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The Bucket Algorithm
We can \answer queries using views" by trying all CQ's with no more (view) subgoals than the query has subgoals.
However, a more organized exploration of the p ossibilities, called the bucket algorithm lo oks at how views can \cover" each of the query subgoals and each variable of the query.
Basic idea due to Levy; improved to consider the query variables as well as the subgoals by Prasenjit Mitra (and indep endently by Levy and Pottinger).
Limits on the Containment Mapping
From Query to Expansion
The key to the bucket algorithm is to understand what can happ en to variables in the query, when they are mapp ed to variables in the expansion.
Each subgoal of the query must b e covered by some view in the solution. That is, the query subgoal must map to some subgoal in some expansion, or the expansion is not contained in the query.
We must di erentiate b etween distinguished variables (those that app ear in the query head or the head of a view de nition) and nondistinguished variables (others).
We must also di erentiate b etween unique nondistinguished variables (those that app ear only once in the query) and shared variables (which app ear more than once).
R1. A distinguished variable in the query must map to a distinguished variable (of some view) in the expansion.
R2. A shared variable X must either map to a distinguished variable of a view expansion or all query subgoals involving X must map to the same nondistinguished variable in the expansion of a single view.
{ The reason is that two view expansions can never have the same nondistinguished variable. { Even if the lo cal variable name is the same in the view de nition, the expansions must use di erent names.
Examples
Figure 0.1 shows one consequence of rules R 1 and R2.
v(X,Y)
p(X,W) q(W,Z) r(Z,Y)
answer(U,V) :−... r(A,B)... s(B,C)
Y
v’(... Y.. .)
Query
Expansion
Solution
Figure 0.1: Covering the query subgoal r (A; B )
A is a unique, nondistinguished variable. It can map to the nondistinguished variable Z in the expansion of view v.
B is a shared, nondistinguished variable. One p ossible treatment, shown in Fig. 0.1, is that B maps to a distinguished variable, Y , and other o ccurrences of B , suggested by the subgoal s(B ; C ), are mapp ed so that o ccurrence of B maps to a distinguished variable of some other view.
{ Note that in designing the solution, we have the freedom to equate the variables of the views, and thus to use the same distinguished variable in two or more view expansions. { This capability makes it p ossible to map the o ccurrences of B to the same variable, coming from expansions of di erent views.
Figure 0.2 suggests what happ ens if a shared variable is forced to map to a nondistinguished variable in a view expansion. The s subgoal now has noplace to go, since there is no s subgoal in the expansion of v , and another expansion could not have Z as a variable.
v(X,Y)
p(X,W) q(W,Z) r(Z,Y)
Query answer(U,V) :−... q(A,B)... s(B,C)
Expansion
Solution
??
Figure 0.2: We cannot map B to lo cal variable Z
On the other hand, if the other subgoal(s) with B can also map to subgoals in the same expansion, then B can b e \covered." Figure 0.3 shows what happ ens when the s subgoal b ecomes and r subgoal.
- If B is a shared variable, then the bucket for B includes each view v and each set of subgoals in the expansion of v such that there is a containment mapping from all the query subgoals containing B to that subset of the subgoals in v 's expansion.
The containment mapping must map distinguished (of the query) to distinguished (of the view de nition).
Example
Consider the following views v and w , and query:
v(X,Y) :- p(X,Z) & p(Z,Y) w(U,V) :- p(U,S) & p(S,T) & p(T,V)
query(A,B) :- p(A,C) & p(C,D) & p(D,E) & p(E,F) & p(F,G) & p(G,B)
Think of p as \parent," v as \grandparent," and w as \great-grandparent."
The bucket for p(A; C ) is empty, b ecause A is distinguished and C is shared. But no view subgoal has two distinguished variables.
The bucket for p(C ; D ) is empty b ecause b oth variables are shared, and again no view subgoal has two distinguished variables.
Similarly, the buckets for the other query subgoals are empty.
The bucket for shared variable C includes fp(X ; Z ); p(Z ; Y )g from v , and fp(U; S ); p(S; T )g from w.
{ It do es not include fp(S; T ); p(T ; V )g from w , b ecause p(A; C ) would have to map to p(S; T ), and A is distinguished, but S is not.
The bucket for shared variable D includes fp(X ; Z ); p(Z ; Y )g from v , and b oth fp(U; S ); p(S; T )g and fp(S; T ); p(T ; V )g from w.
{ Note that we cannot at this p oint guarantee a consistent treatment of the variables C and E that are involved in the mapping of the subgoals with D ; a more careful analysis can check that the requirements of these shared variables can b e met in conjuction with D 's requirements.
We have to put together some views whose expansions cover all the subgoals.
- One p ossibility is to use expansions of v as the buckets for shared variables C , E , and G.
The other shared variables fortunately cna map to distinguished variables of the expansions, so we can handle that sharing by equating the variables in the view subgoals of the solution. Likewise, the distinguished variables of the query can map to distinguished variables of the expansions. The resulting solution:
query(A,B) :- v(A,H) & v(H,I) & v(I,B)
- Another solution uses one expansion of w to cover the buckets for b oth C and D and a second expansion of w for F and G.
Again, the sharing of E can b e handled by using the same variable in the two subgoals of the solution, and the distinguished variables of the query fortuitously map to distinguished variables of the view expansions. The solution:
query(A,B) :- w(A,H) & w(H,B)
- Notice there is no covering with one use of v and one of w. However, we do it, one query subgoal needs to b e covered by a third view expansion, and there is no way to make b oth variables of the \lost" subgoal map to distinguished variables of that expansion.
