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\paragraph { Outline of 1st ICDT Introduction Submission}
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\begin { outline} [enumerate]
\1 Problem Introduction and Background
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\2 Set-\abbrPDB notation, concepts, common (known) results \isIncluded { I. B. vi.}
\3 \notIncluded { Dichotomy} \isIncluded { Can be included in I. B. vi.}
\3 Exact Computation \sharpphard \isIncluded { I. B. vi.}
\2 Formal definition of expected result multiplicity \isIncluded { Can be included in I. C. ii. a.}
\2 \notIncluded { Example} \isIncluded { Can be used potentially sometime after I. C. ii. a.; maybe around D.}
\3 Assumed setting of { \emph set inputs} \isIncluded { I. D.}
\3 \notIncluded { Example based on explaining and motivating formal definition of expected result multiplicity}
\1 \notIncluded { Discussion of set-\abbrPDB \xplural } \isIncluded { Perhaps this might be useful around I. B. vi.}
\2 \notIncluded { Lineage from PosBool$ [ \vct { X } ] $ }
\2 \notIncluded { Encoding of possible worlds via $ \vct { X } $ }
\2 \notIncluded { Computing probability can be done using only the lineage}
\1 \notIncluded { Discussion of bag-\abbrPDB \xplural } \isIncluded { Might be useful around I. C.}
\2 \notIncluded { Link to $ \domN [ \vct { X } ] $ }
\2 \notIncluded { Link to computing the expected count of a lineage polynomial}
\2 \notIncluded { Example to illustrate computing an expected count over a lineage polynomial}
\1 \isIncluded { Computing expected multiplicity for an \abbrSOP representation versus a factorized representation -- I. B. iii. and I. B. iv.}
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\2 Linear for \abbrSOP
\2 Introduce the problem by asking if it's linear in the size of the representation for factorized representation produced by such query optimizations as projection push-down.
\2 State our theoretical results (informally) that it is not linear in general
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\1 \notIncluded { Contributions}
\2 \notIncluded { Hardness results for the expected multiplicity problem}
\3 \notIncluded { Reduction to counting $ \kElem $ -matchings}
\2 \isIncluded { Introduce our approximation algorithm and its guarantees -- I. B. v.}
\2 \notIncluded { Generalization to bag-\abbrPDB \xplural }
\2 \notIncluded { Result over $ \raPlus $ queries }
\2 \isIncluded { Higher moments I. C. e.}
\1 \notIncluded { Overview of our techniques}
\2 \notIncluded { Informal introduction to $ \rpoly $ with example}
\2 \notIncluded { Definition of reduced polynomial}
\2 \notIncluded { Equivalence of $ \rpoly $ and computing $ \expct $ }
\2 \notIncluded { Further details into the technique of obtaining our hardness result}
\1 \notIncluded { Paper organization}
\2 \notIncluded { Also includes evaluation semantics figure}
\end { outline}