decomposition trees  (MathWorks Inc)

Journal: Theory and Applications of Satisfiability Testing – SAT 2020

Article Title: Taming High Treewidth with Abstraction, Nested Dynamic Programming, and Database Technology

doi: 10.1007/978-3-030-51825-7_25

Figure Lengend Snippet: Graph G (left), a TD \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{T}$$\end{document} of graph G (right).

Article Snippet: Compute (some) tree decomposition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}=(T,\chi )$$\end{document} of G . Traverse the nodes of T in post-order (bottom-up tree traversal of T ).

Techniques:

Journal: Theory and Applications of Satisfiability Testing – SAT 2020

Article Title: Taming High Treewidth with Abstraction, Nested Dynamic Programming, and Database Technology

doi: 10.1007/978-3-030-51825-7_25

Figure Lengend Snippet: Tables obtained by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsc {\#}\textsc {Sat}_t$$\end{document} on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{T}_{\text {nice}}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} of Example .

Article Snippet: Compute (some) tree decomposition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}=(T,\chi )$$\end{document} of G . Traverse the nodes of T in post-order (bottom-up tree traversal of T ).

Techniques:

Journal: Theory and Applications of Satisfiability Testing – SAT 2020

Article Title: Taming High Treewidth with Abstraction, Nested Dynamic Programming, and Database Technology

doi: 10.1007/978-3-030-51825-7_25

Figure Lengend Snippet: Selected tables obtained by nested DP on TD \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{T}'$$\end{document} of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_\varphi ^{\{x,y\}}$$\end{document} (left) and on TD \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}''$$\end{document} of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_\varphi ^{\{x\}}$$\end{document} (right) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} and projection variables \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P=\{x,y\}$$\end{document} of Example via \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathtt {HybDP}_{\textsc {\#}\exists \textsc {Sat}_t}$$\end{document} .

Article Snippet: Compute (some) tree decomposition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}=(T,\chi )$$\end{document} of G . Traverse the nodes of T in post-order (bottom-up tree traversal of T ).

Techniques:

Journal: Theory and Applications of Satisfiability Testing – SAT 2020

Article Title: Taming High Treewidth with Abstraction, Nested Dynamic Programming, and Database Technology

doi: 10.1007/978-3-030-51825-7_25

Figure Lengend Snippet: Cactus plot of instances for #Sat , where instances (x-axis) are ordered for each solver individually by runtime[seconds] (y-axis). \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {threshold}_{\text {abstr}}=38$$\end{document} .

Article Snippet: Compute (some) tree decomposition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}=(T,\chi )$$\end{document} of G . Traverse the nodes of T in post-order (bottom-up tree traversal of T ).

Techniques:

Journal: Theory and Applications of Satisfiability Testing – SAT 2020

Article Title: Taming High Treewidth with Abstraction, Nested Dynamic Programming, and Database Technology

doi: 10.1007/978-3-030-51825-7_25

Figure Lengend Snippet: Number of solved \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsc {\#}\exists \textsc {Sat}$$\end{document} insts., grouped by upper bound intervals of treewidth (left), cactus plot (right). time[h] is cumulated wall clock time, timeouts count as 900 s. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {threshold}_{\text {abstr}}\,{=}\,8$$\end{document} .

Article Snippet: Compute (some) tree decomposition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}=(T,\chi )$$\end{document} of G . Traverse the nodes of T in post-order (bottom-up tree traversal of T ).

Techniques:

Journal: Theory and Applications of Satisfiability Testing – SAT 2020

Article Title: Taming High Treewidth with Abstraction, Nested Dynamic Programming, and Database Technology

doi: 10.1007/978-3-030-51825-7_25

Figure Lengend Snippet: Scatter plot of instances for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsc {\#}\exists \textsc {Sat}$$\end{document} , where the x-axis shows runtime in seconds of nestHDB compared to the y-axis showing runtime of projMC (left) and of ganak (right). \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {threshold}_{\text {abstr}}=8$$\end{document} .

Article Snippet: Compute (some) tree decomposition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}=(T,\chi )$$\end{document} of G . Traverse the nodes of T in post-order (bottom-up tree traversal of T ).

Techniques:

Journal: Entropy

Article Title: Optimal Estimation of Wavelet Decomposition Level for a Matching Pursuit Algorithm

doi: 10.3390/e21090843

Figure Lengend Snippet: ( a ) Source signal. ( b ) Entropy (Shannon) vs. decomposition level, wavelet (Daubechies) basis. ( c ) Approximation error vs. decomposition level, wavelet (Daubechies) basis. ( d ) Execution time vs. decomposition level, wavelet (Daubechies) basis. ( e ) Approximation error vs. decomposition level, extended basis. ( f ) Execution time vs. decomposition level, extended basis.

Article Snippet: To optimize the packet wavelet decomposition tree in MATLAB, two optimization functions based on different entropy criteria were presented [ , ].

Techniques:

Journal: Entropy

Article Title: Optimal Estimation of Wavelet Decomposition Level for a Matching Pursuit Algorithm

doi: 10.3390/e21090843

Figure Lengend Snippet: ( a ) Source signal. ( b ) Entropy (Log Energy) vs. decomposition level, wavelet (Daubechies) basis. ( c ) Approximation error vs. decomposition level, wavelet (Daubechies) basis. ( d ) Execution time vs. decomposition level, wavelet (Daubechies) basis. ( e ) Approximation error vs. decomposition level, extended basis. ( f ) Execution time vs. decomposition level, extended basis.

Article Snippet: To optimize the packet wavelet decomposition tree in MATLAB, two optimization functions based on different entropy criteria were presented [ , ].

Techniques:

Journal: Entropy

Article Title: Optimal Estimation of Wavelet Decomposition Level for a Matching Pursuit Algorithm

doi: 10.3390/e21090843

Figure Lengend Snippet: ( a ) Source signal. ( b ) Entropy (Threshold, p = 0.005) vs. decomposition level, wavelet (Daubechies) basis. ( c ) Approximation error vs. decomposition level, wavelet (Daubechies) basis. ( d ) Execution time vs. decomposition level, wavelet (Daubechies) basis. ( e ) Approximation error vs. decomposition level, extended basis. ( f ) Execution time vs. decomposition level, extended basis.

Article Snippet: To optimize the packet wavelet decomposition tree in MATLAB, two optimization functions based on different entropy criteria were presented [ , ].

Techniques:

Journal: Entropy

Article Title: Optimal Estimation of Wavelet Decomposition Level for a Matching Pursuit Algorithm

doi: 10.3390/e21090843

Figure Lengend Snippet: ( a ) Source signal. ( b ) Entropy (SURE, p = 0.005) vs. decomposition level, wavelet (Meyer) basis. ( c ) Approximation error vs. decomposition level, wavelet (Meyer) basis. ( d ) Execution time vs. decomposition level, wavelet (Meyer) basis. ( e ) Approximation error vs. decomposition level, extended basis. ( f ) Execution time vs. decomposition level, extended basis.

Article Snippet: To optimize the packet wavelet decomposition tree in MATLAB, two optimization functions based on different entropy criteria were presented [ , ].

Techniques:

Journal: Entropy

Article Title: Optimal Estimation of Wavelet Decomposition Level for a Matching Pursuit Algorithm

doi: 10.3390/e21090843

Figure Lengend Snippet: ( a ) Source signal. ( b ) Entropy (Norm, p = 2) vs. decomposition level, wavelet (Meyer) basis. ( c ) Approximation error vs. decomposition level, wavelet (Meyer) basis. ( d ) Execution time vs. decomposition level, wavelet (Meyer) basis. ( e ) Approximation error vs. decomposition level, extended basis. ( f ) Execution time vs. decomposition level, extended basis.

Article Snippet: To optimize the packet wavelet decomposition tree in MATLAB, two optimization functions based on different entropy criteria were presented [ , ].

Techniques:

Journal: Entropy

Article Title: Optimal Estimation of Wavelet Decomposition Level for a Matching Pursuit Algorithm

doi: 10.3390/e21090843

Figure Lengend Snippet: Proposed algorithm (optimization, first value) vs. maximum level decomposition (no optimization, second value).

Article Snippet: To optimize the packet wavelet decomposition tree in MATLAB, two optimization functions based on different entropy criteria were presented [ , ].

Techniques: