372 lines
37 KiB
TeX
372 lines
37 KiB
TeX
\section{Missing details from Section~\ref{sec:background}}\label{sec:proofs-background}
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\subsection{Proof of~\Cref{prop:semnx-pdbs-are-a-}}
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\AH{I made small changes to the proof, noteably the summation, the variable definition and the world subscript, the latter of which I am not sure if it is the best notation or not.}
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To prove that $\semNX$-PDBs are complete consider the following construction that for any $\semN$-PDB $\pdb = (\idb, \pd)$ produces an $\semNX$-PDB $\pxdb = (\db, \pd')$ such that $\rmod(\pxdb) = \pdb$. Let $\idb = \{D_1, \ldots, D_{\abs{\idb}}\}$ and let $max(D_i)$ denote $max_{\tup} D_i(\tup)$. For each world $D_i$ we create a corresponding variable $X_i$.
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%variables $X_{i1}$, \ldots, $X_{im}$ where $m = max(D_i)$.
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In $\db$ we assign each tuple $\tup$ the polynomial:
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%
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\[
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\db(\tup) = \sum_{i=1}^{\abs{\idb}} D_i(\tup)\cdot X_{i}
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\]
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The probability distribution $\pd'$ assigns all world vectors zero probability except for $\abs{\idb}$ world vectors (representing the possible worlds) $\vct{w_i}$. All elements of $\vct{w_i}$ are zero except for the positions corresponding to variables $X_{ij}$ for $j \in \{1, \ldots \}$ which are set to $1$. Unfolding definitions it is trivial to show that $\rmod(\pxdb) = \pdb$. Thus, $\semNX$ are a complete representation system.
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The closure under $\raPlus$ queries follows from the fact that an assignment $\vct{X} \to \{0,1\}$ is a semiring homomorphism and that semiring homomorphisms commute with queries over $\semK$-relations.
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\subsection{Proof of~\Cref{prop:expection-of-polynom}}
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\BG{TODO}
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\subsection{Proof for Proposition ~\ref{proposition:q-qtilde}}
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Note that any $\poly$ in factorized form is equivalent to its \abbrSMB expansion. For each term in the expanded form, further note that for all $b \in \{0, 1\}$ and all $e \geq 1$, $b^e = b$. \qed
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\subsection{Proof for Lemma ~\ref{lem:exp-poly-rpoly}}
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Let $\poly$ be the generalized polynomial, i.e., the polynomial of $\numvar$ variables with highest degree $= B$: %, in which every possible monomial permutation appears,
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\[\poly(X_1,\ldots, X_\numvar) = \sum_{\vct{d} \in \{0,\ldots, B\}^\numvar}q_{\vct{d}}\cdot \prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar X_i^{d_i}\].
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Then, assigning $\vct{w}$ to $\vct{X}$, for expectation we have
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\begin{align}
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\expct_{\vct{w}}\pbox{\poly(\vct{w})} &= \sum_{\vct{d} \in \{0,\ldots, B\}^\numvar}q_{\vct{d}}\cdot \expct_{\vct{w}}\pbox{\prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar w_i^{d_i}}\label{p1-s1}\\
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&= \sum_{\vct{d} \in \{0,\ldots, B\}^\numvar}q_{\vct{d}}\cdot \prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar \expct_{\vct{w}}\pbox{w_i^{d_i}}\label{p1-s2}\\
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&= \sum_{\vct{d} \in \{0,\ldots, B\}^\numvar}q_{\vct{d}}\cdot \prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar \expct_{\vct{w}}\pbox{w_i}\label{p1-s3}\\
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&= \sum_{\vct{d} \in \{0,\ldots, B\}^\numvar}q_{\vct{d}}\cdot \prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar \prob_i\label{p1-s4}\\
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&= \rpoly(\prob_1,\ldots, \prob_\numvar)\label{p1-s5}
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\end{align}
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In steps \cref{p1-s1} and \cref{p1-s2}, by linearity of expectation (recall the variables are independent), the expecation can be pushed all the way inside of the product. In \cref{p1-s3}, note that $w_i \in \{0, 1\}$ which further implies that for any exponent $e \geq 1$, $w_i^e = w_i$. Next, in \cref{p1-s4} the expectation of a tuple is indeed its probability.
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Finally, observe \Cref{p1-s5} by construction in \Cref{lem:pre-poly-rpoly}, that $\rpoly(\prob_1,\ldots, \prob_\numvar)$ is exactly the product of probabilities of each variable in each monomial across the entire sum.
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\subsection{Proof For Corollary ~\ref{cor:expct-sop}}
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Note that \cref{lem:exp-poly-rpoly} shows that $\expct\pbox{\poly} =$ $\rpoly(\prob_1,\ldots, \prob_\numvar)$. Therefore, if $\poly$ is already in \abbrSMB form, one only needs to compute $\poly(\prob_1,\ldots, \prob_\numvar)$ ignoring exponent terms (note that such a polynomial is $\rpoly(\prob_1,\ldots, \prob_\numvar)$), which indeed has $O(\smbOf{|\poly|})$ computations.\qed
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\section{Missing details from Section~\ref{sec:hard}}
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\label{app:hard}
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\subsection{Proofs of~\cref{eq:1e}-\cref{eq:2pd-3d}}
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\label{app:easy-counts}
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\cref{eq:1e},~\cref{eq:2p} and~\cref{eq:3s} are immediate.
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A quick argument to why \cref{eq:2m} is true. Note that for edge $(i, j)$ connecting arbitrary vertices $i$ and $j$, finding all other edges in $G$ disjoint to $(i, j)$ is equivalent to finding all edges that are not connected to either vertex $i$ or $j$. The number of such edges is $m - d_i - d_j + 1$, where we add $1$ since edge $(i, j)$ is removed twice when subtracting both $d_i$ and $d_j$. Since the summation is iterating over all edges such that a pair $\left((i, j), (k, \ell)\right)$ will also be counted as $\left((k, \ell), (i, j)\right)$, division by $2$ then eliminates this double counting.
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\cref{eq:2pd-3d} is true for similar reasons. For edge $(i, j)$, it is necessary to find two additional edges, disjoint or connected. As in ~\cref{eq:2m}, once the number of edges disjoint to $(i, j)$ have been computed, then we only need to consider all possible combinations of two edges from the set of disjoint edges, since it doesn't matter if the two edges are connected or not. Note, the factor $3$ of $\threedis$ is necessary to account for the triple counting of $3$-matchings. It is also the case that, since the two path in $\twopathdis$ is connected, that there will be no double counting by the fact that the summation automatically 'disconnects' the current edge, meaning that a two matching at the current vertex will not be counted. The sum over all such edge combinations is precisely then $\numocc{G}{\twopathdis} + 3\numocc{G}{\threedis}$.
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\subsection{Proofs for~\Cref{lem:3m-G2}-\Cref{lem:lin-sys}}\label{subsec:proofs-struc-lemmas}
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Before proceeding, let us introduce a few more helpful definitions.
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\begin{Definition}\label{def:ed-nota}
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For the set of edges in $\graph{\ell}$ we write $E_\ell$. For any graph $\graph{\ell}$, its edges are denoted by the a pair $(e, b)$, such that $b \in \{0,\ldots, \ell-1\}$ and $e\in E_1$, where $(e,0),\dots,(e,\ell-1)$ is the $\ell$-path that replaces the edge $e$.
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\end{Definition}
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\begin{Definition}[$\eset{\ell}$]
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Given an arbitrary subgraph $S\graph{1}$ of $\graph{1}$, let $\eset{1}$ denote the set of edges in $S\graph{1}$. Define then $\eset{\ell}$ for $\ell > 1$ as the set of edges in the generated subgraph $S\graph{\ell}$ (i.e. when we apply~\Cref{def:Gk} to $\graph{1}=S\graph{1}$.
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\end{Definition}
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For example, consider $S\graph{1}$ with edges $\eset{1} = \{e_1\}$. Then the edges of $S\graph{2}$, s defined as $\eset{2} = \{(e_1, 0), (e_1, 1)\}$.
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\begin{Definition}\label{def:ed-sub}
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Let $\binom{S}{t}$ denote the set of subsets in $S$ with exactly $t$ edges. In a similar manner, $\binom{S}{\leq t}$ is used to mean the subsets of $S$ with $t$ or fewer edges.
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\end{Definition}
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The following function $f_\ell$ is a mapping from every $3$-edge shape in $\graph{\ell}$ to its `projection' in $\graph{1}$.
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\begin{Definition}\label{def:fk}
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Let $f_\ell: \binom{E_\ell}{3} \mapsto \binom{E_1}{\leq3}$ be defined as follows. For any $S \in \binom{E_\ell}{3}$, such that $S = \pbrace{(e_1, b_1), (e_2, b_2), (e_3, b_3)}$, define:
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\[ f_\ell\left(\pbrace{(e_1, b_1), (e_2, b_2), (e_3, b_3)}\right) = \pbrace{e_1, e_2, e_3}.\]
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\end{Definition}
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\begin{Definition}[$f_\ell^{-1}$]\label{def:fk-inv}
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The inverse function $f_\ell^{-1}: \binom{E_1}{\leq 3}\mapsto 2^{\binom{E_\ell}{3}}$ takes an arbitrary subset $S\subseteq E_1$ of at most $3$ edges and outputs the set of all subsets of $\binom{\eset{\ell}}{3}$ such that each set $T\in f_\ell^{-1}(S)$ is mapped to the input set $S$ by $f_\ell$, i.e. $f_\ell(T) = S$. %The set returned by $f_\ell^{-1}$ is of size $h$, where $h$ depends on $\abs{s^{(1)}}$, such that $h \leq \binom{\abs{s^{(1)}} \cdot \ell}{3}$.
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\end{Definition}
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Note, importantly, that when we discuss $f_\ell^{-1}$, that each \textit{edge} present in $S$ must have an edge in $T\in f_\ell^{-1}(S)$ that `projects` down to it. In particular, if $|S| = 3$, then it must be the case that each $T\in f_\ell^{-1}(S)$ si given by $T=\{ (e_i, b), (e_j, b), (e_m, b) \}$ where $i,j$ and $m$ are distinct.
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We first note that $f_\ell$ is well-defined:
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\begin{Lemma}\label{lem:fk-func}
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$f_\ell$ is a function.
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\end{Lemma}
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\subsubsection{Proof of~\cref{lem:fk-func}}\label{subsubsec:proof-fk}%[Proof of Lemma \ref{lem:fk-func}]
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Note that $f_\ell$ is properly defined. For any $S \in \binom{E_\ell}{3}$, $|f(S)| \leq 3$, since it has to be the case that any subset of $3$ edges in $E_\ell$ will map to at most three edges in $E_1$. All mappings are in the required range. Then, since for any $b \in \{0,\ldots, \ell-1\}$ the map $(e, b) \mapsto e$ is a function, which %` mapping for which $(e, b)$ maps to no other edge than $e$, and this
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implies that $f_\ell$ is a function.
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We are now ready to prove the structural lemmas. Note that $f_\ell$ maps subsets of three edges in $\graph{\ell}$ to a subset of at most three edges in $E_1$. To prove the structural lemmas, we will use the map $f_\ell^{-1}$. In particular, to count the number of occurrences of $\tri,\threepath,\threedis$ in $\graph{\ell}$ we count for each $S\in\binom{E_1}{\le 3}$, how many of $\tri/\threepath/\threedis$ subgraphs appear in $f_\ell^{-1}(S)$.
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\subsubsection{Proof of Lemma \ref{lem:3m-G2}}
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For each subset $\eset{1}\in \binom{E_1}{\le 3}$, we count the number of $3$-matchings in the $3$-edge subgraphs of $\graph{2}$ in $f_2^{-1}(\eset{1})$. We first consider the case of $\eset{1} \in \binom{E_1}{3}$, where $\eset{1}$ is composed of the edges $e_1, e_2, e_3$ and $f_2^{-1}(\eset{1})$ is the set of all $3$-edge subsets of the set
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\begin{equation*}
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\{(e_1, 0), (e_1, 1), (e_2, 0), (e_2, 1), (e_3, 0), (e_3, 1)\}.
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\end{equation*}
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We do a case analysis based on the `shape' of $\eset{1}$:
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\begin{itemize}
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\item $3$-matching ($\threedis$)
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\end{itemize}
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When $\eset{1} \equiv \threedis$, that edges in $\eset{2}$ are {\em not} disjoint only for the pairs $(e_i, 0), (e_i, 1)$ for $i\in \{1,2,3\}$. All choices for $b_1, b_2, b_3 \in \{0, 1\}$, $(e_1, b_1), (e_2, b_2), (e_3, b_3)$ will compose a 3-matching. One can see that we have a total of two possible choicesi for $b_i$ for each edge $e_i$ in $\graph{1}$ yielding $2^3 = 8$ possible 3-matchings in $f_2^{-1}(\eset{1})$.
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\begin{itemize}
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\item Disjoint Two-Path ($\twopathdis$)
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\end{itemize}
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For $\eset{1} \equiv \twopathdis$ edges $e_2, e_3$ form a $2$-path with $e_1$ being disjoint. This means that $(e_2, 0), (e_2, 1), (e_3, 0), (e_3, 1)$ form a $4$-path while $(e_1, 0), (e_1, 1)$ is its own disjoint $2$-path. We can only pick either $(e_1, 0)$ or $(e_1, 1)$ for $f_2^{-1}(\eset{1})$, and then we need to pick a $2$-matching from $e_2$ and $e_3$. Note that the four path allows there to be 3 possible 2 matchings, specifically,
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\begin{equation*}
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\pbrace{(e_2, 0), (e_3, 0)}, \pbrace{(e_2, 0), (e_3, 1)}, \pbrace{(e_2, 1), (e_3, 1)}.
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\end{equation*}
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Since these two selections can be made independently, there are $2 \cdot 3 = 6$ choices for $3$-matchings in $f_2^{-1}(\eset{1})$.
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\begin{itemize}
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\item $3$-star ($\oneint$)
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\end{itemize}
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When $\eset{1} \equiv \oneint$, the inner edges $(e_i, 1)$ of $\eset{2}$ are all connected, and the outer edges $(e_i, 0)$ are all disjoint. Note that for a valid 3 matching it must be the case that at most one inner edge can be part of the set of disjoint edges. When exactly one inner edge is chosen, there are 3 such possibilities. The remaining possible 3-matching occurs when all 3 outer edges are chosen. Thus, there are $3 + 1 = 4$ many 3-matchings in $f_2^{-1}(\eset{1})$.
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\begin{itemize}
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\item $3$-path ($\threepath$)
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\end{itemize}
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When $\eset{1} \equiv\threepath$ it is the case that all edges beginning with $e_1$ and ending with $e_3$ are successively connected. This means that the edges of $\eset{2}$ form a $6$-path in the edges of $f_2^{-1}(\eset{1})$, where all edges from $(e_1, 0),\ldots,(e_3, 1)$ are successively connected. For a $3$-matching to exist in $f_2^{-1}(\eset{1})$, we cannot pick both $(e_i,0)$ and $(e_i,1)$. % there must be at least one edge separating edges picked from a sequence.
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There are four such possibilities: $\pbrace{(e_1, 0), (e_2, 0), (e_3, 0)}, \pbrace{(e_1, 0), (e_2, 0), (e_3, 1)}, \pbrace{(e_1, 0), (e_2, 1), (e_3, 1)},$\newline $\pbrace{(e_1, 1), (e_2, 1), (e_3, 1)}$ . Thus, there are four possible 3-matchings in $f_2^{-1}(\eset{1})$.
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\begin{itemize}
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\item Triangle ($\tri$)
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\end{itemize}
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For $\eset{1} \equiv \tri$, note that it is the case that the edges in $\eset{2}$ are connected in a successive manner, but this time in a cycle, such that $(e_1, 0)$ and $(e_3, 1)$ are also connected. While this is similar to the discussion of the three path above, the first and last edges are not disjoint, since they are connected. This rules out both subsets of $(e_1, 0), (e_2, 0), (e_3, 1)$ and $(e_1, 0), (e_2, 1), (e_3, 1)$ leaving us with $2$ remaining edge combinations that produce a 3 matching.
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Let us now consider when $S \in \binom{E_1}{\leq 2}$, i.e. patterns among
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\begin{itemize}
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\item $2$-matching ($\twodis$), $2$-path ($\twopath$), $1$ edge ($\ed$)
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\end{itemize}
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When $|\eset{1}| = 2$, we can only pick one from each of two pairs, $\pbrace{(e_1, 0), (e_1, 1)}$ and $\pbrace{(e_2, 0), (e_2, 1)}$. This implies that a $3$-matching cannot exist in $f_2^{-1}(\eset{1})$. The same argument holds for $|\eset{1}| = 1$, where we can only pick one edge from the pair $\pbrace{(e_1, 0), (e_1, 1)}$, thus no $3$-matching exists in $f_2^{-1}(\eset{1})$.
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Observe that all of the arguments above focused solely on the shape/pattern of $S$. In other words, all $S$ of a given shape yield the same number of $3$-matchings in $f_2^{-1}(\eset{1})$, and this is why we get the required identity using the above case analysis.
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\subsubsection{Proof of~\cref{lem:3m-G3}}
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For any $\eset{1} \in \binom{E_1}{\leq3}$, we again then count the number of $3$-matchings in $f_3^{-1}(\eset{1})$ via a case analysis:
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\begin{itemize}
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\item $1$ edge ($\ed$)
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\end{itemize}
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When $\eset{1} \equiv \ed$, $f_3^{-1}(\eset{1})$ has one subset, $(e_1, 0), (e_1, 1), (e_1, 2)$, which clearly does not contain a $3$-matching. Thus there are no $3$-matchings in $f_3^{-1}(\eset{1})$ for this case.
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\begin{itemize}
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\item $2$-path ($\twopath$)
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\end{itemize}
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When $\eset{1} \equiv \twopath$ and now we have all edges in $\eset{3}$ form a $6$-path, and similar to the discussion in the proof of \cref{lem:3m-G2} (when $\eset{1} \equiv \threepath$ in $\graph{2}$), this leads to four $3$-matchings in $f_3^{-1}(\eset{1})$.
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\begin{itemize}
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\item $2$-matching ($\twodis$)
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\end{itemize}
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For $\eset{1} \equiv \twodis$, all edges of $\eset{3}$ are predicated on the fact that $(e_i, b)$ is disjoint with $(e_j, b)$ for $i \neq j\in \{1,2\}$ and $b \in \{0, 1, 2\}$. Pick an aribitrary $e_i$ and note, that $(e_i, 0), (e_i, 2)$ is a $2$-matching, which can combine with any of the $3$ edges in $(e_j, 0), (e_j, 1), (e_j, 2)$ again for $i \neq j$. Since the selections are independent, it follows that there exist $2 \cdot 3 = 6$ many $3$-matchings in $f_3^{-1}(\eset{1})$.
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Now, we consider the 3-edge subgraphs of $\graph{1}$, starting with $\eset{1} = \tri$.
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\begin{itemize}
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\item Triangle ($\tri$)
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\end{itemize}
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As discussed in proof of \cref{lem:3m-G2} for the case of $\tri$, the edges of $\eset{3}$ are a cyclic sequence, and we must be careful not to pair $(e_1, 0)$ with $(e_3, 2)$ in a $3$-matching. For any $T \in f_3^{-1}(\eset{1})$, $T$ is a $3$-matching when we have that for the edges $(e_1, b_1), (e_2, b_2), (e_3, b_3)$ where $b_1, b_2, b_3 \in \{0, 1, 2\}$, such that, for all $i \in [3]$ it is the case that if $b_i = 2$ then $b_{i \mod{3} + 1} \neq 0$. Iterating through all possible choices for $e_1$, we have
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\begin{itemize}
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\item For \textsc{$(e_1, 0)$}, there are five possibilities:
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\begin{itemize}
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\item $\pbrace{(e_1, 0), (e_2, 0), (e_3, 0)}$
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\item $\pbrace{(e_1, 0), (e_2, 0), (e_3, 1)}$
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\item $\pbrace{(e_1, 0), (e_2, 1), (e_3, 0)}$
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\item $\pbrace{(e_1, 0), (e_2, 1), (e_3, 1)}$
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\item $\pbrace{(e_1, 0), (e_2, 2), (e_3, 1)}$
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\end{itemize}
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\item For \textsc{$(e_1, 1)$}, there are eight possibilities:
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\begin{itemize}
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\item $\pbrace{(e_1, 1), (e_2, 0), (e_3, 0)}, \ldots\pbrace{(e_1, 1), (e_2, 1), (e_3, 2)}$
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\item $\pbrace{(e_1, 1), (e_2, 2), (e_3, 1)}$
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\item $\pbrace{(e_1, 1), (e_2, 2), (e_3, 2)}$
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\end{itemize}
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\item For \textsc{$(e_1, 2)$}, there are five possibilities:
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\begin{itemize}
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\item $\pbrace{(e_1, 2), (e_2, 1), (e_3, 0)}$
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\item $\pbrace{(e_1, 2), (e_2, 1), (e_3, 1)}$
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\item $\pbrace{(e_1, 2), (e_2, 1), (e_3, 2)}$
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\item $\pbrace{(e_1, 2), (e_2, 2), (e_3, 1)}$
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\item $\pbrace{(e_1, 2), (e_2, 2), (e_3, 2)}$
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\end{itemize}
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\end{itemize}
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for a total of 18 many 3-matchings in $f_3^{-1}(\eset{1})$.
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\begin{itemize}
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\item $3$-path ($\threepath$)
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\end{itemize}
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When $\eset{1} \equiv \threepath$ and all edges in $\eset{3}$ are successively connected to form a $9$-path. Since $(e_1, 0)$ is disjoint to $(e_3, 2)$, both of these edges can exist in a $3$-matching. This relaxation yields 3 other 3-matchings that couldn't be counted in the case of the $\eset{1} = \tri$, namely
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\begin{equation*}
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\pbrace{(e_1, 0), (e_2, 0), (e_3, 2)},\pbrace{(e_1, 0), (e_2, 1), (e_3, 2)}, \pbrace{(e_1, 0), (e_2, 2), (e_3, 2)}.
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\end{equation*}
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There are therefore $18 + 3 = 21$ $3$-matchings in $f_3^{-1}(\eset{1})$.
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\begin{itemize}
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\item Disjoint Two-Path ($\twopathdis$)
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\end{itemize}
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Assume $\eset{1} = \twopathdis$, then the edges of $\eset{3}$ have successive connectivity from $(e_1, 0)$ through $(e_1, 2)$, and successive connectivity from $(e_2, 0)$ through $(e_3, 2)$. It is the case that the edges in $\eset{3}$ form a 6-path with a disjoint 3-path. There exist $8$ distinct two matchings (with at least one $(e_2,\cdot)$ and at least one $(e_3,\cdot)$ edge) in the $6$-path $(e_2, 0),\ldots, (e_3, 2)$ of the form
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\begin{equation*}
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\pbrace{(e_2, 0), (e_3, 0)},\ldots, \pbrace{(e_2, 1), (e_3, 2)}, \pbrace{(e_2, 2), (e_3, 1)}, \pbrace{(e_2, 2), (e_3, 2)}.
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\end{equation*}
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These matchings can be paired independently with either of the $3$ remaining edges of $(e_1, b)$, for a total of $8 \cdot 3 = 24$ many 3-matchings in $f_3^{-1}(\eset{1})$.
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\begin{itemize}
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\item $3$-star ($\oneint$)
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\end{itemize}
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When $\eset{1} \equiv \oneint$, the edges of $\eset{3}$ are restricted such that the outer edges $(e_i, 0)$ are disjoint from another, the middle edges $(e_i, 1)$ are also disjoint to each other, and only the inner edges $(e_i, 2)$ intersect with one another at exactly one common endpoint. To be precise, any outer edge $(e_i, 0)$ is disjoint to every middle edge $(e_j, 1)$ for $i \neq j$. As previously mentioned in the proof of \cref{lem:3m-G2}, at most one inner edge may appear in a $3$-matching. For arbitrary inner edge $(e_i, 2)$, we have $4$ combinations of the middle and outer edges of $e_j, e_m$, where $i \neq j \neq m$. These choices are independent and we have $4 \cdot 3 = 12$ many 3-matchings. We are not done yet, as we need to consider the middle and outer edge combinations. Notice that for each $e_i$, we have $2$ choices, i.e. a middle or outer edge, contributing $2^3 = 8$ additional $3$-matchings, for a total of $8 + 12 = 20$ many $3$-matchings in $f_3^{-1}(\eset{1})$.
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\begin{itemize}
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\item $3$-matching ($\threedis$)
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\end{itemize}
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When $\eset{1} \equiv \threedis$ subgraph, we have the case that all edges in $\eset{3}$ have the property that $(e_i, b_i)$ is disjoint to $(e_j, b_j)$ for $i \neq j$. For each $e_i$, there are then $3$ choices, independent of each other, and it results that there are $3^3 = 27$ many 3-matchings in $f_3^{-1}(\eset{1})$.
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All of the observations above focused only on the shape of $\eset{1}$, and since we see that for fixed $\eset{1}$, we have a fixed number of $3$-matchings, this implies the identity.
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\subsubsection{Proof of~\cref{lem:3p-G2}}
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For $\mathcal{P} \in f_2^{-1}\inparen{ \eset{2}}$ such that $\mathcal{P} $ is a $3$-path, it \textit{must} be the case by definition of $f_2$ that (i)eall edges in $f_2(\mathcal{P} )$ have at least one mapping from an edge in $\mathcal{P} $ and recall that (ii) $\mathcal{P} $ is connected. These constraint rules out every pattern $\eset{1}$ consisting of $3$ edges (it can be verified that in each three-edge pattern at least one of (i) or (ii) is violated), as well as when $\eset{1} = \twodis$. For $\eset{1} = \ed$, note that $\eset{1}$ doesn't have enough edges to have any output in $f_2^{-1}(\eset{1})$, i.e., there exists no $\eset{1} \in \binom{E_2}{3}$ such that $f_2(\mathcal{P} ) = \eset{1}$. The only surviving pattern is $\eset{1} \equiv \twopath$, where the edges of $\eset{2}$ have successive connectivity from $(e_1, 0)$ to $(e_2, 1)$. There are then two $3$-paths sharing edges $e_1$ and $e_2$ in $f_2^{-1}(\eset{1}), \pbrace{(e_1, 0), (e_1, 1), (e_2, 0)} \text{ and }\{(e_1, 1)$,$ (e_2, 0), (e_2, 1)\}$.
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All of the observations above focused only on the shape of $\eset{1}$, and since we see that for fixed $\eset{1}$, we have a fixed number of $3$-paths, this implies the identity.
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\subsubsection{Proof of~\cref{lem:3p-G3}}
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The argument follows along the same lines as in the proof of \cref{lem:3p-G2}. Given $\mathcal{P} \in f_3^{-1}\inparen{\eset{1}}$, it \textit{must} be that every edge in $f_3(\mathcal{P})$ has at least one edge in $\mathcal{P}$ mapped to it (and $\mathcal{P}$ is connected). Notice again that this cannot be the case for any $\eset{1} \in \binom{E_1}{3}$, nor is it the case when $\eset{1} = \twodis$. This leaves us with two patterns, $\eset{1} = \twopath$ and $\eset{1} = \ed$. For the former, it is the case that we have two $3$-paths across $e_1$ and $e_2$, $\pbrace{(e_1, 1), (e_1, 2), (e_2, 0)}$ and $\pbrace{(e_1, 2), (e_2, 0), (e_2, 1)}$. For the latter pattern $\ed$, it it trivial to see that an edge in $\graph{1}$ becomes a $3$-path in $\graph{3}$, and this proves the identity.
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All of the observations above focused only on the shape of $\eset{1}$, and since we see that for fixed $\eset{1}$, we have a fixed number of $3$-paths, this implies the identity.
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\subsubsection{Proof of~\cref{lem:tri}}
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The number of triangles in $\graph{\ell}$ for $\ell \geq 2$ will always be $0$ for the simple fact that all cycles in $\graph{\ell}$ will have at least six edges.
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\input{lin_sys}
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\section{Missing Details from Section~\ref{sec:algo}}\label{sec:proofs-approx-alg}
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Before proving~\Cref{lem:mon-samp}, we use it to argue our main result,~\Cref{lem:approx-alg}:
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\subsection{Proof of Theorem \ref{lem:approx-alg}}
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Set $\mathcal{E}=\approxq(\etree, (p_1,\dots,p_\numvar),$ $\conf, \error')$, where
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\[\error' = \error \cdot \frac{\rpoly(\prob_1,\ldots, \prob_\numvar)\cdot (1 - \gamma)}{\abs{\etree}(1,\ldots, 1)},\]
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which achieves the claimed accuracy bound on $\mathcal{E}$.
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The claim on the runtime follows since
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\begin{align*}
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\frac 1{\inparen{\error'}^2}\cdot \log\inparen{\frac 1\conf}=&\frac{\log{\frac{1}{\conf}}}{\error^2 \left(\frac{\rpoly(\prob_1,\ldots, \prob_N)}{\abs{\etree}(1,\ldots, 1)}\right)^2}\\
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= &\frac{\log{\frac{1}{\conf}}\cdot \abs{\etree}^2(1,\ldots, 1)}{\error^2 \cdot \rpoly^2(\prob_1,\ldots, \prob_\numvar)},
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\end{align*}
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%and the runtime then follows, thus upholding ~\cref{lem:approx-alg}.
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which completes the proof.
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We now return to the proof of~\Cref{lem:mon-samp}:
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\subsection{Proof of Theorem \ref{lem:mon-samp}}
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Consider now the random variables $\randvar_1,\dots,\randvar_\numvar$, where each $\randvar_i$ is the value of $\vari{Y}_{\vari{i}}$ after~\Cref{alg:mon-sam-product} is executed. In particular, note that we have
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\[Y_i= \onesymbol\inparen{\monom\mod{\mathcal{B}}\not\equiv 0}\cdot \prod_{X_i\in \var\inparen{v}} p_i,\]
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where the indicator variable handles the check in~\Cref{alg:check-duplicate-block}
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Then for random variable $\randvar_i$, it is the case that
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\[\expct\pbox{\randvar_i} = \sum\limits_{(\monom, \coef) \in \expandtree{\etree} }\frac{\onesymbol\inparen{\monom\mod{\mathcal{B}}\not\equiv 0}\cdot c\cdot\prod_{X_i\in \var\inparen{v}} p_i }{\abs{\etree}(1,\dots,1)} = \frac{\rpoly(\prob_1,\ldots, \prob_\numvar)}{\abs{\etree}(1,\ldots, 1)},\]
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where in the first equality we use the fact that $\vari{sgn}_{\vari{i}}\cdot \abs{\coef}=\coef$ and the second equality follows from~\cref{eq:tilde-Q-bi} with $X_i$ substituted by $\prob_i$.
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Let $\empmean = \frac{1}{\samplesize}\sum_{i = 1}^{\samplesize}\randvar_i$. It is also true that
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\[\expct\pbox{\empmean} %\expct\pbox{ \frac{1}{\samplesize}\sum_{i = 1}^{\samplesize}\randvar_i}
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= \frac{1}{\samplesize}\sum_{i = 1}^{\samplesize}\expct\pbox{\randvar_i}
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= \frac{\rpoly(\prob_1,\ldots, \prob_\numvar)}{\abs{\etree}(1,\ldots, 1)}.\]
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Hoeffding's inequality states that if we know that each $\randvar_i$ (which are all independent) always lie in the intervals $[a_i, b_i]$, then it is true that
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\begin{equation*}
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P\left(\left|\empmean - \expct\pbox{\empmean}\right| \geq \error\right) \leq 2\exp{\left(-\frac{2\samplesize^2\error^2}{\sum_{i = 1}^{\samplesize}(b_i -a_i)^2}\right)}.
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\end{equation*}
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Line ~\ref{alg:mon-sam-sample} shows that $\vari{sgn}_\vari{i}$ has a value in $\{-1, 1\}$ that is multiplied with $O(k)$ $p_i\in [0, 1]$, the range for each $\randvar_i$ is $[-1, 1]$.
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Using Hoeffding's inequality, we then get:
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\begin{equation*}
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P\pbox{~\left| \empmean - \expct\pbox{\empmean} ~\right| \geq \error} \leq 2\exp{\left(-\frac{2\samplesize^2\error^2}{2^2 \samplesize}\right)} = 2\exp{\left(-\frac{\samplesize\error^2}{2 }\right)}\leq \conf,
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\end{equation*}
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where the last inequality follows from our choice of $\samplesize$ in~\Cref{alg:mon-sam-global2}.
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This concludes the proof for the first claim of theorem ~\ref{lem:mon-samp}.
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\paragraph{Run-time Analysis}
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The runtime of the algorithm is dominated by~\Cref{alg:mon-sam-onepass} (which by~\Cref{lem:one-pass} takes time $O(size(\etree))$) and the $\samplesize$ iterations of the loop in~\Cref{alg:sampling-loop}. Each iteration's run time is dominated by the call to~\Cref{alg:mon-sam-sample} (which by~\Cref{lem:sample} takes $O(\log{k} \cdot k \cdot depth(\etree))$) and~\Cref{alg:check-duplicate-block}, which by the subsequent argument takes $O(k\log{k})$ time. We sort the $O(k)$ variables by their block IDs and then check if there is a duplicate block ID or not. Adding up all the times discussed here gives us the desired overall runtime.
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\subsection{Proof of~\Cref{cor:approx-algo-const-p}}
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The result follows by first noting that by definition of $\gamma$, we have
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%\AH{Just wondering why you use $\geq$ as opposed to $=$?}
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%\AR{Ah, right-- fixed}
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\[\rpoly(1,\dots,1)= (1-\gamma)\cdot \abs{\etree}(1,\dots,1).\]
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Further, since each $p_i\ge p_0$ and $\poly(\vct{X})$ (and hence $\rpoly(\vct{X})$) has degree at most $k$, we have that
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\[ \rpoly(1,\dots,1) \ge p_0^k\cdot \rpoly(1,\dots,1).\]
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The above two inequalities implies $\rpoly(1,\dots,1) \ge p_0^k\cdot (1-\gamma)\cdot \abs{\etree}(1,\dots,1)$.
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%\AH{This looks really nice!}
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Applying this bound in the runtime bound in~\Cref{lem:approx-alg} gives the first claimed runtime. The final runtime of $O_k\left(\frac 1{\eps^2}\cdot\treesize(\etree)\cdot \log{\frac{1}{\conf}}\right)$ follows by noting that $depth(\etree)\le \treesize(\etree)$ and absorbing all factors that just depend on $k$.
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\subsection{Proof of~\Cref{lem:one-pass}}
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We prove the first part of lemma ~\ref{lem:one-pass}, i.e., correctness, by structural induction over the depth $d$ of the binary tree $\etree$.
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For the base case, $d = 0$, it is the case that the node is a leaf and therefore by definition ~\ref{def:express-tree} must be a variable or coefficient. When it is a variable, \textsc{OnePass} returns $1$, and we have in this case that $\polyf(\etree) = X_i = \polyf(\abs{\etree})$ for some $i$ in $[\numvar]$, and this evaluated at all $1$'s indeed gives $1$, verifying the correctness of the returned value of $\abs{\etree}(1,\ldots, 1) = 1$. When the root is a coefficient, the absolute value of the coefficient is returned, which is indeed $\abs{\etree}(1,\ldots, 1)$. This proves the base case.
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%\AH{The inductive step assumes $k \geq 0$ rather than $k \geq 1$, correct?}
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%\AR{yep!}
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For the inductive hypothesis, assume that for $d \leq k$ for some $k \geq 0$,~\Cref{lem:one-pass} is true for~\Cref{alg:one-pass}.
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Now prove that lemma ~\ref{lem:one-pass} holds for $k + 1$. Notice that $\etree$ has at most two children, $\etree_\lchild$ and $\etree_\rchild$. Note also, that for each child, it is the case that $d \leq k$. Then, by inductive hypothesis, lemma ~\ref{lem:one-pass} holds for each existing child, and we are left with two possibilities for $\etree$. The first case is when $\etree$ is a $+$ node. When this happens,~\Cref{alg:one-pass} computes $|T_\lchild|(1,\ldots, 1) + |T_\rchild|(1,\ldots, 1)$ on line ~\ref{alg:one-pass-plus-add} which by definition is $\abs{\etree}(1,\ldots, 1)$ and hence the inductive hypothesis holds in this case. For the weight computation of the children of $+$, by lines ~\ref{alg:one-pass-plus-add}, ~\ref{alg:one-pass-plus-assign2}, and ~\ref{alg:one-pass-plus-prob} algorithm ~\ref{alg:one-pass} computes $\etree_i.\wght = \frac{|T_i|(1,\ldots, 1)}{|T_\lchild|(1,\ldots, 1) + |T_\rchild|(1,\ldots, 1)}$ which is indeed as claimed. The second case is when the $\etree.\val = \times$. By inductive hypothesis, it is the case that both $\abs{\etree_\lchild}\polyinput{1}{1}$ and $\abs{\etree_\rchild}\polyinput{1}{1}$ have been correctly computed. On line~\ref{alg:one-pass-times-product} algorithm ~\ref{alg:one-pass} then computes the product of the subtree partial values, $|T_\lchild|(1,\ldots, 1) \cdot |T_\rchild|(1,\ldots, 1)$ which by definition is $\abs{\etree}(1,\ldots, 1)$.
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\paragraph{Run-time Analysis}
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The runtime for \textsc{OnePass} is fairly straight forward. Note first that each node is visited at most one time. Second, for each type of node visited, it can be trivially verified that there are only a constant number of operations. This concludes then with a $O\left(\treesize(\etree)\right)$ runtime.
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\subsubsection{Proof of~\Cref{lem:sample}}
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First, we need to show that $\sampmon$ indeed returns a monomial $\monom$,\footnote{Technically it returns $\var(\monom)$ but for less cumbersome notation we will refer to $\var(\monom)$ simply by $\monom$ in this proof.} such that $(\monom, \coef)$ is in $\expandtree{\etree}$, which we do by induction on the depth of $\etree$.
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For the base case, let the depth $d$ of $\etree$ be $0$. We have that the root node is either a constant $\coef$ for which by line ~\ref{alg:sample-num-return} we return $\{~\}$, or we have that $\etree.\type = \var$ and $\etree.\val = x$, and by line ~\ref{alg:sample-var-return} we return $\{x\}$. Both cases sample a monomial%satisfy ~\cref{def:monomial}
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, and the base case is proven.
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For the inductive hypothesis, assume that for $d \leq k$ for some $k \geq 0$, that it is indeed the case that $\sampmon$ returns a monomial.
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For the inductive step, let us take a tree $\etree$ with $d = k + 1$. Note that each child has depth $d \leq k$, and by inductive hypothesis both of them return a valid monomial. Then the root can be either a $+$ or $\times$ node. For the case of a $+$ root node, line ~\ref{alg:sample-plus-bsamp} of $\sampmon$ will choose one of the children of the root. Since by inductive hypothesis it is the case that a monomial is being returned from either child, and only one of these monomials is selected, we have for the case of $+$ root node that a valid monomial is returned by $\sampmon$. When the root is a $\times$ node, lines ~\ref{alg:sample-times-union} and ~\ref{alg:sample-times-product} multiply the monomials returned by the two children of the root, and it is trivial to see that %by definition ~\ref{def:monomial}
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the product of two monomials is also a monomial, which means that $\sampmon$ returns a valid monomial for the $\times$ root node, thus concluding the fact that $\sampmon$ indeed returns a monomial.
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We will next prove by induction on the depth $d$ of $\etree$ that the $(\monom,\coef)$ returned by $\sampmon$ has a probability %`that is in accordance with the monomial sampled,
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$\frac{|\coef|}{\abs{\etree}\polyinput{1}{1}}$.
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For the base case $d = 0$, by definition ~\ref{def:express-tree} we know that the root has to be either a coefficient or a variable. For either case, the probability of the value returned is $1$ since there is only one value to sample from. When the root is a variable $x$ the algorithm correctly returns $(\{x\}, 1 )$. When the root is a coefficient, \sampmon ~correctly returns $(\{~\}, sign(\coef_i))$.
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For the inductive hypothesis, assume that for $d \leq k$ and $k \geq 0$ $\sampmon$ indeed samples $\monom$ in $(\monom, \coef)$ in $\expandtree{\etree}$ with probability $\frac{|\coef|}{\abs{\etree}\polyinput{1}{1}}$.%bove is true.%lemma ~\ref{lem:sample} is true.
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We prove now, that when $d = k + 1$ the inductive step holds. It is the case that the root of $\etree$ has up to two children $\etree_\lchild$ and $\etree_\rchild$. Since $\etree_\lchild$ and $\etree_\rchild$ are both depth $d \leq k$, by inductive hypothesis, $\sampmon$ will sample both monomials $\monom_\lchild$ in $(\monom_\lchild, \coef_\lchild)$ of $\expandtree{\etree_\lchild}$ and $\monom_\rchild$ in $(\monom_\rchild, \coef_\rchild)$ of $\expandtree{\etree_\rchild}$, from $\etree_\lchild$ and $\etree_\rchild$ with probability $\frac{|\coef_\lchild|}{\abs{\etree_\lchild}\polyinput{1}{1}}$ and $\frac{|\coef_\rchild|}{\abs{\etree_\rchild}\polyinput{1}{1}}$.
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Then the root has to be either a $+$ or $\times$ node.
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Consider the case when the root is $\times$. Note that we are sampling a term from $\expandtree{\etree}$. Consider $(\monom, \coef)$ in $\expandtree{\etree}$, where $\monom$ is the sampled monomial. Notice also that it is the case that $\monom = \monom_\lchild \times \monom_\rchild$, where $\monom_\lchild$ is coming from $\etree_\lchild$ and $\monom_\rchild$ from $\etree_\rchild$. The probability that \sampmon$(\etree_{\lchild})$ returns $\monom_\lchild$ is $\frac{|\coef_{\monom_\lchild}|}{|\etree_\lchild|(1,\ldots, 1)}$ and $\frac{|\coef_{\monom_\lchild}|}{\abs{\etree_\rchild}\polyinput{1}{1}}$ for $\monom_\rchild$. Since both $\monom_\lchild$ and $\monom_\rchild$ are sampled with independent randomness, the final probability for sample $\monom$ is then $\frac{|\coef_{\monom_\lchild}| \cdot |\coef_{\monom_R}|}{|\etree_\lchild|(1,\ldots, 1) \cdot |\etree_\rchild|(1,\ldots, 1)}$. For $(\monom, \coef)$ in \expandtree{\etree}, it is indeed the case that $|\coef_i| = |\coef_{\monom_\lchild}| \cdot |\coef_{\monom_\rchild}|$ and that $\abs{\etree}(1,\ldots, 1) = |\etree_\lchild|(1,\ldots, 1) \cdot |\etree_\rchild|(1,\ldots, 1)$, and therefore $\monom$ is sampled with correct probability $\frac{|\coef_i|}{\abs{\etree}(1,\ldots, 1)}$.
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For the case when $\etree.\val = +$, \sampmon ~will sample monomial $\monom$ from one of its children. By inductive hypothesis we know that any $\monom_\lchild$ in $\expandtree{\etree_\lchild}$ and any $\monom_\rchild$ in $\expandtree{\etree_\rchild}$ will both be sampled with correct probability $\frac{|\coef_{\monom_\lchild}|}{\etree_{\lchild}(1,\ldots, 1)}$ and $\frac{|\coef_{\monom_\rchild}|}{|\etree_\rchild|(1,\ldots, 1)}$, where either $\monom_\lchild$ or $\monom_\rchild$ will equal $\monom$, depending on whether $\etree_\lchild$ or $\etree_\rchild$ is sampled. Assume that $\monom$ is sampled from $\etree_\lchild$, and note that a symmetric argument holds for the case when $\monom$ is sampled from $\etree_\rchild$. Notice also that the probability of choosing $\etree_\lchild$ from $\etree$ is $\frac{\abs{\etree_\lchild}\polyinput{1}{1}}{\abs{\etree_\lchild}\polyinput{1}{1} + \abs{\etree_\rchild}\polyinput{1}{1}}$ as computed by $\onepass$. Then, since $\sampmon$ goes top-down, and each sampling choice is independent (which follows from the randomness in the root of $\etree$ being independent from the randomness used in its subtrees), the probability for $\monom$ to be sampled from $\etree$ is equal to the product of the probability that $\etree_\lchild$ is sampled from $\etree$ and $\monom$ is sampled in $\etree_\lchild$, and
|
|
\begin{align*}
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|
&P(\sampmon(\etree) = \monom) = \\
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|
&P(\sampmon(\etree_\lchild) = \monom) \cdot P(SampledChild(\etree) = \etree_\lchild)\\
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&= \frac{|\coef_\monom|}{|\etree_\lchild|(1,\ldots, 1)} \cdot \frac{\abs{\etree_\lchild}(1,\ldots, 1)}{|\etree_\lchild|(1,\ldots, 1) + |\etree_\rchild|(1,\ldots, 1)}\\
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|
&= \frac{|\coef_\monom|}{\abs{\etree}(1,\ldots, 1)},
|
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\end{align*}
|
|
and we obtain the desired result.
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\paragraph{Run-time Analysis}
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We now bound the number of recursive calls in $\sampmon$ by $O\left(k\cdot depth(\etree)\right)$. Note that a sampled monomial corresponds to a subtree of $\etree$. Take an arbitrary sample subgraph of expression tree $\etree$ and note that since every monomial has degree $k$, the subgraph has $O(k)$ leaves and the number of nodes in each layer as one goes from leaves to the root can only go down. Since the sub-graph has depth at most $depth(\etree)$ and that each level has $O(k)$ nodes, the sub-graph as $O(k\cdot depth(\etree))$ nodes in it. Since each node in the sub-graph corresponds to a recursive call we get the desired bound.
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It is easy to check that except for~\Cref{alg:sample-times-union}, all other lines take $O(1)$ time. Thus, overall all lines except for~\Cref{alg:sample-times-union} take $O(k\cdot depth(\etree))$ time. Now consider all executions of~\Cref{alg:sample-times-union} together. We note that at each level we will be adding a given set of variables to some set at most once: since the sum of the sizes of the sets at a given level is at most $k$, each level involves $O(k\log{k})$ time. Thus, overall all executions of~\Cref{alg:sample-times-union} takes $O(k\log{k}\cdot depth(T))$ time, as desired.
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