top- k query evaluation with probabilistic guarantees
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Top-K Query Evaluation with Probabilistic GuaranteesMartin Theobald, Gerhard Weikum, Ralf SchenkelPresenter: Avinandan Sengupta
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• Introduction to Top-k query processing• The threshold algorithm and its variants• Are we solving the right problem?• A probabilistic algorithm• Implementation Details• Results• Conclusion
Presentation Outline
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• Introduction to Top-k query processing• The threshold algorithm and its variants• Are we solving the right problem?• A probabilistic algorithm• Implementation Details• Results• Conclusion
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Data and a Query
Scrip ID Earnings Per Share
P/Eratio
β ... Average Market
Cap (B$)
SNPS 1.27 17.63 0.69 ... 3.27
IBM 12.28 13.85 0.72 ... 200
... … … … ... ...
INFY 2.72 19.51 1.17 30.4
MSFT 2.70 9.32 1.03 210
GOOG 27.73 19.33 1.13 173
Top 10 midcap
stocks with low β
Hypothetical DB of NASDAQ traded stocks. Data collated from Google Finance
Attributes
Objects
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P/ERatio
(norm)
INFY: 1
GOOG: 0.99
SNPS: 0.90
IBM: 0.70
...
MSFT: 0.47
β-1
(norm)
SNPS: 1
IBM: 0.96
MSFT: 0.67
GOOG: 0.61
...
INFY: 0.59
Average MarketCap (B$)
SNPS: 1
INFY : 0.80
...
GOOG: 0.05
IBM: 0.07
MSFT: 0.08
PEj/Highest PE (β-1j /max(β-1
j)) Grades based on how close the market cap is to the midcap median; normalized
Midcap median 4.5B≅
Hypothetical Graded Lists(made fit for consumption by Top-k processors)
f = 0.5*P/E + 1.0*β-1 + 1.0*MCap
weights
Aggregate function
normalization
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Top-kList
SNPS, X
INFY, Y
...
GOOG, Z
Top-k resultsP/E
Ratio(norm)
INFY: 1
GOOG: 0.99
SNPS: 0.90
IBM: 0.70
...
MSFT: 0.47
β-1
(norm)
SNPS: 1
IBM: 0.96
MSFT: 0.67
GOOG: 0.61
...
INFY: 0.59
Average MarketCap (B$)
SNPS: 1
INFY : 0.80
...
GOOG: 0.05
IBM: 0.07
MSFT: 0.08
Top-k Processor
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Presentation Outline
• Introduction to Top-k query processing• The threshold algorithm and its variants• Are we solving the right problem?• A probabilistic algorithm• Implementation Details• Results• Conclusion
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Fagin’s Threshold Algorithm (TA)
• Access the n lists in parallel.• As an object oi is seen, perform a random access
to the other lists to find the complete score for oi.• Do the same for all objects in the current row.• Now compute the threshold τ as the sum of
scores in the current row.• The algorithm stops after k objects have been
found with a score above τ.
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TA with No Random Access (TA-NRA)
• Access the n lists in parallel.• For an item a, compute its (B)est score:
Ba = f { f {scorej | j ∈ seen-attributes(a)}, f {highk | k ∉ seen-attributes(a)}}
highk = last seen score for the kth attribute
and its (W)orst scoreWa = f { f {scorej | j ∈ seen-attributes(a)}, f {0 | k ∉ seen-attributes(a)}}
• Halt when k distinct objects have been seen and there is no object m outside the Top-k list whose Bm ≥ Wk – this means that we also maintain a table of all seen objects with their W/B
scores
Top-kList
SNPS, W1, B1
INFY, W2, B2
...
GOOG, Wk, Bk
Running Top-k list; contains the k objectswith largest W values; ties broken with B values
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Issues with TA and TA-NRA
• High space-time costs• Overly conservative
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Presentation Outline
• Introduction to Top-k query processing• The threshold algorithm and its variants• Are we solving the right problem?• A probabilistic algorithm• Implementation Details• Results• Conclusion
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Are we solving the right problem?
• Is random access possible in most common scenarios?– Web content– XML data, hierarchical data sets
• Does the user need an exact top-k query result?– Or is she satisfied with an approximation?
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How about an approximate solution?
• Can we remove candidates (objects that we think can make it to the top-k list) from consideration early on in the process?– Quickly reach solution
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Pictorially...
Source: www.mpi-inf.mpg.de/~mtb/pub/imprs-topk.pdf (author’s webpage)
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• Introduction to Top-k query processing• The threshold algorithm and its variants• Are we solving the right problem?• A probabilistic algorithm• Implementation Details• Results• Conclusion
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Probabilistic TA-NRA - 1
• Predict the total score of a item for which a partial score is known
• Avoid the overly conservative best-score/worst-score bounds of the original TA-NRA– Instead, calculate the probability that the total
score of the item exceeds a threshold (making the item interesting for the top-k result)
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Probabilistic TA-NRA - 2
• If this probability is sufficiently low (below a threshold), drop the item from the candidate list.
• The probabilistic prediction involves computing the convolution of the score distributions of different index lists.
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Score Distribution of Lists - How?
β-1
(norm)
SNPS: 1
IBM: 0.96
MSFT: 0.67
GOOG: 0.61
...
INFY: 0.59
score0.59 1.0
Median 0.65
Parameter fitting curve fitting
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What it is and What it is not
• Probabilistic guarantees are not about query run-times but about query result quality
• Probabilistic guarantees refers to the approximation of the top-k ranks in a completely scored and exactly ranked result set
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The Math
Source: www.mpi-inf.mpg.de/~mtb/pub/imprs-topk.pdf (author’s webpage)
Set of seen attributes for
an object
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More Math...
Source: www.mpi-inf.mpg.de/~mtb/pub/imprs-topk.pdf (author’s webpage)
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• Introduction to Top-k query processing• The threshold algorithm and its variants• Are we solving the right problem?• A probabilistic algorithm• Implementation Details• Results• Conclusion
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What distributions to consider?
• Uniform distribution– simplest assumptions– convolutions based on moment-generating functions with
generalized Chernoff-Hoeffding bounds• Poisson estimations– efficiently evaluated, provides a reasonable fit for tf*idf
based score distributions for Web corpora• Histograms– when above methods fail– Involves non-trivial computation (done offline per list)
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Solving Convolutions? Difficult
• When the PDF is a uniform distribution, its solution becomes difficult– Use alternate techniques other than convolution– Off-load computation to available probabilistic
engines – OpenMaple, etc
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Queue Management
Source: http://www.mpi-inf.mpg.de/~mtb/pub/imprs-topk-xml_poster.pdf (author’s webpage)
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• Introduction to Top-k query processing• The threshold algorithm and its variants• Are we solving the right problem?• A probabilistic algorithm• Implementation Details• Results• Conclusion
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Results
Source: www.mpi-inf.mpg.de/~mtb/pub/imprs-topk.pdf (author’s webpage)
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Performance as a function of ε
Source: Paper
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Precision of probabilistic predictors for tf*idf, Uniform-, and Zipf-distributed scores
Source: Paper
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• Introduction to Top-k query processing• The threshold algorithm and its variants• Are we solving the right problem?• A probabilistic algorithm• Implementation Details• Results• Conclusion
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• New algorithms were developed based on probabilistic score predictions– Trade-off a small amount of top-k result quality for a
drastic reduction of sorted accesses• Intelligent management of priority queues for
efficient implementation was presented• Assumptions were made regarding the aggregation
function to be summation• Future work to be based on ranked retrieval of XML
data and integrating into XXL search engine
Conclusion
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Thanks!
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