Managing fine-grained provenance is a critical requirement for data stream management systems (DSMS), not only to address complex applications that require diagnostic capabilities and assurance, but also for providing advanced functionality such as revision processing or query debugging. This paper introduces a novel approach that uses operator instrumentation, i.e., modifying the behavior of operators, to generate and propagate fine-grained provenance through several operators of a query network. In addition to applying this technique to compute provenance eagerly during query execution, we also study how to decouple provenance computation from query processing to reduce run-time overhead and avoid unnecessary provenance retrieval. This includes computing a concise superset of the provenance to allow lazily replaying a query network and reconstruct its provenance as well as lazy retrieval to avoid unnecessary reconstruction of provenance. We develop stream-specific compression methods to reduce the computational and storage overhead of provenance generation and retrieval. Ariadne, our provenance-aware extension of the Borealis DSMS implements these techniques. Our experiments confirm that Ariadne manages provenance with minor overhead and clearly outperforms query rewrite, the current state-of-the-art.
DEBS 2013 - "Ariadne: Managing Fine-Grained Provenance on Data Streams"
1. Ariadne: Managing Fine-Grained Provenance on Data
Streams
Boris Glavic1 Kyumars Sheykh Esmaili2 Peter M. Fischer3
Nesime Tatbul4
Illinois Institute of
Technology1
DBGroup
Nanyang
Technological
University2
SANDS
University of
Freiburg3
Web Science
Intel Labs4
Intel Science and
Technology Center
for Big Data
DEBS 2013 - July 7, 2013 - Arlington, USA
3. Fine-grained Provenance for Data Stream Management
Provenance
ā¢ Information about the origin and creation process of data
ā¢ Here: On which inputs does a given output depend on
Fine-grained Provenance
ā¢ At the granularity of tuples
ā¢ Fix a tuple t in an output or intermediate stream
ā¢ On which input tuples does it depend on?
Example
S1 Sout
S2
141 2 3
a b
Slide 1 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Motivation
4. Fine-grained Provenance for Data Stream Management
Provenance
ā¢ Information about the origin and creation process of data
ā¢ Here: On which inputs does a given output depend on
Fine-grained Provenance
ā¢ At the granularity of tuples
ā¢ Fix a tuple t in an output or intermediate stream
ā¢ On which input tuples does it depend on?
Example
S1 Sout
S2
141 2 3
a b
Slide 1 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Motivation
5. Why (Fine-Grained) Stream Provenance?
Use cases
ā¢ Ad hoc inspection
ā¢ DSMS generates alarm event based on sensor readings
ā¢ Understand why alarm was raised to act appropriately
ā¢ Stream query debugging
ā¢ Trace back and forward erroneous data items
ā¢ Auditing
ā¢ Proof of correct operation
ā¢ No access policies violated
Slide 2 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Motivation
6. Challenges and Opportunities
ā¢ Online and inļ¬nite data arrival
ā¢ Traditional methods that reconstruct provenance retroactively not
applicable
ā¢ Ordered data model
ā¢ Order can be exploited for compressing provenance
ā¢ Window-based processing
ā¢ Aggregation prevalent operation that has large provenance
ā¢ Low-latency requirements
ā¢ Provenance overhead should not violate latency requirements
ā¢ Non-determinism
ā¢ Sources: Load-shedding, windowing on system-time
ā¢ Provenance computation can not assume reproducibility of results
Slide 3 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Motivation
7. Related Work
Database Provenance
ā¢ Large body of work on provenance models
ā¢ Query rewrite-based approaches
Workļ¬ow Provenance
ā¢ Many systems support provenance tracking
ā¢ Mostly coarse-grained provenance
Data Stream Provenance
ā¢ Coarse-grained provenance
ā¢ Not detailed enough for most use-cases
ā¢ Fine-grained provenance
ā¢ Stream debugging: One-step (one operator at a time)
ā¢ Approximation techniques (strong assumptions)
Slide 4 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Motivation
8. How to generate provenance?
Inversion
ā¢ Infer provenance from query outputs
ā¢ Usually tracing back one operator at a time
ā¢ No storage overhead
ā¢ Only works for trivial operators
Slide 5 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Motivation
9. How to generate provenance?
Annotation Propagation
ā¢ Annotate data with provenance
ā¢ Propagate and manipulate annotations during query processing
Propagation - Query rewrite
ā¢ Rewrite query network to propagate annotations
ā¢ Use original DSMS operators
ā¢ Easy to implement, applicable to all DSMSā
ā¢ Often results in complex and ineļ¬cient networks
ā¢ Only applicable to deterministic networks
Slide 5 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Motivation
10. How to generate provenance?
Propagation - Operator instrumentation
ā¢ Modify DSMS operators to propagate provenance information
ā¢ More eļ¬cient than query rewrite
ā¢ Tolerates certain types of non-determinism
ā¢ Keeps structure of the original query network intact
ā¢ Requires modiļ¬cation of the engine (DSMS source code needed)
Slide 5 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Motivation
11. Plan of attack
1) Provenance model
ā¢ Annotated tuples
ā¢ Annotated streams
2) Provenance generation and querying
ā¢ Operator instrumentation
ā¢ Buļ¬er inputs + reconstruction for querying
ā¢ Implementation in Ariadne (extension of Borealis)
3) Optimizations
ā¢ Compression
ā¢ Laziness
ā¢ Decoupling provenance computation from query processing
Slide 6 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Motivation
13. Overview
Operator Instrumentation
ā¢ For each operator of the system create new versions
ā¢ Provenance generator (PG): Generates provenance annotations
ā¢ Provenance propagator (PP): Propagates provenance annotations
ā¢ Instrument query (or parts thereof) to generate provenance
ā¢ Replace operators with annotating versions
ā¢ Only in the part of the network we want to track provenance
Reduced Eager
ā¢ Eager
ā¢ Propagate provenance annotations during query processing
ā¢ Reduced Eager
ā¢ Propagate sets of tuple-IDs (TIDs) annotations
ā¢ Temporarily store inputs of relevant streams for restoring full tuples in
provenance
Slide 7 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Reduced Eager Operator Instrumentation
14. Overview
Querying Provenance
ā¢ Translate provenance annotations into regular streaming data
ā¢ For one output tuple t annotated with provenance {t1, . . . , tn}
ā¢ output n tuples (t, t1), (t, t2), . . .
ā¢ New operator (p-join) joins buļ¬ered input data with provenance
annotations
ā¢ Use operators of the DSMS for querying
Slide 7 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Reduced Eager Operator Instrumentation
18. Stream Provenance Model
Provenance Model
ā¢ Provenance is modelled as a set of contributing tuples
ā¢ From input of intermediate streams
ā¢ Provenance set P(t, I)
ā¢ tuple t in intermediate or result stream
ā¢ set of upstream streams I
ā¢ Which tuples belong to provenance?
ā¢ Declarative deļ¬nition that models provenance for single operators
ā¢ Transitivity
Slide 8 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Reduced Eager Operator Instrumentation
19. Stream Provenance Model
Example
ā¢ Query over stream of temperature readings
ā¢ Filter out outliers (temperature above threshold)
ā¢ Compute average temperature over every consecutive window of two
readings
ā¢ P(3:2 , {2}) = {2:2 , 2:3 }
ā¢ 3:2 generated by aggregating values from 2:2 and 2:3
ā¢ P(3:2 , {1}) = {1:2 , 1:4 }
ā¢ 2:2 and 2:3 derived from 1:2 and 1:4
āµ
1 3
2
Slide 8 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Reduced Eager Operator Instrumentation
20. Stream Provenance Model
Provenance Annotated Streams
ā¢ Each tuple in a provenance annotated stream (PAS) is annotated
with its provenance set
ā¢ according to a set of input streams I
ā¢ PAS P(O, I)
ā¢ Provenance annotated stream for stream O according to set of
upstream streams I
Slide 8 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Reduced Eager Operator Instrumentation
21. Stream Provenance Model
Example
ā¢ Query over stream of temperature readings
ā¢ Filter out outliers (temperature above threshold)
ā¢ Compute average temperature over every consecutive window of two
readings
ā¢ Provenance annotated stream (PAS) P(3, {1})
āµ
1 32
Slide 8 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Reduced Eager Operator Instrumentation
22. Implementing Instrumented Operators
Approach
ā¢ Three instrumented versions for each operator
ā¢ Provenance generators: Initialize provenance annotations
ā¢ Provenance propagators: Propagate provenance annotations
ā¢ Provenance dropper: Drop provenance from input
Slide 9 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Reduced Eager Operator Instrumentation
23. Implementing Instrumented Operators
Provenance Generator
ā¢ Computes one-step provenance for an operatorās output based on its
input
ā¢ Attach input TIDs as provenance
ā¢ For windowing operators merge TIDs from window
ā¢ Before instrumentation: I ā op ā O
ā¢ After instrumentation: I ā opPG ā P(O, I)
Slide 9 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Reduced Eager Operator Instrumentation
24. Implementing Instrumented Operators
Provenance Propagator
ā¢ Compute output provenance set by combining provenance sets from
the input
ā¢ Before instrumentation: P(I, I ) ā op ā O
ā¢ After instrumentation: P(I, I ) ā opPP ā P(O, I )
Provenance Dropper
ā¢ Removes provenance annotations
ā¢ Instrumentation: P(I, I ) ā opPD ā O
Slide 9 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Reduced Eager Operator Instrumentation
25. Instrumenting a Network
ā¢ User provides
ā¢ query network q
ā¢ stream O
ā¢ set of streams I upstream from O
ā¢ Output is a network that computes P(O, I)
InstrumentNetwork(q,O,I)
for all op on paths between I and O
if op is connected to I
replace op with PG version
else
replace op with PP version
Slide 10 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Reduced Eager Operator Instrumentation
26. Instrumenting a Network
Example
ā¢ Network that computes P(3, {1})
āµ
1 32
TID time tempature
1:1 0 102
1:2 5 105
1:3 10 399
1:4 15 85
TID avg temp
3:1 103.5 {1:1, 1:2}
3:2 95 {1:2, 1:4}
PPPG
Example
ā¢ Network that computes P(2, {1})
TID avg temp
3:1 103.5
3:2 95
āµ
1 32
TID time temperature
1:1 0 102
1:2 5 105
1:3 10 399
1:4 15 85
TID time temperature
2:1 0 102 {1:1}
2:2 5 105 {1:2}
2:3 15 85 {1:4}
PG PD
Slide 10 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Reduced Eager Operator Instrumentation
27. Ariadne Implementation
Based on Borealis
ā¢ Set of standard DSMS operators
ā¢ Operators connected through queues
ā¢ Fixed-length tuples
Changes to Borealis code
ā¢ Implement annotations
ā¢ Implement operator instrumentation
ā¢ Implement p-join and input buļ¬ering
Slide 11 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Reduced Eager Operator Instrumentation
28. Querying Provenance
Approach
1 Translate provenance annotations into regular stream data to query
using the host language
Slide 12 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Reduced Eager Operator Instrumentation
29. Querying Provenance
Approach
1 Translate provenance annotations into regular stream data to query
using the host language
P-join
ā¢ Joins a PAS P(O, I) with buļ¬ered input tuples from I
ā¢ Combine output t with each tuple in its provenance P(t, I)
Slide 12 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Reduced Eager Operator Instrumentation
30. Querying Provenance
Example
ā¢ Tuple 3:2 with P(3:2, {1}) = {1:2, 1:4}
āµ
1 32
TID time temperature
1:1 0 102
1:2 5 105
1:3 10 399
1:4 15 85
TID avg temp
3:1 103.5 {1:1, 1:2}
3:2 95 {1:2, 1:4}
TID time temperature
2:1 0 102 {1:1}
2:2 5 105 {1:2}
2:3 15 85 {1:4}
PPPG
C1
n
Slide 12 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Reduced Eager Operator Instrumentation
32. Rationale
Compression
ā¢ Reduce load on queues by using concise provenance representation
ā¢ Reduce memory imprint
Lazy Retrieval
ā¢ Avoid reconstructing unneeded provenance
Replay-Lazy
ā¢ Avoid generating unneeded provenance
ā¢ Decouple provenance computation from regular stream processing
Slide 13 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Optimizations
33. Replay-Lazy
Idea
ā¢ Avoid provenance computation if provenance is not needed
ā¢ Propagate concise representation of a super-set of the provenance
ā¢ If provenance is requested replay super-set through provenance
generating network
Slide 14 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Optimizations
34. Replay-Lazy
Idea
ā¢ Avoid provenance computation if provenance is not needed
ā¢ Propagate concise representation of a super-set of the provenance
ā¢ If provenance is requested replay super-set through provenance
generating network
Covering Intervals
ā¢ Interval:
ā¢ Minimal TID in provenance
ā¢ Maximal TID in provenance
ā¢ Constant size super-set of provenance (piggy-back!)
ā¢ Provenance: {1, 2, 4, 5, 10, 16, 65}
ā¢ Covering interval: [1, 65]
Slide 14 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Optimizations
35. Replay-Lazy
Enabling Replay
ā¢ Covering interval operators:
ā¢ CG corresponds to PG
ā¢ CP corresponds to PP
ā¢ C-join operator ā
ā¢ For each input get covering interval
ā¢ Retrieve all tuples from covering interval from connection point
Slide 14 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Optimizations
36. Replay-Lazy
Approach
1 Instrument original network to produce covering interval
2 Filter out results and c-join
3 Feed result into provenance generating copy of the network
PG
āµ
PP
āµ
CPCG
S1 Sout
Provenance
Generating
Network
Filter Provenance
and Fetch Tuples
from Input
Covering
Interval
Generating
Network
ā¦
ā”
PD
P
Slide 14 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Optimizations
41. Conclusions
Reduced-Eager Operator Instrumentation
ā¢ Novel propagation provenance method for DSMS
ā¢ Replace operators in a query network with provenance-aware
(instrumented) versions
ā¢ Keeps original network structure intact
ā¢ Deals well with non-determinism
ā¢ Flexible:
ā¢ Single- and multi-hop provenance
ā¢ Instrument parts of a network
Slide 16 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Conclusions
42. Conclusions
Optimizations
ā¢ Replay-Lazy
ā¢ Propagate only covering intervals - replay through provenance
generating copy of network
ā¢ Reduce cost for low retrieval rates
ā¢ Option to decouple provenance computation
ā¢ Lazy-Retrieval
ā¢ Push ļ¬lters through p-joins
ā¢ Avoids provenance reconstruction
ā¢ Compression
ā¢ Interval
ā¢ Delta
ā¢ Dictionary
Slide 16 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Conclusions
43. Future Work
Optimizing Temporary Input Storage
ā¢ Exploiting additional knowledge in purging temporary input storage
ā¢ Static query network analysis
ā¢ Self-optimizing purging strategies
ā¢ Compression and Approximations
Integration with Distributed Storage and Scaling-out
ā¢ Using distributed write-optimized storage for provenance?
Order-aware Provenance Model
ā¢ Integrate order into the provenance model
ā¢ āTuple t is after u in the result, because tās provenance is before uās
provenance in the input.ā
Slide 17 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Conclusions
44. Questions?
Boris Glavic
IIT
DBGroup
bglavic@iit.edu
Kyumars Sheykh Esmaili
Nanyang University
SANDS
kyumarss@ntu.edu.sg
Peter M. Fischer
University of Freiburg
Web Science
peter.ļ¬scher@
cs.uni-freiburg.de
Nesime Tatbul
Intel Labs
Intel Science and
Technology Center
for Big Data
tatbul@csail.mit.edu
http://cs.iit.edu/~dbgroup/research/ariadne.php
Slide 18 of 18 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne: Conclusions
46. How to Implement Annotations?
Alternatives for sending annotations
ā¢ Borealis models streams as queues of ļ¬xed-length tuples with header
and payload
1 Variable-length tuples: Negates optimizations based on ļ¬xed-length
2 Split provenance sets into ļ¬x-length tuples
3 New information channels: Complex interactions with normal query
processing
Slide 1 of 9 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne - Appendix: Implementation
47. How to Implement Annotations?
Alternatives for sending annotations
ā¢ Borealis models streams as queues of ļ¬xed-length tuples with header
and payload
1 Variable-length tuples: Negates optimizations based on ļ¬xed-length
2 Split provenance sets into ļ¬x-length tuples
ā¢ Send provenance set after each output tuple
ā¢ First tuple has small header to tell downstream operators how many
provenance tuples will follow
ā¢ Additional provenance tuples only store payload (TIDs)
3 New information channels: Complex interactions with normal query
processing
Slide 1 of 9 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne - Appendix: Implementation
48. Implementing Instrumented Operators
Approach
ā¢ Factor out common functionality
ā¢ Serializing/Deserializing provenance to/from queues
ā¢ Caching provenance of windows
ā¢ Merging provenance sets
ā¢ Implemented in Provenance Wrapper
ā¢ Operators access queues through the wrapper
ā¢ Reduces code changes to each operator
ā¢ LOC
ā¢ Provenance wrapper: Ė8000 LOC
ā¢ Instrumented operators: aggregation (largest) 200 LOC
Slide 2 of 9 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne - Appendix: Implementation
49. Temporary Input Storage
Connection Points
ā¢ Borealis feature for storing tuples from a stream
ā¢ Time-based or count based eviction strategies
ā¢ Content can be joined with streams
S1 Sout
S2 n P
Instrumented
Network
Reconstruct
Provenance
Temporary
Input Storage
PG
PG
PP
Slide 3 of 9 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne - Appendix: Implementation
53. Vary Retrieval Frequency
ā¢ Send large batch of 100,000 tuples
ā¢ Measure completion time
ā¢ Instrumentation with Retrieval
ā¢ Replay-Lazy with Retrieval
ā¢ Using a sequence of aggregations
ā¢ Vary Retrieval Frequency - selectivity of ļ¬lter before p-join
ā¢ Slide ļ¬xed to 1, Window size ļ¬xed to 100
Instrumented Version
āµ
PPPG PP
n
Replay Lazy
PPCP PG
āµ
PP
āµ
CPCG
ā¦ n
Slide 5 of 9 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne - Appendix: Additional Experiments
55. Latency
ā¢ In Ariadne inputs are send in batches with a ļ¬xed number of tuples
(parameter)
ā¢ Measure latency
ā¢ Using the Basic Network
ā¢ Vary the load
ā¢ Vary batch size and ļ¬x the delay between consecutive batches
Slide 6 of 9 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne - Appendix: Additional Experiments
59. Compression
Rationale
ā¢ Reduce amount of data that is shipped between operators
ā¢ Requirements
ā¢ Fast compression and decompression
ā¢ Operations such as merging sets on compressed data
Slide 8 of 9 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne - Appendix: Optimizations
60. Compression
Rationale
ā¢ Reduce amount of data that is shipped between operators
ā¢ Requirements
ā¢ Fast compression and decompression
ā¢ Operations such as merging sets on compressed data
Interval Compression
ā¢ Input: {1, 2, 4, 5, 6, 7, 9}
ā¢ Output: {[1 ā 2], [4 ā 6], [9 ā 9]}
Slide 8 of 9 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne - Appendix: Optimizations
61. Compression
Rationale
ā¢ Reduce amount of data that is shipped between operators
ā¢ Requirements
ā¢ Fast compression and decompression
ā¢ Operations such as merging sets on compressed data
Interval Compression
ā¢ Input: {1, 2, 4, 5, 6, 7, 9}
ā¢ Output: {[1 ā 2], [4 ā 6], [9 ā 9]}
Dictionary Compression
ā¢ Input: {1, 2, 4, 5}
ā¢ Output: LZ77({1, 2, 4, 5})
Slide 8 of 9 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne - Appendix: Optimizations
62. Compression
Delta Compression
ā¢ Express provenance as delta to down-stream provenance
ā¢ Once in a while send full provenance
ā¢ Express provenance as delta to previous full provenance
ā¢ Input: {1, 2, 4, 7, 9} ā {4, 7, 9, 10} ā {7, 9, 10, 12, 15, 19}
ā¢ Output: {1, 2, 4, 7, 9} ā 3 ā {10} ā 2 ā {10, 12, 15, 19}
Slide 8 of 9 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne - Appendix: Optimizations
63. Compression
Delta Compression
ā¢ Express provenance as delta to down-stream provenance
ā¢ Once in a while send full provenance
ā¢ Express provenance as delta to previous full provenance
ā¢ Input: {1, 2, 4, 7, 9} ā {4, 7, 9, 10} ā {7, 9, 10, 12, 15, 19}
ā¢ Output: {1, 2, 4, 7, 9} ā 3 ā {10} ā 2 ā {10, 12, 15, 19}
Heuristic Adaptive Compression
ā¢ No one ļ¬ts all
ā¢ Combine compression methods
ā¢ Heuristic rules for when to apply which method
Slide 8 of 9 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne - Appendix: Optimizations
64. Retrieval-Lazy
Approach
ā¢ User query that ļ¬lters out unneeded provenance
āµ
PPPG PP
n
Slide 9 of 9 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne - Appendix: Optimizations
65. Retrieval-Lazy
Approach
ā¢ User query that ļ¬lters out unneeded provenance
ā¢ Avoid expensive provenance reconstruction if provenance is not
needed
ā¢ If possible ļ¬lter before reconstruction (p-join)
ā¢ Push ļ¬lters through p-joins
āµ
PPPG PP
n
Slide 9 of 9 Glavic, Sheykh Esmaili, Fischer, Tatbul - Ariadne - Appendix: Optimizations