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Top k Knapsack Joins and Closure Early Results Witold LITWIN & Thomas Schwarz U. Paris Dauphine, France witold.litwin@dauphine.frwitold.litwin@dauphine.fr Santa Clara U., CA, tjschwarz@scu.edutjschwarz@scu.edu 1

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Knapsack Join (KS-Join) The join defined by the sum of the join attributes being at most some constant Father of 4 kids wishing to buy toys for at most 100€ total A person wishing to buy a computer tower, a screen, a printer and a desk for at most 1000 € …. 2

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Knapsack Join (KS-Join) Traditional join – R1 Join R2 on c1 = c2 KS - join – R1 Join R2 on c1 + c2 ≤ C Syntax legal for FROM clause in Access, SQL Server… 3

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Top k Knapsack Join (KS-Join) Top k items with respect to the descending order on the constant Usually, only a few items the most close to the constant are of interest Select TOP 1 * from Toys T1, Toys T2, Toys T3, Toys T4 Where T1.Price + T2.Price + T3.Price + T4.Price ≤ 100 and T1.Id < T2.Id and T2.Id < T3.Id and T3.Id < T4.Id Order by T1.Price + T2.Price + T3.Price + T4.Price Desc; 4

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Top k Knapsack Join (KS-Join) Top k Knapsack joins are of obvious interest How DBMSs deal with ? Nested loop – To our best knowledge Result: execution time makes the SQL capability useless for a larger data set – Consider our example for just 1000 toys to choose from – FYI, 1K-tuple table & 3-way KS-join killed SQL Server 5

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Our Goal : Optimizing Top k KS-Joins Algorithms provably faster than usual nested loop – Formulate the algorithm – Prove the complexity, storage & processing costs KS-optimized Nested Loop Self-join Nested Loop Sort Merge KS – Join Indices Distributed KS – Join Indices 6

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Our Goal : Optimizing Top k KS-Joins Early Results Only for Top k KS-Joins (TkKS-Joins) Only the formal analysis as yet Many variants of TkKS-Join queries left for future work – See the paper 7

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Knapsack Problem (KP) NP-hard optimization problem Among most studied Input: – A set O of objects {o 1,...o n } – An m-d subspace called knapsack K with – values b i, 1 ≤ i ≤ m, represent each the i-th dimension's capacity of the knapsack – Vector c j represents the benefit of the object j if in the knapsack 8

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Knapsack Problem (KP) Input (continued): – The knapsack's constraints matrix with entries a i,j ; 1 ≤ j ≤ n ; – Each entry stores the constraint value for each object j in each dimension i (price, size, volume...). Output: – A set O' of objects stored in the knapsack. 9

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Knapsack Problem (KP) Binary variable x j ; x j {0, 1}, indicates the selection of the object j into the knapsack – (x j = 1) for object j in and (x j = 0) otherwise – x j is 0–1 decision variable 10

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Knapsack Problem (KP) Select the elements of O’ which maximize the total profit of the selected objects Provided the match of the knapsack constraints 11

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Knapsack Problem (KP) Formally, maximize: Subject to: 12

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Knapsack Problem (KP) The most frequently investigated case is the 1-d one – I.e., i = 1 Often, or perhaps even the most often, the KP concept designates implicitly this case. Frequently, in addition, one also sets every c j to c j = a j. Both conditions are ours below – unless we state otherwise The m-d one is referred to then, if needed, as multidimensional (MKP). 13

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Knapsack Problem (KP) The general research orientation for KP and MKP Find a heuristic providing acceptable approximate result – For the possibly largest data set – In the fastest time, – Or acceptable time – Given necessary constraints on the computer system used. 14

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KP / TkKS -Join Our research orientation follows the database approach Find an exact result – For a reasonably practical problem subspace – For a database size data Say, 1Ktuples per table at least 15

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KP / TkKS -Join Find an exact result (continued) – In the fastest time – Or acceptable time Minutes at most – Given necessary constraints on the computer system used Mainly storage cost 16

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Knapsack Problem (KP) Our reasonably practical problem subspace at present: – As we already stated c j = a j – 1-d space – Fixed # of objects for the knapsack Join instead of closure 17

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Knapsack Problem (KP) Our reasonably practical problem subspace at present (continued): – One tuple = one potential selection – One object = one tuple with distinct ID No objects selected twice in a tuple for the knapsack Closure, MKP… left for the future 18

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Nested loop TkKS-Join Basic cost for tables with n 1 …n m tuples – O (n 1 *…*n m ) To accelerate the calculus start with: – Evaluation of the restrictions t i < C – Evaluation of t i ≤ C – (Min 1 +…+Min j +…+Min m ) for any j ≠ i DBMS may easily maintain the Min j statistics Cost can be O(m) or even O (1) only – Idem for C ≥ Max 1 +…+Max m ? 19

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Nested loop TkKS-Join Self-joint of a table with its copies Since KS-join is commutative one may avoid doubles – E.g. if we have tuple (t 1, t 2 ) then we should not have the tuple tuples (t 2, t 1 ) – In general, we need only one tuple from all its permutations The optimizing cuts the complexity and calculus time by half, at least Final word: we may have – O (n 1 *…*n m /S), where S ≥ 1 20

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Sort-Merge TkKS-Join 2-way join 21 C =150 150

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Sort-Merge TkKS-Join Processing cost of 3-way TkKS-Join – O (n 1 +n 2 ) in general – O ((n 1 +n 2 )/2) for self-join For n-way TkKS-Join – O (n m *…n 3 (n 2 + n 1 )) in general – For self-join ? E.g. For 16K-tuple R1 and R2 tables m-way join accelerates 8K times – 1sec instead of 2+ hours See the paper 22

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KS-Join Index A relational table I KS with at least the attributes (C, t 1.Id,…, t m.Id) – Here C = t 1.c+…+t m.c – Also t 1.Id <… < t m.Id Can be also seen as a materialized view Some or all t i.c should be useful as well E.g. for queries with additional restrictions on individual prices 23

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KS-Join Index I KS should be implemented as file sorted on C first – Then, on other key or non-key attributes of interest – E.g., a B-tree or trie… Storage cost: – O (n 1 *…*n m ) in general – Half of it or less for copies of the same table 3-way indices may be in RAM More should be typically on flash or disk 24

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KS-Join Index Processing cost – O (Log p (n 1 *…*n m ) ) or less, according to the storage cost, where p is the tree fan-out Expected practical figures – ms for RAM, e.g., 3-way KS-Join index for 1K-tuple tables – under 10 ms for flash – under 100 ms for the disk, e.g., 4-way KS-Join index for our 1K-tuple tables 25

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KS-Join Index Maintainance cost – High processing cost – E.g., 1 insert into our 1K tables generates 1M new entries Main drawback of KS-Indices at present Efficient processing is an open problem 26

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Composing KS-Join Indices TkKS-Join calculus can compound existing KS- Indices m-way & n-way indices may speed up (m+n)- way TkKS Join Through the sort-merge algorithm applied to both indices Seconds may suffice for up to 6-way joins – E.g., for our 1K relations 27

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Scalable-Distributed TkKS-Join Index Speeds up the calculus of even larger joins Using the parallel distributed processing Dozens of seconds may suffice for an 8-way join – Over our favorite 1Ktuple relations – With two 4-way KS-Indices – Each being distributed over 1K nodes – Through, e.g., RP* SDDS Maintainance time speeds-up as well 28

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Scalable-Distributed TkKS-Join Index C = 900 ; arrows show nodes to join in parallel 29 100350800 9900 1050450700

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Conclusion TkKS-Joins are potentially useful Our optimizations may speed up the processing by orders of magnitude Queries with TkKS-Joins become then practical With all the usual disclaimers, the results appear ready for mainstream DBMSs 30

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Future Work Deeper formal analysis Experiments More TkKS-Join query types – See the paper Thank You for Your Attention 31

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