Download KDID'03 Keynote Talk

Constraint-Based Model Mining:
Algorithms and Applications
Minos Garofalakis
Internet Management Research Dept.
Bell Labs, Lucent Technologies
[email protected]
http://www.bell-labs.com/~minos/
Lucent Technologies, proprietary and confidential
Outline
• Motivation for mining with constraints
• Algorithmic solutions
– Sequential patterns – SPIRIT
– Decision trees
• Application: Model-Based Semantic Compression SPARTAN
• Future directions
– Cost-based constraint pushing
– Mining continuous data streams
• Conclusions
Lucent Technologies, proprietary and confidential
Motivation
• Data Mining: Extracting concise and interesting models
from large data sets
– I/O and computation intensive, multiple passes over the data
• Lack of user-controlled focus – few “knobs” to specify
models of interest to users
– Patterns/Associations: Only specify a lower bound on support
– Decision trees: Search for some “optimal” tree (e.g., MDL)
– Often, huge numbers of patterns or models that are to large to
comprehend/interpret
• Decision trees with 100s of nodes are not at all “easy to assimilate”…
Lucent Technologies, proprietary and confidential
Motivation (contd.)
• BUT ... users may only be interested in models of specific
form!!
– Patterns conforming to a specific structure
– “Best” decision tree with <= k nodes (to get rough picture of data)
• Problems with traditional, unfocused search:
– Disproportionate cost for “selective” users
– Overwhelming volume of (possibly) useless results
• Sorting through the results to find model(s) of interest can be non-trivial
• Goals of constraint-based data mining:
– Flexible languages for specifying structural and other constraints
on mining models
– Algorithms for “pushing”/exploiting these constraints inside the
mining loop
– Cost should be commensurate with user selectivity
Lucent Technologies, proprietary and confidential
Outline
• Motivation for mining with constraints
• Algorithmic solutions
– Constraint-based sequential pattern discovery – SPIRIT
– Decision trees
• Application: Model-Based Semantic Compression SPARTAN
• Future directions
– Cost-based constraint pushing
– Mining continuous data streams
• Conclusions
Lucent Technologies, proprietary and confidential
Sequential Pattern Discovery
• Discovery of frequent sequential patterns (subsequences) in a
database of sequences
< 1, 2 , 3 , 4 >
< 1, 5 , 6 , 3 , 4 >
<7,1,2,3>
<1,8,2,3>
support >= 50%
<1,2,3>
<1,3,4>
• Scope: market-basket (buying patterns) , page access patterns
(WWW advertising) , alarm correlations (data networks), etc.
– Example (Yahoo! topic hierarchy) - distinct paths for hotels in NY
< travel, yahoo!travel, north america, US, NY, NY city, lodging, hotels >
< travel, lodging, yahoo!lodging, NY, NY cities, NY city, hotels and motels>
Frequency is important, e.g., for advertising
Lucent Technologies, proprietary and confidential
Our Work [VLDB’99,TKDE’02]
• Srikant & Agrawal [ICDE’95,EDBT’96]: Unfocused search
– User only specifies a lower bound on the pattern frequency
(minimum support)
• Our Contributions
– Use of Regular Expressions (REs) as a user-level tool for
specifying interesting pattern structures (natural syntax +
expressive power), e.g.,
travel (lodging | NY | NY city)* (hotels | hotels and motels)
– Novel algorithms (SPIRIT) for pushing REs inside the pattern
mining loop
• Differ in the degree to which the RE is enforced
• Interesting insights into incorporating complex constraints in ad-hoc data
mining
Lucent Technologies, proprietary and confidential
Problem Formulation
• Sequence s = <s1,…, sn> contains pattern t = <t1,…,tm> if
for some j < k < … < p : t1 = sj, t2 = sk, …, tm = sp
s=<1,2,4,5,7,9>
t =<1,
p=<
5,
2,4,5
9>
>
• Regular expression R: language L(R) over the alphabet of
sequence / pattern elements
R = 1*(2 2 | 2 3 4 | 4 4 )
L(R) = zero or more 1’s followed by 2 2 or 2 3 4 or 4 4
Lucent Technologies, proprietary and confidential
Problem Formulation (contd.)
• Given: DB of sequences D , and (user-specified) minimum
support minsup and RE constraint R
• Find: All patterns that are contained in at least minsup
sequences of D and belong to L(R)
Extensions : sequences of itemsets , distance constraints
Lucent Technologies, proprietary and confidential
Definitions
• Well-known fact: REs are equivalent to DFAs
R
1*(2 2 | 2 3 4 | 4 4)
a
Start state
A(R)
2
1
2
b
4
3
c
4
d
Accept state(s)
• A sequence s is :
– Legal wrt state b iff it defines a path in A(R) starting from b
– Valid wrt state b iff it defines a path to an accept state of A(R)
starting from b
– Valid iff it defines a path from the start state to an accept state
of A(R) (i.e., s belongs to L(R) )
Lucent Technologies, proprietary and confidential
Conventional Apriori Mining
Apriori(D, minsup)
repeat {
using F(k-1) generate set of candidate (i.e., potentially
// generation
frequent) k-sequences C(k)
P = { s C(k) : any (k-1)-subsequence of s is not in F(k-1) } // pruning
C(k) = C(k) - P
scan D counting support for all sequences in C(k)
// counting
F(k) = frequent sequences in C(k); F = F + F(k); k = k+1
} until F(k) is empty
• F(k) = frequent sequences of length k (k-sequences)
• C(k) = candidate frequent k-sequences
Lucent Technologies, proprietary and confidential
The SPIRIT Framework
SPIRIT(D , minsup , C)
C’ = relaxation of C
repeat{
using F and C’ generate
// generation
C(k) = { potentially frequent k-sequences that satisfy C’ }
P = { s C(k): s has a subsequence that satisfies C’ // pruning
and does not belong to F }
C(k) = C(k) - P
scan D counting support for sequences in C(k)
// counting
F(k) = frequent sequences in C(k); F = F +F(k); k = k+1
} until TerminatingCondition(F , C’)
output sequences in F that satisfy C
Lucent Technologies, proprietary and confidential
The SPIRIT Framework (contd.)
• Invariant: at the end of pass k , F is the set of frequent
1,2,…,k-sequences that satisfy C’
• “Strength” of the relaxation C’ determines the degree to
which C = R is enforced during mining
• Major cost = candidate counting ( proportional to |C(k)| )
• Goal:
– Exploit minsup and C to restrict |C(k)| as much as
possible
BUT, pushing C “all the way” may not be the best strategy!!
Lucent Technologies, proprietary and confidential
The SPIRIT Framework (contd.)
• SPIRIT employs two types of pruning to restrict C(k)
– Constraint-based Pruning: ensuring candidates in C(k) satisfy C’
(during generation)
– Support-based Pruning: ensuring all subsequences of a candidate
that satisfy C’ are frequent (during pruning)
• If C is anti-monotone (all subsequences of a sequence that
satisfies C also satisfy C) then C’ = C is obviously best
• If C is not anti-monotone (e.g., REs) things are not that
clear-cut
– Aggressive constraint-based pruning can have a negative impact
on support-based pruning
Lucent Technologies, proprietary and confidential
The SPIRIT Algorithms
• Specific instantiations of the framework
• Explore the entire spectrum of choices for employing RE
constraints during pattern mining
Algorithm
Relaxation C’ ( C = R )
SPIRIT(N)
all elements appear in R
Lucent Technologies, proprietary and confidential
SPIRIT(L)
Legal wrt some state of A(R)
SPIRIT(V)
Valid wrt some state of A(R)
SPIRIT(R)
Valid (C’ = C = R)
SPIRIT(N) (“naïve”)
• Employs RE in a trivial manner ( consider only patterns of
elements that appear in the RE )
• Essentially, it’s basic Apriori (only support-based pruning)
with RE applied as an afterthought
Lucent Technologies, proprietary and confidential
SPIRIT(L) (“legal”)
• F(k) = frequent k-sequences that are legal wrt some state
of A(R)
• F(k,b) = frequent k-sequences legal wrt state b
• Candidate Generation
b
s1
c
– Idea: if <s1,…,sk> is frequent and legal wrt b then <s1,…,
s(k-1)> is in F(k-1,b) and <s2,…,sk> is in F(k-1,c)
Join F(k-1)’s across all transitions of the automaton
Lucent Technologies, proprietary and confidential
SPIRIT(L) (“legal”) (contd.)
• Candidate Pruning
– A candidate k-sequence can be pruned if any of its maximal legal
subsequences is not frequent (cannot be found in F)
– We have an efficient Dynamic Programming algorithm that
works off the automaton to find such maximal legal subsequences
– More expensive than Apriori pruning, but we found CPU
overheads to be negligible compared to counting costs
(details in the paper…)
Lucent Technologies, proprietary and confidential
Legal vs. Naive
D: <1 2 3 2>
<1 1 2 2>
<2 4 3 4>
<2 3 4 3>
<1 1 2 3>
minsup = 2
2
1
a
2
b
3
c
4
d accept
4
SPIRIT(N): F(2) = { <1 1> , <1 2>, <1 3>, <2 2>, <2 3>, <2 4>, <3 4>, <4 3> }
C(3) = { <1 1 1> , <1 1 2>, <1 1 3>, <1 2 2>, <1 2 3>,
<2 2 2>, <2 2 3>, <2 2 4>, <2 3 4>, <2 4 3> }
SPIRIT(L):
F(2) = { <1 1> , <1 2>, <2 2>, <2 3>, <3 4>}
C(3) = { <1 1 1> , <1 1 2>, <1 2 2>, <1 2 3>, <2 3 4> }
Lucent Technologies, proprietary and confidential
SPIRIT(V) (“valid”)
• F(k) = frequent k-sequences that are valid wrt some state
of A(R)
• F(k,b) = frequent k-sequences valid wrt state b
• Candidate Generation
b
s1
accept
c
– idea: if <s1,…,sk> is frequent and valid wrt b then
<s2,…,sk> is in F(k-1,c)
Extend F(k-1)’s “backwards” across all transitions
• Candidate Pruning
– a version of our Dynamic Programming strategy developed for
SPIRIT(L) can be used
Lucent Technologies, proprietary and confidential
Valid vs. Legal
D: <1 2 3 2>
<1 1 2 2>
<2 4 3 4>
<2 3 4 3>
<1 1 2 3>
SPIRIT(L):
minsup = 2
2
1
a
2
b
3
c
4
d accept
4
F(2) = { <1 1>, <1 2>, <2 2>, <2 3>, <3 4> }
C(3) = { <1 1 1> , <1 1 2>, <1 2 2>, <1 2 3>, <2 3 4> }
SPIRIT(V):
F(2) = { <2 2>, <3 4> }
C(3) = { <1 2 2>, <2 3 4> }
Lucent Technologies, proprietary and confidential
SPIRIT(R) (“regular”)
• F(k) = frequent k-sequences that are valid ( satisfy R )
• Candidate Generation
– No efficient mechanism -- “brute-force” enumeration of paths in
the automaton
– Some optimizations (e.g., exploiting cycles) to reduce
computation
• Candidate Pruning
– A version of our Dynamic Programming strategy is once again
applicable
Lucent Technologies, proprietary and confidential
Experience with SPIRIT
• Implemented and tested over real-life and synthetic data
• Speedups of more than an order of magnitude possible if
constraints are exploited
• SPIRIT(V) offers a nice tradeoff and provides good
performance over a range of RE constraints
• BUT…
– In general, the choice of the “best” strategy depends critically on
the data-set and RE-constraint characteristics
• E.g., RE “selectivity”
– More on this later…
Lucent Technologies, proprietary and confidential
Outline
• Motivation for mining with constraints
• Algorithmic solutions
– Sequential patterns – SPIRIT
– Constraint-based decision-tree induction
• Application: Model-Based Semantic Compression SPARTAN
• Future directions
– Cost-based constraint pushing
– Mining continuous data streams
• Conclusions
Lucent Technologies, proprietary and confidential
Decision Trees 101
• Decision tree: Tree-structured predictor for a specified categorical
class label using other data attributes
• Built based on training data set
Credit Analysis
salary < 20000
s a la ry
e d u c a t io n
la b e l
10000
h ig h s c h o o l
re je c t
40000
u n d e r g ra d u a t e
ac c ept
education in graduate
15000
u n d e r g ra d u a t e
re je c t
yes
75000
g ra d u a t e
ac c ept
18000
g ra d u a t e
ac c ept
yes
accept
• For numerical class labels: Regression tree
Lucent Technologies, proprietary and confidential
no
accept
no
reject
Conventional Decision-Tree
Induction
• Building phase
– Recursively split nodes using “best” splitting attribute
for node
• E.g., using entropy of the split
• Pruning phase
– Smaller imperfect decision tree generally achieves better
accuracy
– Prune leaf nodes recursively to prevent overfitting
• Typically based on MDL cost of subtrees
Lucent Technologies, proprietary and confidential
The Need for Constraint Support
• MDL-optimal decision tree can be
– Complex with hundreds or thousands of nodes
– Difficult to comprehend
• Users may only be interested in a rough picture of the
patterns in their data
– A simple, quick-to-build, approximate decision tree is much
more useful
Lucent Technologies, proprietary and confidential
Our Work [KDD’00,DMKDJ’03]
• Efficient algorithms for building MDL-optimal decision trees under
– Size Constraints: the number of tree nodes is <= k
– Error Constraints: the total number of misclassified records <= e
• Applying the constraints after tree-building is simple, but …
– A lot of effort is wasted on parts of the tree that are pruned later
• Our algorithms: Integrate constraints in the tree-building phase
– Reduce I/O and improve performance by early removal of useless
parts of the tree
– Guarantee that the final tree is identical to that obtained by
“naïve” late pruning
Lucent Technologies, proprietary and confidential
Key Ideas
• Basic problem: Building works top-down, pruning works bottom-up
– How can we stop expanding “useless” nodes early? Don’t really
know the true MDL cost of the tree underneath them…
• Key Intuition: Branch & Bound based on lower bounds on subtree
cost computed for “yet-to-expand” nodes
tree-building
– Details in the paper…
– Performance speedup of up to an
order of magnitude wrt “late pruning”
N
LB( MDLcost(N, k) )
• Similar ideas for constrained regression-trees
Lucent Technologies, proprietary and confidential
Outline
• Motivation for mining with constraints
• Algorithmic solutions
– Sequential patterns – SPIRIT
– Decision trees
• Application: Model-Based Semantic Compression of
Massive Network-Data Tables - SPARTAN
• Future directions
– Cost-based constraint pushing
– Mining continuous data streams
• Conclusions
Lucent Technologies, proprietary and confidential
Networks Create Data!
• To effectively manage their networks Internet/Telecom Service
Providers continuously gather utilization and traffic data
• Managed IP network elements collect huge amounts of traffic data
– Switch/router-level monitoring (SNMP, RMON, NetFlow, etc.)
– Typical IP router: several 1000s SNMP counters
– Service-Level Agreements (SLAs), Quality-of-Service (QoS) guarantees =>
finer-grain monitoring (per IP flow!!)
• Telecom networks: Call-Detail Records (CDRs) for every call
– CDR = 100s bytes of data with several 10s of fields/attributes (e.g., endpoint
exchanges, timestamps, tarifs)
• End Result: Massive collections of Network-Management (NM)
data (can grow in the order of several TeraBytes/year!!)
Lucent Technologies, proprietary and confidential
Compressing Massive Tables: A New
Direction in Data Compression
• Example table: IP-session measurements
Protocol
http
http
http
http
http
ftp
ftp
ftp
Duration Bytes
12
20K
16
24K
15
20K
19
40K
26
58K
27
100K
32
300K
18
80K
Packets
3
5
8
11
18
24
35
15
• Benefits of data compression are well established
– Optimize storage, I/O, network bandwidth (e.g., data transfers, disconnected
operation for mobile users) over the lifetime of the data
– Faster query processing over synopses
Lucent Technologies, proprietary and confidential
Compressing Massive Tables: A New
Direction in Data Compression
• Several generic compression tools and algorithms(e.g., gzip,
Huffman, Lempel-Ziv)
– Syntactic methods: operate at the byte level, view data as large
byte string
– Lossless compression only
• Effective compression of massive alphanumeric tables
– Need novel methods that are semantic: account for and exploit
the meaning and data dependencies of attributes in the table
– Lossless or lossy compression: flexible mechanisms for users
to specify acceptable information loss
Lucent Technologies, proprietary and confidential
SPARTAN : A Model-Based
Semantic Compressor [SIGMOD’01]
• New, general paradigm: Model-Based Semantic Compression
(MBSC)
– Extract Data Mining models from the data table
– Use the extracted models to construct an effective compression
plan
– Lossless or lossy compression (w/ guaranteed per-attribute error
bounds)
• SPARTAN system: Specific instantiation of MBSC framework
– Key Idea: use carefully-selected collection of Classification and
Regression Trees (CaRTs) to capture strong data correlations and
predict values for entire columns
Lucent Technologies, proprietary and confidential
SPARTAN Example CaRT Models
error=0
Protocol
http
http
http
http
http
ftp
ftp
ftp
error<=3
Duration Bytes
12
20K
16
24K
15
20K
19
40K
26
58K
27
100K
32
300K
18
80K
Packets
3
5
8
11
18
24
35
15
Packets > 10
yes
error = 0
no
Bytes > 60K
yes
Protocol = http
no
Protocol = ftp
Protocol = http
Packets > 16
• Can use two compact trees (one
decision, one regression) to eliminate
two data columns (predicted attributes)
Lucent Technologies, proprietary and confidential
yes
Duration = 29
error <= 3
no
Duration = 15
(outlier: Packets = 11)
SPARTAN Compression Problem
Formulation
• Given: Data table T over set of attributes X and per-attribute error
tolerances
• Find: Set of attributes P to be predicted using CaRT models (and
corresponding CaRTs+outliers) such that
– Each CaRT uses only predictor attributes in X-P
– Each attribute in P is predicted within its specified tolerance
– The overall storage cost is minimized
• Non-trivial problem!
– Space of possible CaRT predictors is exponential in the number of attributes
– CaRT construction is an expensive process (multiple passes over the data)
Lucent Technologies, proprietary and confidential
SPARTAN Architecture
Lucent Technologies, proprietary and confidential
SPARTAN’s CaRTSelector
• “Heart” of the SPARTAN semantic-compression engine
– Uses the constructed Bayesian Network on T to drive the construction and
selection of the “best” subset of CaRT predictors
• Output: Subset of attributes P to be predicted (within tolerance) and
corresponding CaRTs
• Complication: An attribute in P cannot be used as a predictor for
other attributes
– Otherwise, errors will compound!!
• Hard optimization problem -- Strict generalization of Weighted
Maximum Independent Set (WMIS) (NP-hard!!)
• Two solutions
– Greedy heuristic
– Novel heuristic based on WMIS approximation algorithms
Lucent Technologies, proprietary and confidential
The CaRTBuilder Component
• Input: Target predicted attribute Xp; predictor attributes
{X1,…,Xk}; and error tolerance for Xp
• Output: Minimum-storage-cost CaRT for Xp using {X1,…,Xk} as
predictors, within the specified error tolerance
• Repeated calls from CaRTSelector: Efficiency is key!
• Efficient CaRT construction algorithms that exploit error tolerances
– Integrated tree building and error-constraint pruning based on our
cost “lower bounding” ideas
• To keep compression times low, SPARTAN models are built on
samples of the data
Lucent Technologies, proprietary and confidential
Experience with SPARTAN
• Significant improvements in compression ratio over gzip
– Factor of 3 for error tolerances in 5-10% range
• Compression times are also quite reasonable
– E.g., >500K tuples in a few minutes
– Model induction(s) over small samples of the data
• Despite the use of samples, exploiting CaRT constraints and
integrated building/constraint-pruning makes a difference!
– SPARTAN spends 50-75% of its time building CaRTs
– Integrated algorithms can improve performance by 25-30%
• Constraint-based data mining in the context of a “bigger-picture”
problem
Lucent Technologies, proprietary and confidential
Outline
• Motivation for mining with constraints
• Algorithmic solutions
– Sequential patterns – SPIRIT
– Decision trees
• Application: Model-Based Semantic Compression SPARTAN
• Future directions
– Cost-based constraint pushing
– Mining continuous data streams
• Conclusions
Lucent Technologies, proprietary and confidential
Cost-based Constraint Pushing
• Support for user/application-defined constraints is a “must” for
large-scale, ad-hoc data mining
• Interesting early work on simple constraint forms, e.g., for
associations [J. Han et al.]
• Need effective support for more complex constraint forms
– E.g., structural constraints for mining structured/semi-structured
data (sequences, trees, graphs, …)
Lucent Technologies, proprietary and confidential
Cost-based Constraint Pushing (contd.)
• SPIRIT: Broad spectrum of available choices for dealing with
complex, non-antimonotone constraints
• Winning strategy? Not always obvious, and clearly dependent on
constraint and data characteristics
– E.g., “selectivity” of the RE constraint over the data sequences
– Push highly-selective REs more aggressively to maximize the
benefit from constraint-based pruning
• Best strategy may in fact be adaptive
– Different degrees of “constraint pushing” at different stages of
the search
Lucent Technologies, proprietary and confidential
Cost-based Constraint Pushing (contd.)
• Problem very reminiscent of query optimization in DBMSs
– Mining operator with constraints = user query, processing strategy = query
execution plan
• Need a principled, cost-based approach to effectively handling
complex data-mining constraints
– “Optimizer” to evaluate alternatives and pick a “best” plan
• Several issues need to be fundamentally rethought for
pattern/inductive DBs
– Appropriate statistical summaries (“histograms” for mining), plan
optimization strategies, adaptive optimization/execution, …
• Initial steps: “Adaptive SPIRIT” technique [J-F Boulicaut et al.]
• Still, lots more interesting problems here!!
Lucent Technologies, proprietary and confidential
Mining Continuous Data Streams
• Stream-query processing arises naturally in Network Management
– Data records arrive continuously, at high rates from different parts of the
network
– Shipped for archival storage off-site (expensive access)
– Real-time reaction - data-analysis algorithms can only look at the tuples once,
in the fixed order of arrival and with limited available memory
Network Operations
Center (NOC)
Measurements
Alarms
R1
R2
R3
IP Network
Lucent Technologies, proprietary and confidential
Mining Data Streams (contd.)
• A data stream is a (massive) sequence of records: r1 ,..., rn
– General model permits deletion of records as well
Data Streams
Stream Synopses
(in memory)
Stream
Analysis
Engine
(Approximate)
Patterns/Models
• Requirements for stream synopses
– Single Pass: Each record is examined at most once, in fixed (arrival) order
– Small Space: Log or poly-log in data stream size
– Real-time: Per-record processing time (to maintain synopses) must be low
Lucent Technologies, proprietary and confidential
Mining Data Streams (contd.)
• Several interesting recent results: Clustering [Guha et al.], Decision
trees [Domingos et al.], Frequent itemsets [Manku et al.], …
– Overview in data streaming tutorial with J. Gehrke, R. Rastogi
[SIGMOD/VLDB/KDD’02]
• Still, tons of open problems
– Mining structured/semi-structured data streams (sequences, trees,
graphs) is wide open
– How can constraints on the models of interest be exploited in the
streaming context?
• Reduce synopsis size, improve error guarantees, etc…
– “Sliding window” stream mining?
• How do you expire the effects of stale data from your models?
• CVFDT [Domingos et al.] takes some initial steps in this direction
Lucent Technologies, proprietary and confidential
Conclusions
• Personal, biased perspective on constraint-based data mining through
a brief summary of some of my own work in the area
– Sequential patterns, decision/regression trees, and application to model-based
semantic compression
• Similarly-biased discussion of challenging directions for future work
– Principled, cost-based constraint pushing, and data-stream mining
• Effective support for constraints is essential for next-generation adhoc data mining, analysis, and exploration systems
– So far, we’ve only scratched the surface…
– As we move forward towards “Pattern/Inductive DBMSs” (and, DSMSs)
research challenges abound!!
Lucent Technologies, proprietary and confidential
Thank you!
http://www.bell-labs.com/~minos/
[email protected]
Lucent Technologies, proprietary and confidential
The NEMESIS Project
• Massive NM data sets “hide” knowledge that is crucial to key
management tasks
– Application/user profiling, proactive/reactive resource management & traffic
engineering, capacity planning, etc.
• Data Mining research can help!
– Develop novel tools for the effective storage, exploration, and
analysis of massive Network-Management data
• Several challenging research themes
– semantic data compression, approximate query processing, XML,
mining
models for event correlation and fault analysis, network-recommender
systems, . . .
• Loooooong-term goal :-)
– Intelligent, self-tuning, self-”healing” communication networks
Lucent Technologies, proprietary and confidential
SPARTAN’s DependencyFinder
• Input: Random sample of input table T
• Output: A Bayesian Network (BN) identifying strong dependencies
and “predictive correlations” among T’s attributes
• BN Semantics: An attribute is independent of all its non-descendants
given its parents
Season?
Sprinkler on?
It rained?
Ground wet?
Ground slippery?
• Use BN to restrict (huge!) search space of possible CaRT models:
Build CaRTs using “neighboring” attributes as predictors
• SPARTAN uses an (enhanced) constraint-based BN builder
Lucent Technologies, proprietary and confidential
The Greedy CaRTSelector
1. MatSet = {}, PredSet = {}
2. For each attribute Xi in the Bayesian
Network in topological sort order
if Xi has no parents
add Xi to MatSet
else {
2.1 build CaRT for Xi with the
attributes in MatSet as predictors
2.2 if materializationCost(Xi) >
CT * predictionCost(Xi)
add Xi to PredSet
else add Xi to MatSet
}
3. PredSet is the set of attributes to be predicted
Lucent Technologies, proprietary and confidential
X5
X2
X1
X3
X4
X6
X7
The MaxIndependentSet (MIS)
CaRTSelector
• What is wrong with Greedy?
– Myopic strategy: ignores the potential benefits of using an attribute as a
predictor for its descendants
– Effect of the relative benefit parameter CT ?!?
• MIS CaRTSelector
– Addresses both problems of Greedy -- much more “global” view of the
attribute correlations
– Exploits natural mapping of WMIS to the CaRTSelector problem
Lucent Technologies, proprietary and confidential
The MaxIndependentSet (MIS)
CaRTSelector (contd.)
• Key idea: Use a WMIS approximation algorithm iteratively to solve
different instances of WMIS
– “Weight” of a node (attribute) = materializationCost - predictionCost
– Prediction cost using the materialized “predictive neighborhood” of the node
– Each WMIS iteration improves earlier solution by moving a (“near-optimal”)
subset of nodes to the predicted set
– Stop when no improvement is possible
• Number of CaRTs built
– Greedy CaRTSelector: O(n)
– MIS CaRTSelector
Lucent Technologies, proprietary and confidential
: O(n^2/2) in the worst case, O(n logn) “on average”
The RowAggregator Component
• Input: Sub-table TM of materialized data attributes returned by the
CaRTSelector
• Output: Fascicle-based (lossy) compression scheme for TM
• Summary
– Tricky: Attribute errors in TM should not propagate through the
CaRTs to the predicted attributes
– Algorithms based on fascicle algorithms of [JMN99]
Lucent Technologies, proprietary and confidential