c max is a max cell instantiation in Si. sum(nsum/c max ; psum) ≥ σ

19. 19 Separate Monotonicity Consider function rewriting: (E1 + E2) * E into E1 * E + E2 * E. Consider space division divide a space into subspaces, Si. Find approximators using equation rewriting techniques for a subspace, Si.

20. 20 Experimental Studies Consider sum(v) = psum(v) – nsum(v) Three algorithms BUC: push only the minimum support. BUC+: push approximators and mininum support. D&A: push approximators and minimum support.

21. 21 Vary minimum support

22. 22 Without minimum support *) psum(v) >= sigma

23. 23 Scalability

24. 24 Conclusion General aggregate constraints, rather than only well- behaved constraints. SQL-like tuple-based aggregates, rather than item- based aggregates. Constraint independent techniques, rather than constraint specific techniques A new push strategy: divide-and-approximate