Transcript Efficient Query Evaluation on Probabilistic Databases

Efficient Query Evaluation on
Probabilistic Databases
Papers by Nilesh Dalvi, Dan Suciu,
Chris Re
Outline
• Motivation
• Definitions through examples
• Evaluation
• Complexity
Motivation
• Imprecise information on the web
• Partial Information
• Contradictions
• Imprecise queries
Imprecise Querying
Interpreting the ‘~’
• For the actors name we can use edit distance,
frequency similarity measures…
• For the films rating we can use user preferences,
analysis of previous queries,…
• But how to combine them?
• And how to assign a score for a tuple w.r.t. the entire
query?
Probabilistic DB
• Each tuple has a probability of appearing in
the DB
• Assume tuple independence
• Distribution over all possible DB instances
• Possible Worlds Semantics
Example
Semantics
• A query is evaluated on every possible world
• Note that for each concrete world, the query
may have several answers
• In this case, sum, for each answer, the
probabilities of the worlds in which it
appeared in the set of answers
• Example
Example (Join on B=C)
Another Example (join and projection
on A)
Solution attempt
• Obtain a query plan
• Compute intermediate results along with
probabilities
• A plan in our (first) example: First compute
the join, then project on D
Evaluation of the plan
Wrong!
• The tuples in the original DB were
independent
• The tuples in the intermediate DB are not!
• Thus the multiplication (for the projection) is
incorrect.
The problem is hard
• Theorem: Answering a query over a general
probabilistic DB is #P-hard (Data Complexity)
• #P-hard is the “equivalent” of NP-hard for functional
problems
• E.g. #SAT - given a Boolean formula, compute how
many satisfying assignments it has.
• Likely not to have a polynomial solution
Other plans
• Some query plans are OK
• These are plans that preserve independencies
• Let us represent the query as a logical formula
• Tuples that support the answer ‘p’ satisfy:
(s1 or s2) and t1
Plans and formulas
• The query was P((s1 or s2) and t1)
• First join, then project corresponds to
P((s1 and t1) or (s2 and t1)).
This conversion is fine in classic DB
But (s1 and t1), (s2 and t1) are not independent
events!
Safe Plan
• A plan that preserves independencies is called
safe
• In our example: first project s over b, only
then join with t
• = first compute the ‘OR’, then the ‘AND’
Safe Plan
Intuition on evaluation
• Work with probabilistic events
• Carry the events during evaluation
Probabilistic Events
• Atomic events tuples in the original DB
• Complex Events – boolean combination of
events  tuples in intermediate DBs
• Translate a query plan to a complex event
Translation
Translating events to probabilities
(Works iff the DB preserves
independence!)
Safe Plans
• A relational algebra expression has multiple
equivalent expressions
• Each corresponds to a concrete execution
plan.
• Some of these plans may correspond to
correct or incorrect probabilistic computations
• Let us try to detect what makes a plan safe.s
So what can we do?
• 1. Compute a safe plan when there is one
• 2. Compute an approximation when not
Approximation
• Most common is called Monte-carlo
approximation
• Originally by Karp, improved in [suciu07]
• Guarantees convergence
• The error is greater than e with a probability
of less than d after (4*n / e^2)* ln(2/d)
Functional Dependencies (FDs)
• A functional dependency {A1,…An} -> B holds
for a relation R if the values of the A1,…An
decide the value of B
Safe plans using FDs
• Selections and joins (over conjunctive queries) are
always safe (but may cause unsafe successions..)
• Projection of a1,…,ak over the result obtained from
q is safe if for every R, there is an FD a1,...,ak ->
Head(q)
Where Head(q) are the attributes in the result of q
Intuition
• Projection over a1,…,an  OR over all tuples that have the
same values of {a1,…,an}
• To be independent, each atomic event must be sufficient to
distinguish tuples that are ORed (otherwise it appears in more
than one)
• I.e it uniquely determines the other atomic events appearing
in the tuple
• Hence the FD (valid only in combination with a1,…,an)
Conjunctive Queries and
Union thereof
• Whiteboard discussion
Safe Plan algorithm
• Top-Down
• Push all safe projections late in the plan
• When you can’t, split the query q into two
sub-queries q1 and q2 such that their join is q
(when possible)
• If stuck, the query is unsafe
(Union of) Conjunctive Queries
by example
• T(x):- R(x,y),S(y,30)
• T(x):- P(x,y)
• In relational algebra?
– Multiple Possible translations
– Correspond to different ordering of operations
– Each option is called a “query plan”
More notations
• Head(q) is the set of head variables in q,
FreeVar(q) is the set of free variables (i.e. nonhead variables) in q
• R.Key is the set of variables in the key position of
the relation R
• R.NonKey is the set of variables in the non-key
positions of the relation R,
• R.Pred is the predicate that q applies to R. For x in
FreeVar(q), denote qx a new query whose body is
identical with q and where
Head(qx) = Head(q) U {x}.
Conclusion
• Probabilistic DB is a very strong tool
• Combines the exact semantics of classic DB
with capabilities of IR
• Exact evaluation becomes hard sometimes
• But have good approximations (with bounds!)