Added defintion of reduction and example.
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@ -5,7 +5,7 @@
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\secrev{
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This work explores the problem of computing the expectation of a tuple's multiplicity in a bag \abbrTIDB.
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Our analysis specifically considers a restricted form of bag \abbrTIDB which we call a \abbrCTIDB. A \abbrCTIDB,
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$\pdb = \inparen{\worlds, \bpd}$ is a bag of tuples such that each tuple models $\bound$ disjoint events, where each such set of disjoint events is itself independent of the others. A tuple in $\pdb$ has a multiplicity of at most $\bound$. The element $\worlds$ is the set of all worlds and $\bpd$ is the product distribution over the set of all worlds. A given world $\vct{W} = \inset{0,\ldots, \bound}$ can be interpreted for each $\vct{W}\pbox{i}= j$ as denoting that tuple $\tup_i$ appears $j$ times in world $\vct{W}$ for $j \in [0, \bound]$. The resulting product distribution can then be expressed across the $\numvar$ base tuples of the encoding as $\prob_{i, j} = \probOf\pbox{W\pbox{i} = j}$, where each distribution is independent for $i \in [\tupsetsize]$.
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$\pdb = \inparen{\worlds, \bpd}$ encodes a bag of uncertain tuples such that each tuple models $\bound$ disjoint events, where each such set of disjoint events is itself independent of the others. A tuple in $\pdb$ has a multiplicity of at most $\bound$. The set of all worlds is encoded in $\worlds$, which is the set of all vectors of length $\abs{\tupset}$ such that each index corresponds to a distinct $\tup \in \tupset$ storing its multiplicity. $\bpd$ is the product distribution over the set of all worlds. A given world $\worldvec = \inset{0,\ldots, \bound}$ can be interpreted for each $\worldvec\pbox{i}= j$ as denoting that tuple $\tup_i$ appears $j$ times in world $\worldvec$ for $j \in [0, \bound]$. The resulting product distribution can then be expressed across the $\numvar$ base tuples of the encoding as $\prob_{i, j} = \probOf\pbox{W\pbox{i} = j}$, where each distribution is independent for $i \in [\tupsetsize]$.
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}
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%\mypar{For a later section}
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%\sout{
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@ -20,6 +20,7 @@ We can formally state this problem as:
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\begin{Problem}\label{prob:expect-mult}
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Given a \abbrCTIDB $\pdb = \inparen{\worlds, \bpd}$, $\raPlus$ query $\query$, and result tuple $\tup$, compute the expected multiplicity of $\tup$: $\expct_{\randDB\sim\bpd}\pbox{\query\inparen{\randDB}\inparen{\tup}}$.
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\end{Problem}
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\AH{I \emph{think} we use $\randDB$ to denote something different in one of the proofs. Have to keep an eye open for this to avoid overloading notation.}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%Example of product distribution of c-TIDB
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@ -117,7 +118,7 @@ Set query evaluation semantics over $1$-\abbrTIDB\xplural have been studied exte
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One of the main theoretical points in this work is to discern whether or not bag \abbrCTIDB query semantics are indeed linear in the runtime of an equivalent deterministic query. If this is true, then this would open up the way for deployment of \abbrCTIDB\xplural in practice.
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Unfortunately, we prove that this is not the case. To analyze this question we denote by $\timeOf{}^*(Q,\pdb)$ the optimal runtime complexity of computing ~\cref{prob:expect-mult} over \abbrCTIDB $\pdb$. Let $\gentupset$ denote the set of tuples in $\pdb$, i.e.,
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\begin{Definition}[$\gentupset$]
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Define $\gentupset$ to be the set of tuples appearing across all the possible worlds of a $\abbrCTIDB$, formally $\gentupset = \inset{\tup ~|~ \forall \db \in \pdb, \tup \in \db}$. When a specific $\pdb = \inparen{\worlds, \bpd}$ is being referred to, we will use $\tupset$ to denote the set of tuples.
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Define $\gentupset$ to be the set of tuples appearing across all the possible worlds of a $\abbrCTIDB$, formally $\gentupset = \inset{\tup_i ~|~ \forall \worldvec \in \worlds,~\forall i \in \abs{\tupset}:~\worldvec\pbox{i} > 0}$. When a specific $\pdb = \inparen{\worlds, \bpd}$ is being referred to, we will use $\tupset$ to denote the set of tuples.
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\end{Definition}
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Let $\qruntime{\query, \dbbase}$ be the optimal runtime (with some caveats; discussed in~\cref{sec:gen}) of query $\query$ on a comparable deterministic database $\dbbase$ defined next.
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@ -199,7 +200,97 @@ is equivalent to computing \Cref{prob:bag-pdb-poly-expected} (see \Cref{prop:exp
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}
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\secrev{
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\input{two-step-model}
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\subsection{Our Machinery}
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\mypar{Lower Bound Proof Techniques}
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All of our results rely on working with a {\em reduced} form of the lineage polynomial $\poly$. In fact, it turns out that for the $1$-\abbrCTIDB case, computing the expected multiplicity (over bag query semantics) is {\em exactly} the same as evaluating this reduced polynomial over the probabilities that define the \abbrTIDB. This is also true when the query input(s) is a block independent disjoint probabilistice database (with tuple multiplicity of at most $1$), which we refer to as a $1$-\abbrBIDB. For our results to be applicable to \abbrCTIDB\xplural, we introduce the following reduction.
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\begin{Definition}
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Any \abbrCTIDB $\pdb$, can be reduced to an equivalent $1$-\abbrBIDB $\pdb'$ in the following manner. For each $\tup_i \in \tupset$, create a block of $\bound + 1$ disjoint \abbrBIDB tuples in $\pdb'$ such that each tuple in the newly formed block is mapped to its own boolean variable $X_{i, j}$ for $i \in \abs{D}$ and $j \in \pbox{c+1}$. Then, given $\worldvec \in \worlds$, the equivalent world in $\pdb'$ will set each variable $X_{i, j} = 1$ for each $\worldvec\pbox{i} = j$, while $\inparen{\text{for }\ell \neq j}$ all other $X_{i, \ell} \in \vct{X}$ of $\pdb'$ are set to $0$.
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\end{Definition}
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\begin{Example}
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Consider the $\boldsymbol{Route}$ relation of~\cref{fig:two-step} and query $\query = \project_{\text{City}_1}\inparen{\boldsymbol{Route}}$. The output relation $\query$ is $\inset{\intup{Chicago, X}, \intup{Chicago, Y}}$ and can be represented as a \abbrCTIDB $\query' = \inset{\intup{Chicago, X', 2}}$, where the following probabilities are true: $\probOf\pbox{X' = 0} = \probOf\pbox{\neg X \wedge \neg Y}$, $\probOf\pbox{X' = 1} = \probOf\pbox{\inparen{X \vee Y}\wedge\inparen{\neg X \vee \neg Y}}$, and $\probOf\pbox{X' = 2} = \probOf\pbox{X\wedge Y}$. $\query'$ can then be reduced to a $1$-\abbrBIDB by creating a block of the following disjoint tuples: $\query'' = \inset{\intup{\text{Chicago}, X'_0}, \intup{\text{Chicago}, X'_1}, \intup{\text{Chicago}, X'_2}}$ such that $\probOf\pbox{X'_i = 1} = \probOf\pbox{X' = i}$.
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\end{Example}
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\AH{Left to do for this subsubsection: read through the rest and update def and lemma; start and complete second subsubsection \textbf{Upper Bound Proof and Algorithm Techniques}.}
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Next, we motivate this reduced polynomial.
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Consider the query $\query$ defined as follows over the bag relations of \Cref{fig:two-step}:
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}
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\begin{lstlisting}
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SELECT 1 FROM OnTime a, Route r, OnTime b
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WHERE a.city = r.city1 AND b.city = r.city2
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\end{lstlisting}
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\secrev{
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It can be verified that $\poly\inparen{A, B, C, E, X, Y, Z}$ for the sole result tuple (i.e. the count) of $\query$ is $AXB + BYE + BZC$. Now consider the product query $\query^2 = \query \times \query$.
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The lineage polynomial for $Q^2$ is given by $\poly^2\inparen{A, B, C, E, X, Y, Z}$
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$$
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=A^2X^2B^2 + B^2Y^2E^2 + B^2Z^2C^2 + 2AXB^2YE + 2AXB^2ZC + 2B^2YEZC.
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$$
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By exploiting linearity of expectation, further pushing expectation through independent \abbrTIDB variables and observing that for any $\randWorld\in\{0, 1\}$, we have $\randWorld^2=\randWorld$, the expectation is
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$\expct\limits_{\vct{\randWorld}\sim\pdassign}\pbox{\poly^2\inparen{\vct{\randWorld}}}$ (where $\randWorld_A$ is the random variable corresponding to $A$, distributed by $\pdassign$).
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%Atri: Combined the the first step below with the next one to save space.
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%\begin{footnotesize}
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%\begin{multline*}
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%\expct\pbox{\randWorld_A^2}\expct\pbox{\randWorld_X^2}\expct\pbox{\randWorld_B^2} + \expct\pbox{\randWorld_B^2}\expct\pbox{\randWorld_Y^2}\expct\pbox{\randWorld_E^2} + \expct\pbox{\randWorld_B^2}\expct\pbox{\randWorld_Z^2}\expct\pbox{\randWorld_C^2} + 2\expct\pbox{\randWorld_A}\expct\pbox{\randWorld_X}\expct\pbox{\randWorld_B^2}\expct\pbox{\randWorld_Y}\expct\pbox{\randWorld_E}\\
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%+ 2\expct\pbox{\randWorld_A}\expct\pbox{\randWorld_Y}\expct\pbox{\randWorld_B^2}\expct\pbox{\randWorld_Z}\expct\pbox{\randWorld_C} + 2\expct\pbox{\randWorld_B^2}\expct\pbox{\randWorld_Y}\expct\pbox{\randWorld_E}\expct\pbox{\randWorld_Z}\expct\pbox{\randWorld_C}.
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%\end{multline*}
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%\end{footnotesize}
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%\noindent Since for any $\randWorld\in\{0, 1\}$, we have $\randWorld^2=\randWorld$,
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%then for any $k > 0$, $\expct\pbox{\randWorld^k} = \expct\pbox{\randWorld}$, which means that
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%$\expct\limits_{\vct{\randWorld}\sim\pdassign}\pbox{\poly^2\inparen{\vct{\randWorld}}}$ simplifies to:
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\begin{footnotesize}
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\begin{multline*}
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\expct\pbox{\randWorld_A}\expct\pbox{\randWorld_X}\expct\pbox{\randWorld_B} + \expct\pbox{\randWorld_B}\expct\pbox{\randWorld_Y}\expct\pbox{\randWorld_E} + \expct\pbox{\randWorld_B}\expct\pbox{\randWorld_Z}\expct\pbox{\randWorld_C} + 2\expct\pbox{\randWorld_A}\expct\pbox{\randWorld_X}\expct\pbox{\randWorld_B}\expct{\randWorld_Y}\expct\pbox{\randWorld_E} \\
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+ 2\expct\pbox{\randWorld_A}\expct\pbox{\randWorld_Y}\expct\pbox{\randWorld_B}\expct\pbox{\randWorld_Z}\expct\pbox{\randWorld_C} + 2\expct\pbox{\randWorld_B}\expct\pbox{\randWorld_Y}\expct\pbox{\randWorld_E}\expct\pbox{\randWorld_Z}\expct\pbox{\randWorld_C}.
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\end{multline*}
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\end{footnotesize}
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\noindent This property leads us to consider a structure related to the lineage polynomial.
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\begin{Definition}\label{def:reduced-poly}
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For any polynomial $\poly(\vct{X})$ corresponding to a \abbrTIDB (henceforth, \abbrTIDB-lineage polynomial),
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%\BG{Better introduce the notion of TIDB lin poly before here, then it iis more clear?},
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%Atri: Done
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define the \emph{reduced polynomial} $\rpoly(\vct{X})$ to be the polynomial obtained by setting all exponents $e > 1$ in the \abbrSMB form of $\poly(\vct{X})$ to $1$.
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\end{Definition}
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With $\poly^2\inparen{A, B, C, E, X, Y, Z}$ as an example, we have:
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\begin{align*}
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&\widetilde{\poly^2}(A, B, C, E, X, Y, Z) = AXB + BYE + BZC + 2AXBYE + 2AXBZC + 2BYEZC.
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%&\; = AXB + BYD + BZC + 2AXBYD + 2AXBZC + 2BYDZC
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\end{align*}
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Note that we have argued that for our specific example the expectation that we want is $\widetilde{\poly^2}(\probOf\inparen{A=1},$ $\probOf\inparen{B=1}, \probOf\inparen{C=1}), \probOf\inparen{E=1}, \probOf\inparen{X=1}, \probOf\inparen{Y=1}, \probOf\inparen{Z=1})$.
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%It can be verified that the reduced polynomial parameterized with each variable's respective marginal probability is a closed form of the expected count (i.e., $\expct\limits_{\vct{\randWorld}\sim\pd}\pbox{\Phi^2\inparen{\vct{X}}} = \widetilde{\Phi^2}(\probOf\pbox{A=1},$ $\probOf\pbox{B=1}, \probOf\pbox{C=1}), \probOf\pbox{D=1}, \probOf\pbox{X=1}, \probOf\pbox{Y=1}, \probOf\pbox{Z=1})$).
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\Cref{lem:tidb-reduce-poly} generalizes the equivalence to {\em all} $\raPlus$ queries on \abbrTIDB\xplural (proof in \Cref{subsec:proof-exp-poly-rpoly}).
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\begin{Lemma}\label{lem:tidb-reduce-poly}
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Let $\pdb$ be a \abbrTIDB over $n$ input tuples
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such that the probability distribution $\pdassign$ over $\vct{W}\in\{0,1\}^\numvar$ (the set of possible worlds) is induced by the probability vector $\probAllTup = \inparen{\prob_1,\ldots,\prob_\numvar}$ where $\prob_i=\probOf\inparen{W_i=1}$.
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For any \abbrTIDB-lineage polynomial
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%\BG{Term has not been introduced yet.}
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%Atri: fixed
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$\poly\inparen{\vct{X}}=\apolyqdt(\vct{X})$, it holds that $
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\expct_{\vct{W} \sim \pdassign}\pbox{\poly\inparen{\vct{W}}} = \rpoly\inparen{\probAllTup}.
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$
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\end{Lemma}
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To prove our hardness result we show that for the same $Q$ from the example above, for an arbitrary `product width' $k$, the query $Q^k$ is able to encode various hard graph-counting problems (assuming $\bigO{\numvar}$ tuples rather than the $O(1)$ tuples in \Cref{fig:two-step}).
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We do so by considering an arbitrary graph $G$ (analogous to the $Route$ relation of $\query$) and analyzing how the coefficients in the (univariate) polynomial $\widetilde{\poly}\left(p,\dots,p\right)$ relate to counts of subgraphs in $G$ that are isomorphic to various graphs with $k$ edges. E.g., we exploit the fact that the leading coefficient in $\poly$ corresponding to $\query^k$ is proportional to the number of $k$-matchings in $G$, a known hard problem in parameterized/fine-grained complexity literature.
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For an upper bound on approximating the expected count, it is easy to check that if all the probabilties are constant then $\poly\left(\prob_1,\dots, \prob_n\right)$ (i.e. evaluating the original lineage polynomial over the probability values) is a constant factor approximation. For example, using $\query^2$ from above, using $\prob_A$ to denote $\probOf\pbox{A = 1}$ (and similarly for the other variables), we can see that
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\begin{footnotesize}
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\begin{align*}
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\hspace*{-3mm}
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\poly^2\inparen{\probAllTup} &= \prob_A^2\prob_X^2\prob_B^2 + \prob_B^2\prob_Y^2\prob_E^2 + \prob_B^2\prob_Z^2\prob_C^2 + 2\prob_A\prob_X\prob_B^2\prob_Y\prob_E + 2\prob_A\prob_X\prob_B^2\prob_Z\prob_C + 2\prob_B^2\prob_Y\prob_E\prob_Z\prob_C\\
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&\leq\prob_A\prob_X\prob_B + \prob_B\prob_Y\prob_E + \prob_B\prob_Z\prob_C +
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2\prob_A\prob_X\prob_B\prob_Y\prob_E + 2\prob_A\prob_X\prob_B\prob_Z\prob_C + 2\prob_B\prob_Y\prob_E\prob_Z\prob_C
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= \rpoly\inparen{\vct{p}}
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%\inparen{0.9\cdot 1.0\cdot 1.0 + 0.5\cdot 1.0\cdot 1.0 + 0.5\cdot 1.0\cdot 0.5}^2 = 2.7225 < 3.45 = \rpoly^2\inparen{\probAllTup}
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\end{align*}
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\end{footnotesize}
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If we assume that all seven probability values are at least $p_0>0$,
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%Choose the least factor that is reduced in $\rpoly^2\inparen{\vct{X}}$, in this case $\prob_A\prob_X\prob_B$, and
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we get that $\poly^2\inparen{\vct{\prob}}$ is in the range $[\inparen{p_0}^3\cdot\rpoly\inparen{\vct{\prob}}, \rpoly\inparen{\vct{\prob}}]$.
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%
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To get an $(1\pm \epsilon)$-multiplicative approximation we uniformly sample monomials from the \abbrSMB representation of $\poly$ and `adjust' their contribution to $\widetilde{\poly}\left(\cdot\right)$.
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Here is where we introduce the specifics for producing our results and algorithm.
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\begin{itemize}
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@ -414,90 +505,6 @@ Given one circuit $\circuit$ that encodes $\apolyqdt$ for all result tuples $\tu
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mypar{Overview of our Techniques} All of our results rely on working with a {\em reduced} form of the lineage polynomial $\poly$. In fact, it turns out that for the \abbrTIDB (and \abbrBIDB) case, computing the expected multiplicity is {\em exactly} the same as evaluating this reduced polynomial over the probabilities that define the \abbrTIDB/\abbrBIDB. Next, we motivate this reduced polynomial.
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Consider the query $\query$ defined as follows over the bag relations of \Cref{fig:two-step}:
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\begin{lstlisting}
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SELECT 1 FROM OnTime a, Route r, OnTime b
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WHERE a.city = r.city1 AND b.city = r.city2
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\end{lstlisting}
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%$Q()\dlImp$$OnTime(\text{City}), Route(\text{City}, \text{City}'),$ $OnTime(\text{City}')$
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It can be verified that $\poly\inparen{A, B, C, E, X, Y, Z}$ for the sole result tuple (i.e. the count) of $\query$ is $AXB + BYE + BZC$. Now consider the product query $\query^2 = \query \times \query$.
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%\BG{$\query(\db)$ is a query result, so I changed it to this}
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%Atri: Sounds good.
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The lineage polynomial for $Q^2$ is given by $\poly^2\inparen{A, B, C, E, X, Y, Z}$:%\AR{Changed the variable $D$ to $E$ to avoid conflict with use of $D$ as a DB.}
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$$%\begin{multline*}
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%\inparen{AXB + BYE + BZC}^2\\
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=A^2X^2B^2 + B^2Y^2E^2 + B^2Z^2C^2 + 2AXB^2YE + 2AXB^2ZC + 2B^2YEZC.
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$$%\end{multline*}
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By exploiting linearity of expectation, further pushing expectation through independent \abbrTIDB variables and observing that for any $\randWorld\in\{0, 1\}$, we have $\randWorld^2=\randWorld$, the expectation is
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%\AH{If we choose to use $\pd$ in \Cref{prob:bag-pdb-poly-expected}, then we either need to follow the same convention here OR introduce the notation $\pdassign$ before using it.}
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$\expct\limits_{\vct{\randWorld}\sim\pdassign}\pbox{\poly^2\inparen{\vct{\randWorld}}}$ (where $\randWorld_A$ is the random variable corresponding to $A$, distributed as $\pdassign$).
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%Atri: Combined the the first step below with the next one to save space.
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%\begin{footnotesize}
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%\begin{multline*}
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%\expct\pbox{\randWorld_A^2}\expct\pbox{\randWorld_X^2}\expct\pbox{\randWorld_B^2} + \expct\pbox{\randWorld_B^2}\expct\pbox{\randWorld_Y^2}\expct\pbox{\randWorld_E^2} + \expct\pbox{\randWorld_B^2}\expct\pbox{\randWorld_Z^2}\expct\pbox{\randWorld_C^2} + 2\expct\pbox{\randWorld_A}\expct\pbox{\randWorld_X}\expct\pbox{\randWorld_B^2}\expct\pbox{\randWorld_Y}\expct\pbox{\randWorld_E}\\
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%+ 2\expct\pbox{\randWorld_A}\expct\pbox{\randWorld_Y}\expct\pbox{\randWorld_B^2}\expct\pbox{\randWorld_Z}\expct\pbox{\randWorld_C} + 2\expct\pbox{\randWorld_B^2}\expct\pbox{\randWorld_Y}\expct\pbox{\randWorld_E}\expct\pbox{\randWorld_Z}\expct\pbox{\randWorld_C}.
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%\end{multline*}
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%\end{footnotesize}
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%\noindent Since for any $\randWorld\in\{0, 1\}$, we have $\randWorld^2=\randWorld$,
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%then for any $k > 0$, $\expct\pbox{\randWorld^k} = \expct\pbox{\randWorld}$, which means that
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%$\expct\limits_{\vct{\randWorld}\sim\pdassign}\pbox{\poly^2\inparen{\vct{\randWorld}}}$ simplifies to:
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\begin{footnotesize}
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\begin{multline*}
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\expct\pbox{\randWorld_A}\expct\pbox{\randWorld_X}\expct\pbox{\randWorld_B} + \expct\pbox{\randWorld_B}\expct\pbox{\randWorld_Y}\expct\pbox{\randWorld_E} + \expct\pbox{\randWorld_B}\expct\pbox{\randWorld_Z}\expct\pbox{\randWorld_C} + 2\expct\pbox{\randWorld_A}\expct\pbox{\randWorld_X}\expct\pbox{\randWorld_B}\expct{\randWorld_Y}\expct\pbox{\randWorld_E} \\
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+ 2\expct\pbox{\randWorld_A}\expct\pbox{\randWorld_Y}\expct\pbox{\randWorld_B}\expct\pbox{\randWorld_Z}\expct\pbox{\randWorld_C} + 2\expct\pbox{\randWorld_B}\expct\pbox{\randWorld_Y}\expct\pbox{\randWorld_E}\expct\pbox{\randWorld_Z}\expct\pbox{\randWorld_C}.
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\end{multline*}
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\end{footnotesize}
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\noindent This property leads us to consider a structure related to the lineage polynomial.
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\begin{Definition}\label{def:reduced-poly}
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For any polynomial $\poly(\vct{X})$ corresponding to a \abbrTIDB (henceforth, \abbrTIDB-lineage polynomial),
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%\BG{Better introduce the notion of TIDB lin poly before here, then it iis more clear?},
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%Atri: Done
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define the \emph{reduced polynomial} $\rpoly(\vct{X})$ to be the polynomial obtained by setting all exponents $e > 1$ in the \abbrSMB form of $\poly(\vct{X})$ to $1$.
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\end{Definition}
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With $\poly^2\inparen{A, B, C, E, X, Y, Z}$ as an example, we have:
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\begin{align*}
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&\widetilde{\poly^2}(A, B, C, E, X, Y, Z) = AXB + BYE + BZC + 2AXBYE + 2AXBZC + 2BYEZC.
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%&\; = AXB + BYD + BZC + 2AXBYD + 2AXBZC + 2BYDZC
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\end{align*}
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Note that we have argued that for our specific example the expectation that we want is $\widetilde{\poly^2}(\probOf\inparen{A=1},$ $\probOf\inparen{B=1}, \probOf\inparen{C=1}), \probOf\inparen{E=1}, \probOf\inparen{X=1}, \probOf\inparen{Y=1}, \probOf\inparen{Z=1})$.
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%It can be verified that the reduced polynomial parameterized with each variable's respective marginal probability is a closed form of the expected count (i.e., $\expct\limits_{\vct{\randWorld}\sim\pd}\pbox{\Phi^2\inparen{\vct{X}}} = \widetilde{\Phi^2}(\probOf\pbox{A=1},$ $\probOf\pbox{B=1}, \probOf\pbox{C=1}), \probOf\pbox{D=1}, \probOf\pbox{X=1}, \probOf\pbox{Y=1}, \probOf\pbox{Z=1})$).
|
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\Cref{lem:tidb-reduce-poly} generalizes the equivalence to {\em all} $\raPlus$ queries on \abbrTIDB\xplural (proof in \Cref{subsec:proof-exp-poly-rpoly}).
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\begin{Lemma}\label{lem:tidb-reduce-poly}
|
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Let $\pdb$ be a \abbrTIDB over $n$ input tuples
|
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such that the probability distribution $\pdassign$ over $\vct{W}\in\{0,1\}^\numvar$ (the set of possible worlds) is induced by the probability vector $\probAllTup = \inparen{\prob_1,\ldots,\prob_\numvar}$ where $\prob_i=\probOf\inparen{W_i=1}$.
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||||
For any \abbrTIDB-lineage polynomial
|
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%\BG{Term has not been introduced yet.}
|
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%Atri: fixed
|
||||
$\poly\inparen{\vct{X}}=\apolyqdt(\vct{X})$, it holds that $
|
||||
\expct_{\vct{W} \sim \pdassign}\pbox{\poly\inparen{\vct{W}}} = \rpoly\inparen{\probAllTup}.
|
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$
|
||||
\end{Lemma}
|
||||
|
||||
To prove our hardness result we show that for the same $Q$ from the example above, for an arbitrary `product width' $k$, the query $Q^k$ is able to encode various hard graph-counting problems (assuming $\bigO{\numvar}$ tuples rather than the $O(1)$ tuples in \Cref{fig:two-step}).
|
||||
We do so by considering an arbitrary graph $G$ (analogous to the $Route$ relation of $\query$) and analyzing how the coefficients in the (univariate) polynomial $\widetilde{\poly}\left(p,\dots,p\right)$ relate to counts of subgraphs in $G$ that are isomorphic to various graphs with $k$ edges. E.g., we exploit the fact that the leading coefficient in $\poly$ corresponding to $\query^k$ is proportional to the number of $k$-matchings in $G$, a known hard problem in parameterized/fine-grained complexity literature.
|
||||
|
||||
For an upper bound on approximating the expected count, it is easy to check that if all the probabilties are constant then $\poly\left(\prob_1,\dots, \prob_n\right)$ (i.e. evaluating the original lineage polynomial over the probability values) is a constant factor approximation. For example, using $\query^2$ from above, using $\prob_A$ to denote $\probOf\pbox{A = 1}$ (and similarly for the other variables), we can see that
|
||||
\begin{footnotesize}
|
||||
\begin{align*}
|
||||
\hspace*{-3mm}
|
||||
\poly^2\inparen{\probAllTup} &= \prob_A^2\prob_X^2\prob_B^2 + \prob_B^2\prob_Y^2\prob_E^2 + \prob_B^2\prob_Z^2\prob_C^2 + 2\prob_A\prob_X\prob_B^2\prob_Y\prob_E + 2\prob_A\prob_X\prob_B^2\prob_Z\prob_C + 2\prob_B^2\prob_Y\prob_E\prob_Z\prob_C\\
|
||||
&\leq\prob_A\prob_X\prob_B + \prob_B\prob_Y\prob_E + \prob_B\prob_Z\prob_C +
|
||||
2\prob_A\prob_X\prob_B\prob_Y\prob_E + 2\prob_A\prob_X\prob_B\prob_Z\prob_C + 2\prob_B\prob_Y\prob_E\prob_Z\prob_C
|
||||
= \rpoly\inparen{\vct{p}}
|
||||
%\inparen{0.9\cdot 1.0\cdot 1.0 + 0.5\cdot 1.0\cdot 1.0 + 0.5\cdot 1.0\cdot 0.5}^2 = 2.7225 < 3.45 = \rpoly^2\inparen{\probAllTup}
|
||||
\end{align*}
|
||||
\end{footnotesize}
|
||||
If we assume that all seven probability values are at least $p_0>0$,
|
||||
%Choose the least factor that is reduced in $\rpoly^2\inparen{\vct{X}}$, in this case $\prob_A\prob_X\prob_B$, and
|
||||
we get that $\poly^2\inparen{\vct{\prob}}$ is in the range $[\inparen{p_0}^3\cdot\rpoly\inparen{\vct{\prob}}, \rpoly\inparen{\vct{\prob}}]$.
|
||||
%
|
||||
To get an $(1\pm \epsilon)$-multiplicative approximation we uniformly sample monomials from the \abbrSMB representation of $\poly$ and `adjust' their contribution to $\widetilde{\poly}\left(\cdot\right)$.
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
|
|
|
|||
10
macros.tex
10
macros.tex
|
|
@ -18,7 +18,7 @@
|
|||
\newcommand{\BG}[1]{\todo{\textbf{Boris says:$\,$} #1}}
|
||||
\newcommand{\SF}[1]{\todo{\textbf{Su says:$\,$} #1}}
|
||||
\newcommand{\OK}[1]{\todo[color=gray]{\textbf{Oliver says:$\,$} #1}}
|
||||
\newcommand{\AH}[1]{\todo[backgroundcolor=cyan, caption={}]{\textbf{Aaron says:$\,$} #1}}
|
||||
\newcommand{\AH}[1]{\todo[inline, backgroundcolor=cyan, caption={}]{\textbf{Aaron says:$\,$} #1}}
|
||||
\newcommand{\AR}[1]{\todo[inline,color=green]{\textbf{Atri says:$\,$} #1}}
|
||||
\else
|
||||
\newcommand{\BG}[1]{}
|
||||
|
|
@ -134,13 +134,15 @@
|
|||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%Bag c-TIDB Notation %
|
||||
%Bag c-TIDB Notation %
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\newcommand{\bound}{c}
|
||||
\newcommand{\tupsetsize}{n}
|
||||
\newcommand{\tupset}{D}
|
||||
\newcommand{\gentupset}{\overline{D}}
|
||||
\newcommand{\worlds}{\inset{0,\ldots, c}^\tupsetsize}
|
||||
\newcommand{\world}{\inset{0,\ldots, c}}
|
||||
\newcommand{\worldvec}{\vct{W}}
|
||||
\newcommand{\worlds}{\world^\tupset}
|
||||
\newcommand{\bpd}{\mathcal{P}}%bpd for bag probability distribution
|
||||
%BIDB
|
||||
\newcommand{\block}{b}
|
||||
|
|
@ -213,7 +215,7 @@
|
|||
\newcommand{\kElem}{k}%the kth element<---where and how are we using this?
|
||||
%Random Variables
|
||||
\newcommand{\randWorld}{W}
|
||||
\newcommand{\randDB}{\overline{\db}}
|
||||
\newcommand{\randDB}{\vct{\db}}
|
||||
\newcommand{\rvW}{W}%\rvW for random variable of type World<---this is the same as \randWorld
|
||||
%One of these needs to go...I think...
|
||||
|
||||
|
|
|
|||
|
|
@ -37,7 +37,7 @@
|
|||
Chicago & Bremen & $Z$ & 1.0 \\
|
||||
\end{tabular}};
|
||||
%label below cylinder
|
||||
\node[below=0.2 cm of cylinder]{{\LARGE$ \dbbase$}};
|
||||
\node[below=0.2 cm of cylinder]{{\LARGE$ \tupset$}};
|
||||
%First arrow
|
||||
\node[single arrow, right=0.25 of cylinder, draw=black, fill=black!65, text=white, minimum height=0.75cm, minimum width=0.25cm](arrow1) {\textbf{\abbrStepOne}};
|
||||
\node[above=of arrow1](arrow1Label) {$\query$};
|
||||
|
|
@ -106,7 +106,7 @@
|
|||
\end{tabular}
|
||||
};
|
||||
%label below rectangle
|
||||
\node[below=0.2cm of rect]{{\LARGE $\query(\dbbase)\inparen{\tup}\equiv \poly\inparen{\vct{X}}$}};
|
||||
\node[below=0.2cm of rect]{{\LARGE $\query(\tupset)\inparen{\tup}\equiv \poly\inparen{\vct{X}}$}};
|
||||
%Second arrow
|
||||
\node[single arrow, right=0.25 of rect, draw=black, fill=black!65, text=white, minimum height=0.75cm, minimum width=0.25cm](arrow2) {\textbf{\abbrStepTwo}};
|
||||
%Expectation computation; (output of step 2)
|
||||
|
|
|
|||
Loading…
Reference in a new issue