Approximate Lineage for Probabilistic Databases

Christopher Ré and Dan SuciuUniversity of Washington

Approximate Lineage in One Slide

• Lineage (Provenance) – In QP used to track correlations– Explain query/view results

• VLDBs have lots of lineage– Chokes QP– Hard for users to understand

• Obs: lineage contains a lot of redundancy!This work: Approximate the lineage, by keeping only the most important correlations

In a view, lineage is all derivations of a tuple

Especially with complex queries/views

probabilistic databases

Overview

• Motivation & Preliminaries

• An apx lineage approach: Sufficient Lineage

• Experiments

• Conclusions

id Description PX1 “Dr. Z told me” 0.9X2 “PubMed:123” 0.8X3 “Lab Experiment” 0.3

A Protein DatabaseProtein ProcessAGO2 “Cell Death”AGO2 “Embryonic Devel.”AGO2 “Glands”Aac11 “Cell Death”Aac11 “Embryonic Devel.”

Data are from somewhere

Lineage (l) is important

id DescriptionX1 “Dr. Z told me”

X2 “PubMed:123”

X3 “Lab Experiment”

Protein Process l

AGO2 “Cell Death” X1

AGO2 “Embryonic Devel.” X1

AGO2 “Glands” X2

Aac11 “Cell Death” X2

Aac11 “Embryonic Devel.” X3

AtomsManually Created

Machine inferredLineage from a wide variety of sources – not all trusted the same

Some with confidence, too!

Inspired by the Geneontology (GO) Database

Process (P)

Standard pDB, e.g. Mystiq, Trio

Protein lAGO2Protein lAGO2 (X1 ˄ X2)

Review: Lineage tracking

How do we derive the lineage ?

Protein Process l

AGO2 “Cell Death” X1

AGO2 “Embryonic Devel.” X1

AGO2 “Glands” X2

Aac11 “Cell Death” X2

Aac11 “Embryonic Devel.” X3

V(y) :- P(x,y),P(`Aac11’, y), x `Aac11’

Lineage propagates with queries /views

Protein l

AGO2 (X1 ˄ X2)˅ (X1 ˄ X3)

PRA[Fuhr&Rolleke 97], Trio [Widom 05], Mystiq [R,Dalvi,S07]

Lineage tracks all derivations

(GO) 1 tuple 10MB lineage!

Big DB = Big Lineage

Big Lineage chokes the engine!

Process (P)

“Proteins related to same process as `Aac11’”

Prob QP: Pr[V(‘AGO2’)] = Pr[l1]

l1

Problems with Large Lineage in pDB

• Lineage is used to:– Process Queries

– Give explanations to users

– Find influential atoms

On VLDBs, helpful to shrink (approximate) the lineage

Large: Many redundant explanations

Large: chokes QP

Large: Needle in a haystack

This talk

Approximate Lineage Approach

Original VLDB

Level 1 Database(Big lineage)

Level 2 Database(Small lineage)

Protein lAGO2 (X1 ˄ X2)

˅ (X1 ˄ X3)

lProtein aAGO2 (X1 ˄ X2)

asmaller, approximate formula

All (most) querying on Level 2 database (using a instead of l)

Focus is on the Level 2 database

Protein aAGO2 0.5*x1 + 0.3

error, e

Overview

• Motivation & Preliminaries

• An apx lineage approach: Sufficient Lineage

• Experiments

• Conclusions

Sufficient lineage (SL)

Protein aAGO2 (X1 ˄ X2)

• Represent as?

• Use as to:– Answer queries?– Provide explanations?– Find influential tuples?

• Build good a, efficiently?

DNF formulae, that logically imply l

The remainder of this talk

Protein lAGO2 (X1 ˄ X2)

˅ (X1 ˄ X3)Reuse existing systems!

See paper

Nugget: An algorithm that always finds small, good SL

a is a lower bound l

Formalizing “good as”

id Description PX1 “Dr. Z told me” 0.9X2 “PubMed:123” 0.8X3 “Lab Experiment” 0.3

Atoms

Choosing an approximation a for a lineage function, l

Protein l

AGO2 (X1 ˄ X2)˅ (X1 ˄ X3)

Intuition: a should agree on most worlds

An atom is a Boolean proposition. A world is a set of the true atoms.

Formalizing this,

E[l – a] e Expectation of difference over all worlds, should be small

NB: really standard ℓ2 distance

Illustrating Good Lineage

id Description PX1 “Dr. Z told me” 0.9X2 “PubMed:123” 0.8X3 “Lab Experiment” 0.3

Protein lAGO2 (X1 ˄ X2)

˅ (X1 ˄ X3)

Protein aAGO2 (X1 ˄ X2)

0.9 *(1 - (1 - 0.8)(1-0.3)) 0.9 * 0.8 = 0.72= 0.9 *0.86 = .774

e = 0.054

E[l – a] = E[ ]l – E[a] e

Intuition: Pr[ ] a high means good lineage

1st step: Lineage DNFs to “graphs”

X1

X2

Xn

Y1

Ym

Y2

(X1 ˄ Y1) ˅(X2 ˄ Y1)

We can think of DNFs as graphs (k-DNF a k-hypergraph)

Goal: Given error e, find a subset of edges with error smaller than e and small size, i.e. a best lower bound;

Atoms = nodes

Monomials = edges

Trick: matching is an SL formula.

How big a matching could we need?

Matching of size 9 implies Pr[M] > .9

X1

X2

Xn

Y1

Ym

Y2Pr[M] = 1- (1-0.25)|M|

Assume Pr[Xi ] = Pr[Yj ] = 0.5

If l has a small good matching, take a to be matching. Call this a “good enough matching”

Subtle: size bound depends on k, e and Pr[Xi] – not # of tuples

For any e > 0.1 ; M can always < 9

Size Pr[M]

9 0.9

17 0.99

25 0.999

33 0.9999

There is not always a good-enough matching

X1 Y1

Ym

Y2

Obs: no “good-enough matching”, then cover must be small

nodes in any maximal matching

(Y1 ˅ Y2 ˅ … Ym) – a (k-1)-DNF

Formally, {X1,X2} is a small cover

Best matching is 0.4 , but formula very close to 0.625!

X1 ˄ APX(Y1 ˅ Y2 ˅ … Ym) ˅ (X2 ˅ Z)

Must apx the (k-1)-DNF w. smaller e to account for correlations

ZX2

Y5

SL is always small

Two Cases:1. Small-good matching 2. Small-cover of important nodes

We’re done!Recurse on k-1 DNF

THM (SL is always small) Size of SL is constant in data.

Problem: Maximum matching in general hypergraphs is NP-hard

Apx NP-hard!need a maximal matching – pick greedily!

Requires “non-vanishing” probs In datasets, usually, Pr > 10-3

Exponential in query Similar to data-complexity

Summary of Constructing SL

• For SL, good lineage = big lineage– Not true in general.

• Gave an algorithm that always finds small SL– Constant in the data– Exponential in almost everything else

• Main trick: Don’t try to find optimal solutions, when sloppy is good enough!

Other fun results in the paper

• Sufficient Lineage (SL)– Error bounds for QP– Finding influential tuples

• Polynomial Lineage (PL): DNF to polynomial– Use Taylor/Fourier approximation of poly– Algos for QP, explanations and influential tuples

Leverage extensive prior art!

PL smaller than SL, but not usable in pDBs (Mystiq, Trio).

Overview

• Motivation & Preliminaries

• An apx lineage approach: Sufficient Lineage

• Experiments

• Conclusions

Experiments

• Geneontology Database– Publically available– Predefined views– Atoms = “evidence codes”

• Discuss a single view– 6 tables– 2 sources of evidence– 1119 tuples – 141MB

Similar results on IMDB data not presented

“All proteins associated with a single protein”

Compression Ratio v. Error

0.5 0.1

0.05000000000000010.01

0.0010

100200300400

e, error level (smaller is more conservative)

Compress Ratio

Good compression ratio even for

stringent error

30x compression

141MB to 4MB

Effect on QP

0.1 0.01 0.00110

100

1000

10000

100000

Running TimeSeconds

(Log10 Scale)

e, error level (smaller is more conservative)

Sufficient Lineage

Original Lineage

Compute each tuple in the view

Which ls give the biggest gain?

1 37 73 1091451812172532893253613974334695055411

10

100

1000

10000

100000

1000000

Top 500 formula in descending size (# is rank)

# Terms

Original Lineage

Sufficient Lineage

Win: Compressing big terms

Compressing Single View

Conclusion

• Discussed approximate lineage approach– Goal: Fast QP, Explanations

• Sufficient Lineage– Can be used by standard QPs– Improves QP dramatically

• Apx lineage is more general, e.g. Polynomial