Merge branch 'master' of gitlab.odin.cse.buffalo.edu:ahuber/SketchingWorlds

circuits-model-runtime.tex

@@ -104,7 +104,7 @@ Given a $\semNX$-PDB $\pxdb$ and query plan $Q$, the runtime of $Q$ over $\bagdb
 We now have all the pieces to argue that using our approximation algorithm,  the expected multiplicities of a SPJU query can be computed in essentially the same runtime as deterministic query processing for the same query:
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{Corollary}
-  Given an SPJU query $Q$ over a \ti $\pxdb$ and let $\db_{max}$ denote the world containing all tuples of $\pxdb$, we can compute a $(1\pm\eps)$-approximation of the expectation for each output tuple with probability at least $1-\delta$ in time
+  Given an SPJU query $Q$ over a \ti $\pxdb$ and let $\db_{max}$ denote the world containing all tuples of $\pxdb$, we can compute a $(1\pm\eps)$-approximation of the expectation for each output tuple in $\query(\pxdb)$ with probability at least $1-\delta$ in time
 %
   \[
     O_k\left(\frac 1{\eps^2}\cdot\qruntime{Q,\db_{max}}\cdot \log{\frac{1}{\conf}}\cdot \log(n)\right)

intro.tex

@@ -206,15 +206,14 @@ To see why computing this probability is hard, observe that the clauses of the d
 Conversely, in Bag-PDBs, correlations between clauses of the SOP polynomial are not problematic thanks to linearity of expectation.
 The expectation computation over the output lineage is simply the sum of expectations of each clause.
 For \Cref{ex:intro}, the expectation is simply
-{\small
-\begin{align*}
-\expct\pbox{\poly(W_a, W_b, W_c)} &= \expct\pbox{W_aW_b} + \expct\pbox{W_bW_c} + \expct\pbox{W_cW_a}\\
-\intertext{\normalsize
+\begin{equation*}
+\expct\pbox{\poly_{bag}(W_a, W_b, W_c)} = \expct\pbox{W_aW_b} + \expct\pbox{W_bW_c} + \expct\pbox{W_cW_a}
+\end{equation*}
 In this particular lineage polynomial, all variables in each product clause are independent, so we can push expectations through.
-}
-&= \expct\pbox{W_a}\expct\pbox{W_b} + \expct\pbox{W_b}\expct\pbox{W_c} + \expct\pbox{W_c}\expct\pbox{W_a}
-\end{align*}
-}
+\begin{equation*}
+= \expct\pbox{W_a}\expct\pbox{W_b} + \expct\pbox{W_b}\expct\pbox{W_c} + \expct\pbox{W_c}\expct\pbox{W_a}
+\end{equation*}
+
 Computing such expectations is indeed linear in the size of the SOP as the number of operations in the computation is \textit{exactly} the number of multiplication and addition operations of the polynomial.
 As a further interesting feature of this example, note that $\expct\pbox{W_i} = \probOf[W_i = 1]$, and so taking the same polynomial over the reals:
 \begin{multline}
@@ -307,7 +306,7 @@ With $\poly^2$ as an example, we have:
 Note that the reduced polynomial is a closed form of the expected count (i.e., $\expct\pbox{\poly^2} = \rpoly(\probOf\pbox{W_a=1}, \probOf\pbox{W_b=1}, \probOf\pbox{W_c=1})$).
 Also note that the $\poly$ in~\Cref{ex:bag-vs-set} is already in reduced form.
 
-The reduced form of a polynomial can be obtained in a linear scan over the clauses of a SOP encoding of the polynomial.
+The reduced form of a polynomial can be obtained in a linear scan over the clauses of an SOP encoding of the polynomial.
 In prior work on lineage-based Bag-PDBs~\cite{kennedy:2010:icde:pip,DBLP:conf/vldb/AgrawalBSHNSW06,yang:2015:pvldb:lenses} where this encoding is implicitly assumed, computing the expected count is linear in the size of the encoding.
 In general however, compressed encodings of the polynomial can be exponentially smaller in $k$ for $k$-products --- the query $\poly^k$ obtained by taking the Cartesian product of $k$ copies of $\poly$ has a factorized encoding of size $6\cdot k$, while the SOP encoding is of size $2\cdot 3^k$.
 This leads us to the \textbf{central question of this paper}:

poly-form.tex

@@ -22,7 +22,7 @@ A monomial is a product of variable terms, each raised to a non-negative integer
   \[
     \sum_{i=1}^n c_i \cdot m_i
   \]
-where each $c_i$ is a positive integer and each $m_i$ is a monomial and $m_i \neq m_j$ for $i \neq j$. The \abbrSMB of a polynomial $\poly$ is $\smbOf{\poly}$.
+where each $c_i$ is an integer and each $m_i$ is a monomial and $m_i \neq m_j$ for $i \neq j$. The \abbrSMB of a polynomial $\poly$ is $\smbOf{\poly}$.
 %  fully expanded out such that no product of sums exist and where each unique monomial appears exactly once.
 \end{Definition}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -94,7 +94,9 @@ Given the set of BIDB variables $\inset{X_{b,i}}$, define
 \end{Definition}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %
-Intuitively, in the reduced form, all exponents $e > 1$ are reduced to $e = 1$ by $\text{mod } \mathcal T$, and all monomials with multile variables from the same block $\block$ are dropped by $\text{mod } \mathcal B$ (i.e., any world containing more than one tuple from a block has $0$ probability and can be ignored). 
+
+Intuitively, in the reduced form, all exponents $e > 1$ are reduced to $e = 1$ by $\text{mod } \mathcal T$, and all monomials with multiple variables from the same block $\block$ are dropped by $\text{mod } \mathcal B$ (i.e., any world containing more than one tuple from a block has $0$ probability and can be ignored). 
+
 For the special case of \tis, the second step is not necessary since every block contains a single tuple.
 %Alternatively, one can think of $\rpoly$ as the \abbrSMB of $\poly(\vct{X})$ when the product operator is idempotent.
 %
@@ -126,7 +128,7 @@ Consider $\poly(X, Y) = (X + Y)(X + Y)$ where $X$ and $Y$ are from different blo
 %
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{Definition}[Valid Worlds]
-For probability distribution $\probDist$ and its corresponding probability mass function $\probOf$, the set of valid worlds $\eta$ is the worlds with probability value greater than $0$; i.e., for variable vector $\vct{W}$
+For probability distribution $\probDist$ and its corresponding probability mass function $\probOf$, the set of valid worlds $\eta$ consists of all the worlds with probability value greater than $0$; i.e., for variable vector $\vct{W}$
 \[
 \eta = \{\vct{w}\st \probOf[\vct{W} = \vct{w}] > 0\}
 \]

ra-to-poly.tex

@@ -13,7 +13,7 @@ Denote the schema of $\db$ as $\sch(\db)$. A \textit{probabilistic database} $\p
 For a probabilistic  database $\pdb = (\idb, \pd)$,  the result of a query is the pair $(\query(\idb), \pd')$ where $\pd'$ is a probability distribution over $\query(\idb)$  that assigns to each possible query result the sum of the probabilities of the worlds that produce this answer:
 \[\forall \db \in \query(\idb): \probOf'(\db) = \sum_{\db' \in \idb: \query(\db') = \db} \probOf(\db') \]
 
-Note that in this work we consider multisets, i.e., each possible world is a set of multiset relations and queries are evaluated using bag semantics. We will use K-relations to model multisets. A \emph{K-relation}~\cite{DBLP:conf/pods/GreenKT07} is a relation whose tuples are annotated with elements from a commutative semiring $\semK = (\domK, \addK, \multK, \zeroK, \oneK)$.  A commutative semiring is a structure with a domain $\domK$ and associative and commutative binary operations $\addK$ and $\multK$ such that $\multK$ distributes over $\addK$, $\zeroK$ is the identity of $\addK$, $\oneK$ is the identity of $\multK$, and $\zeroK$ annihilates all elements of $\domK$ when combined by $\multK$.
+Note that in this work we consider multisets, i.e., each possible world is a set of multiset relations and queries are evaluated using bag semantics. We will use $\domK$-relations to model multisets. A \emph{$\domK$-relation}~\cite{DBLP:conf/pods/GreenKT07} is a relation whose tuples are annotated with elements from a commutative semiring $\semK = (\domK, \addK, \multK, \zeroK, \oneK)$.  A commutative semiring is a structure with a domain $\domK$ and associative and commutative binary operations $\addK$ and $\multK$ such that $\multK$ distributes over $\addK$, $\zeroK$ is the identity of $\addK$, $\oneK$ is the identity of $\multK$, and $\zeroK$ annihilates all elements of $\domK$ when combined by $\multK$.
 Let $\udom$ be a countable domain of values.
 Formally, an n-ary $\semK$-relation over $\udom$ is a function $\rel: \udom^n \to \domK$ with finite support $\support{\rel} = \{ \tup \mid \rel(\tup) \neq \zeroK \}$.
 A $\semK$-database is a set of $\semK$-relations. It will be convenient to also interpret a $\semK$-database as a function from tuples to annotations. Thus, $\rel(t)$ (resp., $\db(t)$) denotes the annotation associated by $\semK$-relation $\rel$ ($\semK$-database $\db$) to $t$.