section 4 added
This commit is contained in:
rriegel
@@ -10,24 +10,26 @@
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\newtheorem{corollary} {Corollary}
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\newtheorem{definition} {Definition}
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\newcommand{\OpSym}{\mathrm{O\!p}}
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\newcommand{\opsym}{{\scriptstyle \mathrm{o\!p}}}
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% \newcommand{\OpSym}{\bigoplus\nolimits}
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% \newcommand{\opsym}{\oplus}
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\newcommand{\OpCurry}[3][]{\mathop{\OpSym^{#1}_{#2}{#3}}}
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\newcommand{\opcurry}[3][]{\mathop{\opsym^{#1}_{#2}{#3}}}
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\newcommand{\Op}[2][]{\OpCurry[#1]{#2}{}}
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\newcommand{\op}[2][]{\opcurry[#1]{#2}{}}
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\newcommand{\VecOp}[2][]{\mathop{\overrightarrow{\OpSym^{#1}_{#2}}}}
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\newcommand{\vecop}[2][]{\mathop{\overrightarrow{\opsym^{#1}_{#2}}}}
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\newcommand{\comp}{\mathop{\circ}\nolimits}
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\newcommand{\GNP}{\psi_{\Theta}}
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\newcommand{\GNP}[1][\psi]{{#1}_{\Theta}}
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\newcommand{\GNPvec}[1][\psi]{{#1}_{\overrightarrow{\Theta}}}
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\newcommand{\otimesvec}{\mathbin{\overrightarrow{\otimes}}}
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\newcommand{\bigotimesvec}{\mathop{\overrightarrow{\bigotimes}}}
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\newcommand{\otimeshat}{\mathbin{\widehat{\otimes}}}
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\newcommand{\odothat}{\mathbin{\widehat{\odot}}}
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\newcommand{\otimestilde}{\mathbin{\widetilde{\otimes}}}
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\newcommand{\odottilde}{\mathbin{\widetilde{\odot}}}
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\newcommand{\bigotimestilde}{\mathop{\widetilde{\bigotimes}}}
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\newcommand{\bigodottilde}{\mathop{\widetilde{\bigodot}}}
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\DeclareMathOperator*{\argmin}{argmin}
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\DeclareMathOperator*{\argmax}{argmax}
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\DeclareMathOperator*{\map}{map}
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\newcommand{\comp}{\mathbin{\circ}}
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\newcommand{\st}{{\rm~s.t.~}}
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\title{Some Awesome Title}
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@@ -363,18 +365,18 @@ presented in the next section, though we will later introduce a
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transform to help work around them.
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\begin{definition}
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A second-order reduce problem $\Theta$ is {\em regular} if $g(y,a) =
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a$ for all $y \in \mathcal{Y}$ and $a \in \mathcal{A}$. Such a
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problem is given by $\Psi_{\Theta} = h \comp \psi_{\Theta}$, where
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a$ for all $y \in \mathcal{Y}$ and $a \in \mathcal{A}$, and is thus
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given by $\Psi_{\Theta} = h \comp \psi_{\Theta}$, where
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$\psi_{\Theta}(Y,X) = \bigodot_{y \in Y} \bigotimes_{x \in X}
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f(x,y)$.
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f(x,y)$. Note that $\mathcal{B} = \mathcal{A}$.
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\end{definition}
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\begin{definition}
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A regular second-order reduce problem $\Theta$ is {\em block
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decomposable} if, for all nonempty partitions $X^L \cup X^R = X
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decomposable} if, for all nonempty partitions $X^{\!L} \cup X^{\!R} = X
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\subset \mathcal{X}$ and nonempty $Y \in \mathcal{Y}$, $\GNP(Y,X) =
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\GNP(Y,X^L) \otimes \GNP(Y,X^R)$. Such a problem\footnote{Observe
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\GNP(Y,X^{\!L}) \otimes \GNP(Y,X^{\!R})$. Such a problem\footnote{Observe
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that commutativity and associativity ensure that $\GNP(Y,X) =
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\GNP(Y^L,X) \odot \GNP(Y^R,X)$.} is known as a {\em second-order
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\GNP(Y^{\!L},X) \odot \GNP(Y^{\!R},X)$.} is known as a {\em second-order
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generalized $N$-body problem}.
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\end{definition}
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@@ -392,14 +394,13 @@ vectorization.
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equivalent to $\overrightarrow{\Theta}$ with
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\[ \begin{array}{rclrcl}
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\overrightarrow{f}(x,y) & = & \{(y, f(x,y))\}, & \overrightarrow{g}(A) & = & \{(y, g(y,v)) | (y,v) \in A\}, \\
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A \mathop{\overrightarrow{\otimes}}\nolimits B & = & \{(y, u \otimes v) | (y,u) \in A, (y,v) \in B\}, & A \odot B & = & A \cup B.
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A \otimesvec B & = & \{(y, u \otimes v) | (y,u) \in A, (y,v) \in B\}, & A \odot B & = & A \cup B.
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\end{array} \]
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\end{lemma}
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\begin{proof}
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During na\"{\i}ve computation, arguments presented to
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$\mathop{\overrightarrow{\otimes}}\nolimits$ and
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$\overrightarrow{g}$ are singleton sets for some $y \in Y$. Vector
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operations then trivially match the original version of the
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During na\"{\i}ve computation, arguments presented to $\otimesvec$
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and $\overrightarrow{g}$ are singleton sets for some $y \in Y$.
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Vector operations then trivially match the original version of the
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algorithm.
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\end{proof}
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\noindent Vectorized operations become more interesting after the
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@@ -420,11 +421,11 @@ regions, possibly with permuted orderings of rows or columns.
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\begin{eqnarray*}
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\begin{array}{ccccccccc}
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\scriptstyle ( \!\!\!&\scriptstyle\!\!\! f(x_1,y_1) \!\!\!&\scriptstyle\!\!\! \otimes \!\!\!&\scriptstyle\!\!\! f(x_2,y_1) \!\!\!&\scriptstyle\!\!\! \otimes \!\!\!&\scriptstyle\!\!\! \cdots \!\!\!&\scriptstyle\!\!\! \otimes \!\!\!&\scriptstyle\!\!\! f(x_N,y_1) \!\!\!&\scriptstyle\!\!\! ) \\
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&\scriptstyle \odot \\
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\multicolumn{9}{c}{\scriptstyle \odot} \\
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\scriptstyle ( \!\!\!&\scriptstyle\!\!\! f(x_1,y_2) \!\!\!&\scriptstyle\!\!\! \otimes \!\!\!&\scriptstyle\!\!\! f(x_2,y_2) \!\!\!&\scriptstyle\!\!\! \otimes \!\!\!&\scriptstyle\!\!\! \cdots \!\!\!&\scriptstyle\!\!\! \otimes \!\!\!&\scriptstyle\!\!\! f(x_N,y_2) \!\!\!&\scriptstyle\!\!\! ) \\
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&\scriptstyle \odot \\
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&\scriptstyle \vdots \\
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&\scriptstyle \odot \\
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\multicolumn{9}{c}{\scriptstyle \odot} \\
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\multicolumn{9}{c}{\scriptstyle \vdots} \\
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\multicolumn{9}{c}{\scriptstyle \odot} \\
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\scriptstyle ( \!\!\!&\scriptstyle\!\!\! f(x_1,y_M) \!\!\!&\scriptstyle\!\!\! \otimes \!\!\!&\scriptstyle\!\!\! f(x_2,y_M) \!\!\!&\scriptstyle\!\!\! \otimes \!\!\!&\scriptstyle\!\!\! \cdots \!\!\!&\scriptstyle\!\!\! \otimes \!\!\!&\scriptstyle\!\!\! f(x_N,y_M) \!\!\!&\scriptstyle\!\!\! )
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\end{array}
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& = &
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@@ -441,11 +442,11 @@ regions, possibly with permuted orderings of rows or columns.
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\!\!\!&\scriptstyle\!\!\! \otimes \!\!\!&\!\!\!
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\left( \begin{array}{ccccccc}
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\scriptstyle ( \!\!\!&\scriptstyle\!\!\! f(x_2,y_1) \!\!\!&\scriptstyle\!\!\! \otimes \!\!\!&\scriptstyle\!\!\! \cdots \!\!\!&\scriptstyle\!\!\! \otimes \!\!\!&\scriptstyle\!\!\! f(x_N,y_1) \!\!\!&\scriptstyle\!\!\! ) \\
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&\scriptstyle \odot \\
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\multicolumn{7}{c}{\scriptstyle \odot} \\
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\scriptstyle ( \!\!\!&\scriptstyle\!\!\! f(x_2,y_2) \!\!\!&\scriptstyle\!\!\! \otimes \!\!\!&\scriptstyle\!\!\! \cdots \!\!\!&\scriptstyle\!\!\! \otimes \!\!\!&\scriptstyle\!\!\! f(x_N,y_2) \!\!\!&\scriptstyle\!\!\! ) \\
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&\scriptstyle \odot \\
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&\scriptstyle \vdots \\
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&\scriptstyle \odot \\
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\multicolumn{7}{c}{\scriptstyle \odot} \\
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\multicolumn{7}{c}{\scriptstyle \vdots} \\
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\multicolumn{7}{c}{\scriptstyle \odot} \\
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\scriptstyle ( \!\!\!&\scriptstyle\!\!\! f(x_2,y_M) \!\!\!&\scriptstyle\!\!\! \otimes \!\!\!&\scriptstyle\!\!\! \cdots \!\!\!&\scriptstyle\!\!\! \otimes \!\!\!&\scriptstyle\!\!\! f(x_N,y_M) \!\!\!&\scriptstyle\!\!\! )
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\end{array} \right)
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\end{array}
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@@ -466,8 +467,8 @@ operator pairs that gaurantee block decomposability.
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\begin{proof}
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By commutativity and associativity, we may rearrange
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\[ \begin{array}{ll}
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\multicolumn{2}{l}{\displaystyle \GNP(Y,X) = \bigotimes_{y \in Y} \bigotimes_{x \in X} f(x,y) = \bigotimes_{x \in X} \bigotimes_{y \in Y} f(x,y) = \bigotimes_{x \in X^L} \bigotimes_{y \in Y} f(x,y) \otimes \bigotimes_{x \in X^R} \bigotimes_{y \in Y} f(x,y)} \\
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& \displaystyle = \bigotimes_{y \in Y} \bigotimes_{x \in X^L} f(x,y) \otimes \bigotimes_{y \in Y} \bigotimes_{x \in X^R} f(x,y) = \GNP(Y,X^L) \otimes \GNP(Y,X^R).
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\multicolumn{2}{l}{\displaystyle \GNP(Y,X) = \bigotimes_{y \in Y} \bigotimes_{x \in X} f(x,y) = \bigotimes_{x \in X} \bigotimes_{y \in Y} f(x,y) = \bigotimes_{x \in X^{\!L}} \bigotimes_{y \in Y} f(x,y) \otimes \bigotimes_{x \in X^{\!R}} \bigotimes_{y \in Y} f(x,y)} \\
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& \displaystyle = \bigotimes_{y \in Y} \bigotimes_{x \in X^{\!L}} f(x,y) \otimes \bigotimes_{y \in Y} \bigotimes_{x \in X^{\!R}} f(x,y) = \GNP(Y,X^{\!L}) \otimes \GNP(Y,X^{\!R}).
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\end{array} \]
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\end{proof}
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\begin{lemma}\label{lem:map}
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@@ -475,13 +476,13 @@ operator pairs that gaurantee block decomposability.
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$\odot = \map$.
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\end{lemma}
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\begin{proof}
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For $\psi_{\Theta}(Y,X) = \map_{y \in Y} \bigotimes_{x \in X} f(x,y)
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\equiv \bigcup_{y \in Y} \mathop{\overrightarrow{\bigotimes}}_{x \in
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X} \{(y,f(x,y))\} = \psi_{\overrightarrow{\Theta}}(Y,X)$ and by
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commutativity, associativity, and the definition of map, we have
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For $\GNP(Y,X) = \map_{y \in Y} \bigotimes_{x \in X} f(x,y) \equiv
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\bigcup_{y \in Y} \bigotimesvec_{x \in X} \{(y,f(x,y))\} =
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\GNPvec(Y,X)$ and by commutativity, associativity, and the
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definition of map, we have
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\[ \begin{array}{ll}
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\multicolumn{2}{l}{\displaystyle \psi_{\overrightarrow{\Theta}}(Y,X) \equiv \Big\{ \!\Big( y,\bigotimes_{x \in X} f(x,y) \Big)\! \Big| y \in Y \Big\} = \Big\{ \!\Big( y,\bigotimes_{x \in X^L} f(x,y) \otimes \bigotimes_{x \in X^R} f(x,y) \Big)\! \Big| y \in Y \Big\}} \\
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& \displaystyle = \Big\{ \!\Big( y,\bigotimes_{x \in X^L} f(x,y) \Big)\! \Big| y \in Y \Big\} \mathop{\overrightarrow{\otimes}}\nolimits \Big\{ \!\Big( y,\bigotimes_{x \in X^R} f(x,y) \Big)\! \Big| y \in Y \Big\} \equiv \psi_{\overrightarrow{\Theta}}(Y,X^L) \mathop{\overrightarrow{\otimes}}\nolimits \psi_{\overrightarrow{\Theta}}(Y,X^R).
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\multicolumn{2}{l}{\displaystyle \GNPvec(Y,X) \equiv \Big\{ \!\Big( y,\bigotimes_{x \in X} f(x,y) \Big)\! \Big| y \in Y \Big\} = \Big\{ \!\Big( y,\bigotimes_{x \in X^{\!L}} f(x,y) \otimes \bigotimes_{x \in X^{\!R}} f(x,y) \Big)\! \Big| y \in Y \Big\}} \\
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& \displaystyle = \Big\{ \!\Big( y,\bigotimes_{x \in X^{\!L}} f(x,y) \Big)\! \Big| y \in Y \Big\} \otimesvec \Big\{ \!\Big( y,\bigotimes_{x \in X^{\!R}} f(x,y) \Big)\! \Big| y \in Y \Big\} \equiv \GNPvec(Y,X^{\!L}) \otimesvec \GNPvec(Y,X^{\!R}).
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\end{array} \]
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\end{proof}
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@@ -520,112 +521,230 @@ to the na\"{\i}ve computation,
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\[
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\GNP(Y,X) = \left\{ \begin{array}{lrr}
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f(x,y) & \multicolumn{2}{r}{\mbox{if } X = \{x\} \mbox{ and } Y = \{y\},} \\
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\multicolumn{2}{l}{\GNP(Y^L,X) \odot \GNP(Y^R,X)} & \mbox{if } |Y| \geq |X|, \\
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\multicolumn{2}{l}{\GNP(Y,X^L) \otimes \GNP(Y,X^R)} & \mbox{otherwise}.
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\multicolumn{2}{l}{\GNP(Y^{\!L},X) \odot \GNP(Y^{\!R},X)} & \mbox{if } Y \succ X, \\
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\multicolumn{2}{l}{\GNP(Y,X^{\!L}) \otimes \GNP(Y,X^{\!R})} & \mbox{otherwise},
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\end{array} \right.
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\]
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Recursion forms a binary tree with one leaf per element of $X \times
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Y$. Exhuastive computation thus requires time $O(N^2)$, the same as
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the na\"{\i}ve algorithm. It may be possible, however, to obtain
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results for some components of the recursive block decomposition
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without computing them exhaustively. Exploiting this, we hope to
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drive the expected running time down.
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where $Y \succ X$ is some means of deciding what to split first, such
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as $|Y| \geq |X|$. Recursion forms a binary tree with one leaf per
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element of $X \times Y$. Exhuastive computation thus requires time
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$O(N^2)$, the same as the na\"{\i}ve algorithm. It may be possible,
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however, to obtain results for some components of the recursive block
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decomposition without computing them exhaustively (i.e.~to {\em prune}
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them). Exploiting this, we hope to drive the expected running time
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down.
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% \subsection{Summaries and Statistics}
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{\bf Summaries and Statistics}
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{\bf Summaries and Statistics.} In order to accelerate computation,
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we need some quick means of summarizing the possible results of
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$\GNP(Y,X)$. This typically involves consideration of $X$ and $Y$ at
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the abstract level formed by a concise (and ideally precomputed) set
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of statistics on the two.
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\begin{definition}
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Given statistics functions $\sigma_x \colon 2^{\mathcal{X}} \to
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\mathcal{S}_x$ and $\sigma_y \colon 2^{\mathcal{Y}} \to
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\mathcal{S}_y$, let summary function $\GNP[\sigma] \colon
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\mathcal{S}_y \times \mathcal{S}_x \to 2^\mathcal{A}$ be such that
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\[
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\GNP[\sigma](\sigma_y(Y),\sigma_x(X)) \supseteq \{\GNP(Y^*,X^*) | X^* \st \sigma(X^*) = \sigma(X), Y^* \st \sigma(Y^*) = \sigma(Y)\}.
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\]
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\end{definition}
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\noindent Intuitively, $\GNP[\sigma](\sigma_y(Y),\sigma_x(X))$
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represents all possible results of $\GNP(Y,X)$ given what we know
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about $X$ and $Y$. It is permitted to be a superset of such results
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because the exact set may be costly or impossible to represent. Note
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that functions $\sigma_x$, $\sigma_y$, and $\GNP[\sigma]$ are not
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unique for a given GNP; indeed, chosing the right statistics can
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significantly impact running time. A common example of statistics is
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finding bounding boxes of data in Euclidean space. Summaries computed
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from these may be represented with upper and lower bounds on distances
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between points in $X$ and $Y$.
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% \subsection{Expansion Patterns}
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% \subsection{Intrinsic Pruning}
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% \subsection{All Manner of Pruning}
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{\bf Intrinsic Pruning.} Summarization leads directly to our first
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form of pruning.
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\begin{lemma}
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We may perform {\em intrinsic pruning} when summary results form a
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singleton set.
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\end{lemma}
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\begin{proof}
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Setting $X^* = X$ and $Y^* = Y$ meets the requirements for inclusion
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in $\GNP[\sigma](\sigma_y(Y),\sigma_x(X))$. Thus, summary results
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contain the exact result. Thus, singleton
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$\GNP[\sigma](\sigma_y(Y),\sigma_x(X)) = \{\GNP(Y,X)\}$ and we are
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free to shortcut all further computation on $X$ and $Y$ with this
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value.
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\end{proof}
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\begin{corollary}
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We may perform intrinsic pruning for the 2-point correlation.
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\end{corollary}
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\begin{proof}
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Let $\sigma_x$ and $\sigma_y$ find bounding boxes for points in $X$
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and $Y$ and let $D^{\!U}$ and $D^{\!L}$ find upper and lower bounds
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between bounding boxes. For $x \in X$ and $y \in Y$, the bounds on
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$I(d(x,y) \leq r)$ are then $\left[
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I(D^{\!U}(\sigma_x(X),\sigma_y(Y)) \leq r),
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I(D^{\!L}(\sigma_x(X),\sigma_y(Y)) \leq r) \right]$. We may thus
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define
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\[
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\GNP[\sigma](Y,X) = \left[ |Y| \cdot |X| \cdot I(D^{\!U}(\sigma_x(X),\sigma_y(Y)) \leq r), |Y| \cdot |X| \cdot I(D^{\!L}(\sigma_x(X),\sigma_y(Y)) \leq r) \right].
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\]
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This set is $\{|Y| \cdot |X|\}$ when
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$D^{\!U}(\sigma_x(X),\sigma_y(Y)) \leq r$ and $\{0\}$ when
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$D^{\!L}(\sigma_x(X),\sigma_y(Y)) > r$.
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\end{proof}
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% \subsection{Iterative Refinement.}
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{\bf Iterative Refinement.} Further pruning is possible in some
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problems by considering results gathered from other parts of
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computation. This is assisted by the ability to compose summary
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results.
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\begin{definition}
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Given summary results $A, B \subset \mathcal{A}$, let operators
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$\otimeshat, \odothat \colon 2^{\mathcal{A}} \times 2^{\mathcal{A}}
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\to 2^{\mathcal{A}}$ be such that
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\[ \begin{array}{rcl}
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A \otimeshat B \supseteq \{a \otimes b | a \in A, b \in B\} & \mbox{ and, likewise, } & A \odothat B \supseteq \{a \odot b | a \in A, b \in B\}.
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\end{array} \]
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\end{definition}
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\noindent We develope a notion of itertive refinement by means of
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replacing summary result sets for the various components of
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computation with composed summary results for their left and right
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subcomponents.
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\begin{definition}
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To perform {\em iterative refinement}, first initialize
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$\GNP[\Sigma](Y,X) \leftarrow
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\GNP[\sigma](\sigma_y(Y),\sigma_x(X))$, and then repeatedly select
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some $\GNP[\sigma](\sigma_y(Y'),\sigma_x(X'))$ present in
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$\GNP[\Sigma](Y,X)$ and replace it with
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\[
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\GNP[\sigma](\sigma_y(Y'),\sigma_x(X')) \leftarrow \left\{ \begin{array}{lrr}
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\{f(x,y)\} & \multicolumn{2}{r}{\mbox{if } X' = \{x\} \mbox{ and } Y' = \{y\},} \\
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\multicolumn{2}{l}{\GNP[\sigma](\sigma_y(Y^{\!L}),\sigma_x(X')) \odothat \GNP[\sigma](\sigma_y(Y^{\!R}),\sigma_x(X'))} & \mbox{if } Y \succ X, \\
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\multicolumn{2}{l}{\GNP[\sigma](\sigma_y(Y'),\sigma_x(X^{\!L})) \otimeshat \GNP[\sigma](\sigma_y(Y'),\sigma_x(X^{\!R}))} & \mbox{otherwise}.
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\end{array} \right.
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\]
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Futher, define $\GNP[\Sigma](Y',X')$ to refer to any component
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$\GNP[\sigma](\sigma_y(Y'),\sigma_x(X'))$ having been introduced or
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updated during refinement. The values of all $\GNP[\Sigma](Y',X')$
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are understood to reflect any changes made to their subcomponents.
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\end{definition}
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\noindent Like the recursive formulation, iterative refinement
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constructs a binary tree that grows by one node per replacement, and
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thus must terminate after $O(N^2)$ steps. It is useful to speak of
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this tree directly, with nodes $\GNP[\Sigma](Y',X')$ and functions
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$left$, $right$, $op$, $parent$, and $sibling$ defined intuitively.
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Refinement need not be performed in any particular pattern.
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Depth-first is often a good choice due to its low overhead, though
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pruning in some problems strongly favors other expansion patterns.
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% \subsection{Extrinsic Pruning}
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{\bf Extrinsic Pruning.} Iterative refinement allows us to prune
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components when more precise knowledge of their results cannot affect
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the global result.
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\begin{lemma}
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For node $\GNP[\Sigma](Y',X')$ at depth $D$ of tree
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$\GNP[\Sigma](Y,X)$ and path $A_0,\ldots,A_D$ given by $A_D =
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\GNP[\Sigma](Y',X')$ and $A_{d-1} = parent(A_{d})$ for $1 \leq d
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\leq D$, construct sets of summary result sets
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\[ \begin{array}{rcl}
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R_D = \{\GNP[\Sigma](Y',X')\} & \mbox{ and } & R_{d-1} = \{\{a\} \mathbin{op(A_{d-1})} B | a \in sibling(A_d), B \in R_d\}
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\end{array} \]
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for $1 \leq d \leq D$. We may then perform {\em extrinsic pruning}
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if $B$ is singleton for all $B \in R_0$.
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\end{lemma}
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\begin{proof}
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As constructed, $R_0$ is the set of all possible summary result sets
|
||||
for $\GNP[\Sigma](Y,X)$ given summary results of
|
||||
$\GNP[\Sigma](Y',X')$ and one possible result of all other
|
||||
components. Singleton $B \in R_0$ then implies that all possible
|
||||
results of $\GNP[\Sigma](Y',X')$ lead to the same global result
|
||||
under some valuation of the other components. If all such $B$ are
|
||||
singleton, then any $p \in \GNP[\Sigma](Y',X')$ must lead to the
|
||||
same global result as any other regardless of the valuation of the
|
||||
other components and we are free to shortcut all further compuation
|
||||
on $X'$ and $Y'$ with this value.
|
||||
\end{proof}
|
||||
|
||||
\begin{corellary}
|
||||
We may perform extrinsic pruning for all-nearest-neighbors.
|
||||
\end{corellary}
|
||||
\begin{proof}
|
||||
|
||||
\end{proof}
|
||||
|
||||
% \subsection{Thresholded Pruning}
|
||||
|
||||
{\bf Thresholded Pruning.} Postprocessing function $h$ introduces a
|
||||
third form of pruning.
|
||||
\begin{lemma}
|
||||
We may perform {\em thresholded pruning} if $\{h(b) | b \in B\}$ is
|
||||
singleton for all $B \in R_0$.
|
||||
\end{lemma}
|
||||
\begin{proof}
|
||||
Similar to the above, with all $p \in \GNP[\Sigma](Y',X')$ leading
|
||||
to the same postprocesed result.
|
||||
\end{proof}
|
||||
|
||||
% \subsection{Approximation Pruning}
|
||||
|
||||
% {\bf Approximation Pruning.} Some problems do not lend themselves to
|
||||
% any of the above forms of pruning. For these, we may still be able to
|
||||
% find approximate results with bounded error more quickly than
|
||||
% exhaustive computation. We must first establish a notion of error.
|
||||
% \begin{definition}
|
||||
% Let $div \colon 2^\mathcal{A} \to \mathcal{A} \times \mathcal{A} \to
|
||||
% \mathbb{R}$ be some measure of divergence between results given the
|
||||
% summary results of the full computation, written $div(a,b |
|
||||
% \GNP[\Sigma](Y,X))$. Define $err \colon 2^\mathcal{A} \to
|
||||
% 2^\mathcal{A} \to \mathbb{R}$ to be $err(\GNP[\Sigma](Y',X') |
|
||||
% \GNP[\Sigma](Y,X)) = \min_{\widehat{a} \in \mathcal{A}} \max_{b \in
|
||||
% \GNP[\Sigma](Y',X')} div(\widehat{a},b | \GNP[\Sigma](Y,X))$.
|
||||
% \end{definition}
|
||||
% \noindent For example, we might have $div(a,b | \GNP[\Sigma](Y,X)) =
|
||||
% |a - b| / \min_{c \in \GNP[\Sigma](Y,X)} |c|$, or relative error.
|
||||
%
|
||||
% Given a desired $\epsilon$, iterative refinement may terminate once
|
||||
% $err(\GNP[\Sigma](Y,X) | \GNP[\Sigma](Y,Z)) < \epsilon$, returning the
|
||||
% minimizing $\widehat{a}$ found for $err$. A simple algorithm might
|
||||
% then check the error of $\GNP[\Sigma](Y,X)$ after each step, refining
|
||||
% components in the order of descending error in attempt to make the
|
||||
% most of its work. This approach must use a priority queue to manage
|
||||
% expansion, incuring significant overhead.
|
||||
%
|
||||
% An alternate approach distributes error to the various components of
|
||||
% computation.
|
||||
% \begin{lemma}
|
||||
% Given summary results $A, B \subset \mathcal{A}$, let operators
|
||||
% $\otimestilde, \odottilde \colon \mathbb{R} \times \mathbb{R}
|
||||
% \to \mathbb{R}$ be such that
|
||||
% \begin{eqnarray*}
|
||||
% err(A | \GNP[\Sigma](Y,X)) \otimestilde err(B | \GNP[\Sigma](Y,X)) & = & err(A \otimeshat B | \GNP[\Sigma](Y,X)) \\
|
||||
% err(A | \GNP[\Sigma](Y,X)) \odottilde err(B | \GNP[\Sigma](Y,X)) & = & err(A \odothat B | \GNP[\Sigma](Y,X)).
|
||||
% \end{eqnarray*}
|
||||
% Given some desired $\epsilon$, let $\epsilon^*$ be such that
|
||||
% $\bigodottilde_{y \in Y} \bigotimestilde_{x \in X} \epsilon^* =
|
||||
% \epsilon$ and let $\epsilon' = \bigodottilde_{y \in Y'}
|
||||
% \bigotimestilde_{x \in X'} \epsilon^*$. We may perform {\em
|
||||
% approximation pruning} when $err(\GNP[\Sigma](Y',X') |
|
||||
% \GNP[\Sigma](Y,X)) < \epsilon'$.
|
||||
% \end{lemma}
|
||||
|
||||
% \subsection{Practical Considerations: Trees, Bounds}
|
||||
|
||||
\subsection{Abstraction}
|
||||
|
||||
We denote abstract results with the set $\GNP^A(X'_1,\ldots,X'_n)$ of
|
||||
all possible results of $\GNP(X'_1,\ldots,X'_n)$ given some selection
|
||||
of descriptive statistics on input $X'_i$, $1 \leq i \leq n$. This
|
||||
set must contain the exact result; accordingly, it cannot be empty.
|
||||
Operations on abstract results yield the set of all possible outcomes
|
||||
\[
|
||||
\begin{array}{rcl}
|
||||
\lefteqn{\GNP^A(X'_1,\ldots,X^L_i,\ldots,X'_n) \op{i} \GNP^A(X'_1,\ldots,X^R_i,\ldots,X'_n)} \\
|
||||
& \equiv & \{a \op{i} b | a \in \GNP^A(X'_1,\ldots,X^L_i,\ldots,X'_n), b \in \GNP^A(X'_1,\ldots,X^R_i,\ldots,X'_n)\} \\
|
||||
& & \mbox{} \cap \GNP^A(X'_1,\ldots,X^L_i \cup X^R_i,\ldots,X'_n).
|
||||
\end{array}
|
||||
\]
|
||||
[[Intersecting with abstract results for the composed region is
|
||||
optional, as the composed abstract results for the subregions is
|
||||
almost always tighter. Perhaps also indicate that exact results
|
||||
composed with abstract results work how you'd expect.]]
|
||||
|
||||
\subsection{The Algorithm}
|
||||
|
||||
[[Initialize frontier $\GNP^F$ to $\GNP^A(X_1,\ldots,X_n)$. Then
|
||||
perform the following update procedure until all abstract results have
|
||||
been elminated:]]
|
||||
|
||||
\begin{itemize}
|
||||
\item Select some $\GNP^A(X'_1,\ldots,X'_n)$ from $\GNP^F$
|
||||
\item If prune possible, replace with $a \in \GNP^A(X'_1,\ldots,X'_n)$
|
||||
\item If leaf, replace with $f(x_1,\ldots,x_n)$
|
||||
\item Otherwise, replace with $\GNP^A(X'_1,\ldots,X^L_i,\ldots,X'_n) \op{i} \GNP^A(X'_1,\ldots,X^R_i,\ldots,X'_n)$
|
||||
\end{itemize}
|
||||
|
||||
\[
|
||||
\begin{array}{rcl}
|
||||
\lefteqn{\GNP^A(X'_1,\ldots,X'_n)} \\
|
||||
& \leftarrow & \left\{
|
||||
\begin{array}{lr}
|
||||
a \in \GNP^A(X'_1,\ldots,X'_n) & \mbox{ if prune} \\
|
||||
f(x'_1,\ldots,x'_n) & \mbox{ if leaf} \\
|
||||
\GNP^A(X'_1,\ldots,X^L_i,\ldots,X'_n) \op{i} \GNP^A(X'_1,\ldots,X^R_i,\ldots,X'_n) & \mbox{ otherwise}
|
||||
\end{array}
|
||||
\right.
|
||||
\end{array}
|
||||
\]
|
||||
|
||||
\section{Exact Pruning}
|
||||
|
||||
\subsection{Intrinsic}
|
||||
|
||||
\subsection{Extrinsic}
|
||||
|
||||
\subsection{Global}
|
||||
|
||||
\section{Approximate Pruning}
|
||||
|
||||
\subsection{Global}
|
||||
|
||||
\subsection{Intrinsic}
|
||||
|
||||
\subsection{Extrinsic}
|
||||
|
||||
\section{Practical Considertaions}
|
||||
|
||||
[[Abstract results must be represented somehow. When intermediate
|
||||
results form a lattice, upper and lower bounds may be used. Further
|
||||
conditions on operators allow for easy maintenance of bounds.]]
|
||||
|
||||
[[Too expensive to decide how to decompose blocks on an individual
|
||||
basis, so reused precomputed tree splits when breaking up work. Trees
|
||||
also handy to find bottom-up statistics and store intermediate
|
||||
results.]]
|
||||
|
||||
[[We can't afford to recompute the global bounds on $\GNP^F$ after
|
||||
each update from scratch, but instead need some means of updating a
|
||||
running total, so to speak. For invertible operators, it is
|
||||
sufficient to apply and undo changes directly to some globally
|
||||
accessible value. Some noninvertible operators (such as min) don't
|
||||
need to be undone, but others can pose a challenge (resolved below).]]
|
||||
|
||||
[[Any expansion pattern may be used for choosing the next abstract
|
||||
result to refine in $\GNP^F$. Which to use is a trade-off between
|
||||
overhead and improved pruning information. Also, some expansion
|
||||
patterns (e.g.~depth-first) can eliminate the ``undo'' problem by
|
||||
allowing partial results to be stored at low cost.]]
|
||||
|
||||
\appendix
|
||||
|
||||
% \section{Full Permutability}
|
||||
@@ -644,18 +763,18 @@ allowing partial results to be stored at low cost.]]
|
||||
% \end{theorem}
|
||||
%
|
||||
% \begin{proof}
|
||||
% ($\Rightarrow$) Given selected split $i$ and partitions $X^L_i \cup
|
||||
% X^R_i = X_i$, we have
|
||||
% ($\Rightarrow$) Given selected split $i$ and partitions $X^{\!L}_i \cup
|
||||
% X^{\!R}_i = X_i$, we have
|
||||
% \[
|
||||
% \begin{array}{rcl}
|
||||
% \lefteqn{\GNP(X'_1,\ldots,X'_i,\ldots,X'_n)} \\
|
||||
% & = & \displaystyle \Op{1}_{x_1 \in X'_1}\cdots\Op{i}_{x_i \in X'_i}\cdots\Op{n}_{x_n \in X'_n}f(x_1,\ldots,x_n) \\
|
||||
% & = & \displaystyle \Op{i}_{x_i \in X'_i}\Op{1}_{x_1 \in X'_i}\cdots\Op{i-1}_{x_{i-1} \in X'_{i-1}}\Op{i+1}_{x_{i+1} \in X'_{i+1}}\cdots\Op{n}_{x_n \in X'_n}f(x_1,\ldots,x_n) \\
|
||||
% & = & \displaystyle \Op{i}_{x_i \in X^L_i}\Op{1}_{x_1 \in X'_i}\cdots\Op{i-1}_{x_{i-1} \in X'_{i-1}}\Op{i+1}_{x_{i+1} \in X'_{i+1}}\cdots\Op{n}_{x_n \in X'_n}f(x_1,\ldots,x_n) \\
|
||||
% & & \displaystyle \mbox{} \op{i} \Op{i}_{x_i \in X^R_i}\Op{1}_{x_1 \in X'_i}\cdots\Op{i-1}_{x_{i-1} \in X'_{i-1}}\Op{i+1}_{x_{i+1} \in X'_{i+1}}\cdots\Op{n}_{x_n \in X'_n}f(x_1,\ldots,x_n) \\
|
||||
% & = & \displaystyle \Op{1}_{x_1 \in X'_i}\cdots\Op{i}_{x_i \in X^L_i}\cdots\Op{n}_{x_n \in X'_n}f(x_1,\ldots,x_n) \\
|
||||
% & & \displaystyle \mbox{} \op{i} \Op{1}_{x_1 \in X'_i}\cdots\Op{i}_{x_i \in X^R_i}\cdots\Op{n}_{x_n \in X'_n}f(x_1,\ldots,x_n) \\
|
||||
% & = & \displaystyle \GNP(X'_1,\ldots,X^L_i,\ldots,X'_n) \op{i} \GNP(X'_1,\ldots,X^R_i,\ldots,X'_n).
|
||||
% & = & \displaystyle \Op{i}_{x_i \in X^{\!L}_i}\Op{1}_{x_1 \in X'_i}\cdots\Op{i-1}_{x_{i-1} \in X'_{i-1}}\Op{i+1}_{x_{i+1} \in X'_{i+1}}\cdots\Op{n}_{x_n \in X'_n}f(x_1,\ldots,x_n) \\
|
||||
% & & \displaystyle \mbox{} \op{i} \Op{i}_{x_i \in X^{\!R}_i}\Op{1}_{x_1 \in X'_i}\cdots\Op{i-1}_{x_{i-1} \in X'_{i-1}}\Op{i+1}_{x_{i+1} \in X'_{i+1}}\cdots\Op{n}_{x_n \in X'_n}f(x_1,\ldots,x_n) \\
|
||||
% & = & \displaystyle \Op{1}_{x_1 \in X'_i}\cdots\Op{i}_{x_i \in X^{\!L}_i}\cdots\Op{n}_{x_n \in X'_n}f(x_1,\ldots,x_n) \\
|
||||
% & & \displaystyle \mbox{} \op{i} \Op{1}_{x_1 \in X'_i}\cdots\Op{i}_{x_i \in X^{\!R}_i}\cdots\Op{n}_{x_n \in X'_n}f(x_1,\ldots,x_n) \\
|
||||
% & = & \displaystyle \GNP(X'_1,\ldots,X^{\!L}_i,\ldots,X'_n) \op{i} \GNP(X'_1,\ldots,X^{\!R}_i,\ldots,X'_n).
|
||||
% \end{array}
|
||||
% \]
|
||||
% ($\Leftarrow$) Given permutation $p_1,\ldots,p_n$ of the numbers
|
||||
|
||||
Reference in New Issue
Block a user