slides01-14 - Magic Sets Optimization technique for...

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Unformatted text preview: Magic Sets Optimization technique for recursive Datalog. Also a win on some nonrecursive SQL Mumick, Finkelstein, Pirahesh, and Ramakrishnan, 1990 SIGMOD, pp. 247 258. Combines bene ts of both top-down backward chaining, recursive tree search and bottom-up forward chaining, naive, seminaive processing of logic, without disadvantages of either. Example of Nonrecursive Use Find the programmers who are making less than the average salary for their department. SELECT FROM Emps e1 WHERE e1.job = 'programmer' AND e1.sal  SELECT AVGe2.sal FROM Emps e2 WHERE e2.dept = e1.dept ; Naive implementation computes the average salary for all departments. Magic-sets" implementation rst determines the departments that have programmers perhaps very few. It can then use an index on Emps.dept to avoid accessing the entire Emps relation. Recursive Example ancX,Y :- parX,Y ancX,Y :- parX,Z & ancZ,Y Query: anc0; W . Top-down search e.g., Prolog would: 1. Query the EDB for par0; Y . 2. By the rst rule: return all such answers, say f0; 1; 0; 2g. 3. The same parent facts are also useful in the second rule to set up calls" to anc1; Y  and anc2; Y . 4. Recursively solve these queries. 1 Advantage of Top-Down We never even ask about individuals that are not in the ancestry of individual 0. Advantage of Bottom-Up i.e., naive, seminaive We don't go into in nite recursive loops. Example Both of the following Datalog programs loop if evaluated top-down: ancX,Y :- parX,Y ancX,Y :- ancX,Z & parZ,Y ancX,Y :- parX,Y ancX,Y :- ancX,Z & ancZ,Y Key Magic-Sets Ideas 1. Introduce magic predicates" to represent the bound arguments in queries that a top-down search would ask. 2. Introduce supplementary predicates" to represent how answers are passed from leftto-right through a rule. 3. Technical details to get right: a Predicate splitting : an IDB predicate must be called" in top-down search with only one binding pattern. b Subgoal recti cation : avoid IDB subgoals with repeated variables. Rule Goal Graphs Needed to assure unique binding patterns for IDB predicates. Composed of rule and goal nodes, as follows. Goal Nodes Predicate + adornment." Adornment = list of b's and f 's, indicating which arguments are bound, which are free. Example: pbfb. First and third arguments of p are bound. 2 Rule Nodes riS T represents the point in rule r after seeing i subgoals, with variables in set S bound, those in T free. j Children of Goal Nodes Children of goal node p are those rule nodes r0S T such that 1. Rule r has head predicate p. 2. S is the set of variables that appear in those arguments of the head that says are bound. 3. T is the other variables of r. j Children of Rule Nodes Children of the rule node rjS T are: 1. The goal node of the j + 1st subgoal of r, with adornment that binds those arguments whose only variables are in S . 2. The rule node rjS T , where S = S + +1 variables appearing the in j + 1st subgoal; T is the other variables. Exceptions: no rj +1 rule node if r has only j + 1 subgoals. No goal child if j = 0 and r has no subgoals. j 0 j 0 0 0 Constructing the RGG Start with goal node whose adornment matches bindings of query. Add nodes by constructing children as required by rules from previous slides. Reordering of subgoals of a rule is allowed: helps maximize bound" arguments. Reordering may be di erent for di erent rule nodes. Example Here is a nonrecursive example, where the RGG is a tree. r1: pX,Y :- qX,Z & rZ,Y r2: rA,B :- sA,B r3: rA,B :- tA,B 3 Query form pbf , e.g., p0; W ? pbf r1X0 Y;Z : j r1X1;Z Y : qbf j rbf r2A0B : r3A0B : sbf tbf j j Recursive Example r1: ancX,Y :- parX,Y r2: ancX,Y :- ancX,Z & ancZ,Y Query; ancbb, e.g., ancjoe; sue? ancbb r1X0Y : r2X0;Y Z : j j parbb r2X1;Y;Z : ancbf X r1:0Y j r2X0 Y;Z : j j r2X1;Z Y : parbf j Splitting Predicates For magic-sets to work, there must be a unique binding pattern associated with each IDB predicate. No constraint on EDB predicates. Key idea: For each adornment such that p appears in the RGG, make a new predicate 4 p . Rules for p are the same as for p, but predicates of IDB subgoals are the version with the correct binding pattern. RGG helps us gure out the needed binding patterns. Example For RGG above: anc bbX,Y :- parX,Y anc bbX,Y :- anc bfX,Z & anc bbZ,Y anc bfX,Y :- parX,Y anc bfX,Y :- anc bfX,Z & anc bfZ,Y Rectifying Subgoals All IDB subgoals must have arguments that are distinct variables. Feasible for datalog no function symbols. Fixes some problems where RGG knows about fewer bound arguments than the top-down expansion does. 3 See p. 801 of PDKS-II. Trick: replace an IDB subgoal G with variables appearing in more than one argument and or constant arguments by a new predicate whose arguments are single copies of the variables appearing in G. Create rules for the new predicate by unifying G with heads of rules for G's predicate. Repetition may be needed because the resulting rules may have unrecti ed subgoals. Example r1: pX,Y :- aX,Y r2: pX,Y :- bX,Z & pZ,Z & bZ,Y pZ; Z  is unrecti ed. Create qZ  = pZ; Z . Unify heads of rules with pZ; Z . Careful! Z in body of r2 must be renamed. r1 becomes pZ,Z :- aZ,Z or qZ :- aZ,Z 5 r2 becomes pZ,Z :- bZ,W & pW,W & bW,Z or qZ :- bZ,W & qW & bW,Z Finally, in the original r2 we replace subgoal pZ; Z  by qZ . The resulting rules, with variables renamed: pX,Y :- aX,Y pX,Y :- bX,Z & qZ & bZ,Y qX :- aX,X qX :- bX,Y & qY & bY,X Magic Sets Transformation Start with a program and a binding pattern for a query. 1. Split predicates to get unique binding patterns. 2. Rectify subgoals. 3. Introduce magic and supplementary predicates as follows. Magic Predicates For each IDB predicate p, introduce m p. Arguments of m p correspond to bound arguments of p in its unique binding pattern. Intuition: m p is true of exactly those tuples that are members of queries to some p-node in the top-down expansion. Supplementary Predicates For each rule r of n subgoals, introduce supplementary predicates supr:j for 0  j n. Arguments are the bound and active variables before the j + 1st subgoal of r. 3 A variable is active i it appears either in the head or a subgoal from j + 1 on. Intuition: true for a tuple i that tuple represents a possible binding for the bound, active variables at that point. 6 ...
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