1. A Constraint Satisfaction Problem (CSP) is a combinatorial decision problem defined by a set of variables, a set of domain values for these variables, and a set of constraints restricting the allowable combinations of values for variables, where the task is to find a solution (i.e., an assignment of a value to each variable satisfying all constraints), or to find all such solutions. Dynamic Bundling for Non-binary CSPs and Databases Anagh Lal and Berthe Y. Choueiry Constraint Systems Laboratory Computer Science & Engineering University of Nebraska-Lincoln alal, choueiry@cse.unl.edu S cd, e, f d V1V1 V2V2 Dynamic bundling [1] : Interchangeability sets are updated during search Static bundling [3] : Interchangeability sets are computed before search ce, fd d V1V1 V2V2 S Search without bundling cefd d V1V1 V2V2 S 1.Interchangeability: An algorithm for computing interchangeability in non-binary CSPs. 2.Dynamic bundling: Integration of the above with backtrack search for solving non-binary CSPs. 3.Modeling: A new representation of a database join-query as a CSP. 4. A new sorting-based bundling algorithm. 5. A new sort-merge join algorithm for producing bundled tuples. Contributions C4C4 {1, 2, 3, 4, 5, 6} {1, 2, 3} C2C2 C1C1 C3C3 Constraint Variable Non-binary CSP (nb-CSP) SELECT R1.A,R1.B,R1.C FROM R1,R2 WHERE R1.A=R2.A AND R1.B=R2.B AND R1.C=R2.C Result: 10 tuples in 3 nested tuples  Use new join algorithm when materializing join queries  Exploit bundled results in data-analysis/data-mining packages  Assist in query size estimation  Improve accuracy of sampling operators Interchangeability is about finding values that are equivalent in all solutions of a CSP [2]. Full Interchangeability (FI): d, e, and f can be swapped for V 2 in any solution. Neighborhood Interchangeability (NI): finds e and f but misses d. It efficiently approximates FI. Dynamic bundling is guaranteed never less efficient than non-bundling and static bundling [1]. Interchangeability & Bundling Modeling the join query as a CSP Attributes  variables Attribute values  domains Relations  relational constraints Join conditions  join-condition constraints Dynamic Bundling for Databases Dynamic bundling = Search + dynamic interchangeability 1.Dynamic bundling was thought to be an overkill and not practical. We show it is worthwhile:  Finding all solutions: theoretically best  Finding first solutions: empirical evidence –Bundles solutions –Bundles no-goods (i.e., bundles of inconsistent partial solutions) 2.We exploit nb-DTs in Forward Checking (FC) 3.Search with dynamic bundling yields families of robust and compact solutions. Experiments : Compare the performance of search in FC and DynBndl using dynamic least domain as a variable ordering strategy. For low tightness values:  FBS is large: 1200 solutions in 1 bundle  However, the cost of DynBndl (CPU time) exceeds the gains due to savings in nodes visited. Around phase-transition (most critical region):  DynBundl shows considerable improvement in #NV and CPU time  FBS remains significant in most data-sets. For the data-set shown, DynBndl produces 26 robust solutions in less time than FC needs to produce 1 solution. Effect of domain size:  Gains of DynBndl increase with domain size.  Idea: Explore the use of DynBndl in databases where large domains are typical. Relational constraint Join-condition constraint R1.AR1.BR1.C R2.AR2.BR2.C R1 R2 Sorting-based bundling algorithm Sort-merge join algorithm based on dynamic bundling  Computes the join by finding all solutions of the CSP using DynBndl.  Reduces number of tuples compared in the main memory.  Is memory efficient and produces compacted results, saving –I/O for the next operator and –disk space (and network bandwidth in distributed databases). Experiments  Real-world problem: 3 relations; 4 attributes; largest table: 300 tuples –Compaction rate achieved: 2.26 (69 tuples in 32 nested tuples)  Random data-set: Relations with same schema as example; each relation: 10’000 tuples; memory size: 4’000 tuples; page size 200 tuples. –Compaction rate achieved: 1.48 Task: Join query V3V3 {d} {a, b, d}{a, b, c} {c, d, e, f} V4V4 V2V2 V1V1 Binary CSP Partition Unequal partitions Symmetric partitions Bundle for R1.A: {1, 5} R1 A B C 1 12 23 1 13 23 1 14 23 2 10 25 5 12 23 5 13 23 5 14 23 1.Choueiry, B.Y., Davis, A.M.: Dynamic bundling: Less Effort for More Solutions. SARA 02. 2.Freuder, E.C.: Eliminating Interchangeable Values in Constraint Satisfaction Problems. AAAI 91. 3.Haselböck, A: Exploiting Interchangeabilities in Constraint Satisfaction Problems. IJCAI 93. 4.Lal, A., Choueiry, B.Y., Freuder, E.C.: Neighborhood Interchangeability and Dynamic Bundling for Non-Binary Finite CSPs. CSCLP 04. 5.Lal, A., Choueiry, B.Y.: Constraint Processing Techniques for Improving Join Computation. CDB 04. {1, 2} {3} {1, 2} {1, 3} {3} No-good bundle V D C A B Solution bundle Constraint Satisfaction Problems {c, d, e, f } V3V3 {d} {a, b, d}{a, b, c} V4V4 V2V2 V1V1 Bundling is about exploiting interchangeability relations in backtrack search to produce robust, multiple solutions. This research is supported by a Maude Hammond Fling Faculty Research Fellowship and NSF CAREER Award #0133568. Experiments were conducted utilizing the Research Computing Facility of the University of Nebraska-Lincoln. 0 200 400 600 800 1000 1200 0.350.40.450.50.550.60.650.7 # NV in DynBndl # NV in FC First Bundle Size Tightness n = 20, a = 10 p 2 = 0.25 c 3 = 3 c 4 = 2 0 500 1000 1500 2000 0.350.40.450.50.550.60.650.7 DynBndl FC n = 20, a = 10 p 2 = 0.25 c 3 = 3 c 4 = 2 Tightness Overhead due to large FBS (upto 1200) at low tightness Maximum gains occur around phase- transition area CPU time [ms] Dynamic Bundling for Non-Binary CSPs Interchangeability in non-binary CSPs We show how to compute NI for non-binary constraints by [4]: 1.Building a data-structure, called the non-binary discrimination tree (nb- DT), determines the NI sets of a variable given a constraint that applies to the variable. 2.Intersecting the NI sets from the nb-DTs of a set of constraints yields the domain partition of the variable given the constraints.  Tests carried out on random CSPs  Parameters: n (number of variables), a (domain size), p 2 (proportion of binary constraints), c 3 / c 4 (number of ternary / quaternary constraints).  Criteria: FBS (First Bundle Size), CPU time, number of nodes visited (NV), and number of constraint checks.  partitions domains in a memory efficient manner.  fits into the iterator model of databases and produces one bundle at a time. Future Research Directions References