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lazysptalk.tex
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\documentclass[serif]{beamer}
\def\gridopacity{100}
\def\gridopacity{0}
% what is this for?
\usepackage{multimedia}
% see http://tex.stackexchange.com/a/24491
% for how to use enumitem with beamer
\usepackage{enumitem}
\setitemize{label=\usebeamerfont*{itemize item}%
\usebeamercolor[fg]{itemize item}
\usebeamertemplate{itemize item}}
% default page size is:
% \beamer@paperwidth 12.80cm%
% \beamer@paperheight 9.60cm%
\setbeamersize{text margin left=0.4cm,text margin right=0.4cm}
\usepackage{tikz}
\usetikzlibrary{calc,decorations.pathreplacing,positioning}
\usepackage{pgfplots}
\usepackage{pgfplotstable}
\usepackage[clock,misc]{ifsym} % for \showclock
% pretty tables
\usepackage{multirow}
\usepackage{booktabs}
% this is from http://tex.stackexchange.com/a/103691
% taken from manual
\makeatletter
\pgfdeclareshape{document}{
\inheritsavedanchors[from=rectangle] % this is nearly a rectangle
\inheritanchorborder[from=rectangle]
\inheritanchor[from=rectangle]{center}
\inheritanchor[from=rectangle]{north}
\inheritanchor[from=rectangle]{south}
\inheritanchor[from=rectangle]{west}
\inheritanchor[from=rectangle]{east}
% ... and possibly more
\backgroundpath{% this is new
% store lower right in xa/ya and upper right in xb/yb
\southwest \pgf@xa=\pgf@x \pgf@ya=\pgf@y
\northeast \pgf@xb=\pgf@x \pgf@yb=\pgf@y
% compute corner of ‘‘flipped page’’
\pgf@xc=\pgf@xb \advance\pgf@xc by-5pt % this should be a parameter
\pgf@yc=\pgf@yb \advance\pgf@yc by-5pt
% construct main path
\pgfpathmoveto{\pgfpoint{\pgf@xa}{\pgf@ya}}
\pgfpathlineto{\pgfpoint{\pgf@xa}{\pgf@yb}}
\pgfpathlineto{\pgfpoint{\pgf@xc}{\pgf@yb}}
\pgfpathlineto{\pgfpoint{\pgf@xb}{\pgf@yc}}
\pgfpathlineto{\pgfpoint{\pgf@xb}{\pgf@ya}}
\pgfpathclose
% add little corner
\pgfpathmoveto{\pgfpoint{\pgf@xc}{\pgf@yb}}
\pgfpathlineto{\pgfpoint{\pgf@xc}{\pgf@yc}}
\pgfpathlineto{\pgfpoint{\pgf@xb}{\pgf@yc}}
\pgfpathlineto{\pgfpoint{\pgf@xc}{\pgf@yc}}
}
}
\makeatother
\usepackage{algorithm}
\usepackage[noend]{algpseudocode}
% symbols
\usepackage{amsmath} % assumes amsmath package installed
\usepackage{amssymb} % for \square
% non-italicized math subscripts
\newcommand{\ms}[1]{\mbox{\scriptsize #1}}
% from http://tex.stackexchange.com/a/5255
\DeclareMathOperator*{\argmin}{arg\,min}
% fonts
\renewcommand\rmdefault{pplx}
\renewcommand\mathfamilydefault{cmr}
\setbeamertemplate{navigation symbols}{}
\setbeamertemplate{footline}{%
\raisebox{7pt}{\makebox[\paperwidth]{%
\hfill\makebox[10pt]{\textcolor{gray}{%
\small\insertframenumber\hspace{0.1in}}}}}}
\title{Pathfinding with Expensive Edge Evaluations: \\
Lazy Best-First Path Search with Edge Selectors}
\author{Christopher M. Dellin and Siddhartha S. Srinivasa\\
{\usebeamercolor[fg]{structure} Laura Barbulescu}}
\date{June 15, 2016}
\titlegraphic{
\includegraphics[height=1.5cm]{figs/logo-ri.png}
}
\begin{document}
\begin{frame}[plain]
%\begin{columns}
% \column{1.2\textwidth}
\maketitle
%\end{columns}
\end{frame}
%\begin{frame}
% \frametitle{Pathfinding on Graphs}
%
% The \emph{shortest-path problem} is prevalent and fundamental.
%
% Lots of applications.
%
% Graph $G=(V,E)$, edge weight function $w: E \rightarrow \mathbb{R}$.
%
% Single-pair, all-pairs, single-source/single-sink, etc.
%
% Bounded-suboptimal, incremental, anytime, etc ...
%\end{frame}
%\iffalse
\begin{frame}
\frametitle{The Shortest Path Problem}
\begin{tikzpicture}[font=\small]
\tikzset{>=latex} % arrow heads
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\only<2->{
\node at (1,7) {\includegraphics[width=1.5cm]{figs/rubik.png}};
\node at (3,7) {\includegraphics[width=1.5cm]{figs/15puzzle.png}};
}
\only<3->{
\node at (5,7) {\includegraphics[width=1.5cm]{figs/internet-routers.jpg}};
}
\only<4->{
\node at (7,7) {\includegraphics[width=1.5cm]{figs/lazysp-herbarm/herbarm-path33.png}};
\node at (9,7) {\includegraphics[width=1.5cm]{figs/quad.png}};
}
\only<5->{
\node at (11,7) {\includegraphics[width=1.5cm]{figs/military-truck.jpg}};
}
\only<6->{
% graph
\coordinate (va) at ( 2.0,4.0);
\coordinate (vb) at ( 3.5,4.7);
\coordinate (vc) at ( 4.0,5.9);
\coordinate (vd) at ( 5.5,4.5);
\coordinate (ve) at ( 8.0,5.0);
\coordinate (vf) at (10.0,4.0);
\coordinate (vg) at ( 1.5,5.0);
\coordinate (vh) at ( 4.0,3.5);
\coordinate (vi) at ( 7.0,3.2);
\coordinate (vj) at ( 6.5,5.5);
\coordinate (vk) at ( 9.0,3.5);
\only<9->{
\node[circle,fill=black!20,inner sep=0.1cm] at (va) {};
\node[circle,fill=black!20,inner sep=0.1cm] at (vf) {};
}
\only<10->{
\draw[line width=0.2cm,color=black!30,line cap=round]
(va) -- (vb) -- (vd) -- (ve) -- (vf);
}
\node[circle,fill=black,inner sep=0.05cm] at (va) {};
\node[circle,fill=black,inner sep=0.05cm] at (vb) {};
\node[circle,fill=black,inner sep=0.05cm] at (vc) {};
\node[circle,fill=black,inner sep=0.05cm] at (vd) {};
\node[circle,fill=black,inner sep=0.05cm] at (ve) {};
\node[circle,fill=black,inner sep=0.05cm] at (vf) {};
\node[circle,fill=black,inner sep=0.05cm] at (vg) {};
\node[circle,fill=black,inner sep=0.05cm] at (vh) {};
\node[circle,fill=black,inner sep=0.05cm] at (vi) {};
\node[circle,fill=black,inner sep=0.05cm] at (vj) {};
\node[circle,fill=black,inner sep=0.05cm] at (vk) {};
\draw (va) -- (vb) coordinate [midway] (eab);
\draw (vb) -- (vc) coordinate [midway] (ebc);
\draw (vb) -- (vd) coordinate [midway] (ebd);
\draw (vc) -- (vd) coordinate [midway] (ecd);
\draw (vd) -- (ve) coordinate [midway] (ede);
\draw (ve) -- (vf) coordinate [midway] (eef);
\draw (va) -- (vg) coordinate [midway] (eag);
\draw (vb) -- (vg) coordinate [midway] (ebg);
\draw (va) -- (vh) coordinate [midway] (eah);
\draw (vd) -- (vh) coordinate [midway] (edh);
\draw (vb) -- (vh) coordinate [midway] (ebh);
\draw (vd) -- (vi) coordinate [midway] (edi);
\draw (ve) -- (vi) coordinate [midway] (eei);
\draw (vc) -- (vj) coordinate [midway] (ecj);
\draw (vd) -- (vj) coordinate [midway] (edj);
\draw (ve) -- (vj) coordinate [midway] (eej);
\draw (vf) -- (vk) coordinate [midway] (efk);
\draw (vi) -- (vk) coordinate [midway] (eik);
\draw (ve) -- (vk) coordinate [midway] (eek);
\only<7->{
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eab) {5};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebc) {5};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebd) {6};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ecd) {7};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ede) {9};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eef) {7};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eag) {3};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebg) {6};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eah) {6};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (edh) {7};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebh) {5};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (edi) {8};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eei) {8};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ecj) {8};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (edj) {6};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eej) {4};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (efk) {3};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eik) {6};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eek) {6};
}
}
\only<6->{
\node[draw,align=center,minimum height=1.0cm,minimum width=2.9cm,thick]
at (4.0,2.05) {Graph\\$G=(V,E)$};
}
\only<7->{
\node[draw,align=center,minimum height=1.0cm,minimum width=2.9cm,thick]
at (8.0,2.05) {Weight Function\\$w:E \rightarrow \mathbb{R}$};
%\node[align=center,color=black!50] at (10.75,2) {
% $w(e) \geq 0$\\
% $w(e) > 0$\\
% $w(e) \geq w_{\ms{est}}(e)$\\
%};
}
\only<8->{
\node[draw,align=center,minimum height=1.0cm,minimum width=4cm,thick]
(alg) at (6,0.55) {Shortest Path\\Algorithm};
\draw[->] (4.5,1.55) -- (4.5,1.05);
\draw[->] (7.5,1.55) -- (7.5,1.05);
}
\only<9->{
\node[draw,align=center,shape=document,minimum width=3.0cm,ultra thin]
(query) at (2,0.55) {Query $v_s$, $v_t$\;\;};
\draw[->] (query.east) -- (alg.west);
\node[left=0.1cm of va] {$v_s$};
\node[right=0.1cm of vf] {$v_t$};
}
\only<10->{
\node[draw,align=center,shape=document,minimum width=2.0cm,ultra thin]
(path) at (10,0.55) {Path $p^*$\;};
\draw[->] (alg.east) -- (path.west);
}
\end{tikzpicture}
\end{frame}
% application -> graph representation -> algorithm -> path -> application solution
% [ lots of them] G=(V,E) [lots of them]
% w: e -> R
% root vertices (v_s, v_t)
% (vertex heuristic?)
% problem
% - single-pair
% - optimal
% - bounded-suboptimal
% - bounded-cost
%\begin{frame}
% \frametitle{Pathfinding on Graphs 2}
% \begin{tikzpicture}
% \tikzset{>=latex} % arrow heads
%
% \draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
%
% \node at (1.5,7.5) {Applications};
%
% \node at (1.0,6.5) {\includegraphics[width=1.5cm]{figs/rubik.png}};
% \node at (2.0,6.5) {\includegraphics[width=1.5cm]{figs/15puzzle.png}};
% \node at (1.5,4.5) {\includegraphics[width=1.5cm]{figs/internet-routers.jpg}};
% \node at (1.5,3.0) {Traffic Routing};
% \node at (1.5,2.5) {Motion Planning};
% %\node at (1.5,2.0) {Optimal Motion Planning};
%
% \node at (6.0,7.5) {Representation};
%
% \node at (10.5,7.5) {Algorithms};
%
% \node[align=center] at (7,3) {
% $w(e) \geq 0$\\
% $w(e) > 0$\\
% $w(e) \geq w_{\ms{est}}(e)$\\
% };
%
% %\draw[step=1cm,black!10,very thin] (0,0) grid (8,4);
% \node[draw,align=center,minimum height=1.0cm,thick]
% at (6.0,6.0) {Graph\\$G=(V,E)$};
% \node[draw,align=center,minimum height=1.0cm,thick]
% at (6.0,4.5) {Weight Function\\$w:E \rightarrow [0,+\infty]$};
% \node[draw,align=center,shape=document,minimum width=3.0cm,ultra thin]
% (query) at (6,1) {Query $v_s$, $v_t$\;\;};
%
% \node at (10.5,6.5) {Dijkstra's};
% \node at (10.5,5.5) {A*};
% \node at (10.5,4.5) {IDA*};
% \node at (10.5,3.5) {PEA*};
%
% \end{tikzpicture}
%\end{frame}
\begin{frame}
\frametitle{The Shortest Path Problem}
\begin{tikzpicture}[font=\small]
\tikzset{>=latex} % arrow heads
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
% c-space obstacles
\only<8-12>{
\fill[black!20] (4.5,3.25) rectangle (6.0,5.0);
\fill[black!20] (8.5,4.00) rectangle (9.5,5.5);
\only<8-10>{
\node[color=black!70,font=\scriptsize] at (5.25,3.5) {$CO_1$};
\node[color=black!70,font=\scriptsize] at (9.0,5.25) {$CO_2$};
}
\only<11->{
\node[color=black!70,font=\scriptsize] at (5.25,3.5) {$SO_1$};
\node[color=black!70,font=\scriptsize] at (9.0,5.25) {$SO_2$};
}
}
% graph start
\coordinate (va) at ( 2.0,4.0);
\coordinate (vb) at ( 3.5,4.7);
\coordinate (vc) at ( 4.0,5.9);
\coordinate (vd) at ( 5.5,4.5);
\coordinate (ve) at ( 8.0,5.0);
\coordinate (vf) at (10.0,4.0);
\coordinate (vg) at ( 1.5,5.0);
\coordinate (vh) at ( 4.0,3.5);
\coordinate (vi) at ( 7.0,3.2);
\coordinate (vj) at ( 6.5,5.5);
\coordinate (vk) at ( 9.0,3.5);
%\node[circle,fill=black!20,inner sep=0.1cm] at (va) {};
%\node[circle,fill=black!20,inner sep=0.1cm] at (vf) {};
\node[circle,fill=black,inner sep=0.05cm] at (va) {};
\node[circle,fill=black,inner sep=0.05cm] at (vb) {};
\node[circle,fill=black,inner sep=0.05cm] at (vc) {};
\node[circle,fill=black,inner sep=0.05cm] at (vd) {};
\node[circle,fill=black,inner sep=0.05cm] at (ve) {};
\node[circle,fill=black,inner sep=0.05cm] at (vf) {};
\node[circle,fill=black,inner sep=0.05cm] at (vg) {};
\node[circle,fill=black,inner sep=0.05cm] at (vh) {};
\node[circle,fill=black,inner sep=0.05cm] at (vi) {};
\node[circle,fill=black,inner sep=0.05cm] at (vj) {};
\node[circle,fill=black,inner sep=0.05cm] at (vk) {};
\draw (va) -- (vb) coordinate [midway] (eab);
\draw (vb) -- (vc) coordinate [midway] (ebc);
\draw (vb) -- (vd) coordinate [midway] (ebd);
\draw (vc) -- (vd) coordinate [midway] (ecd);
\draw (vd) -- (ve) coordinate [midway] (ede);
\draw (ve) -- (vf) coordinate [midway] (eef);
\draw (va) -- (vg) coordinate [midway] (eag);
\draw (vb) -- (vg) coordinate [midway] (ebg);
\draw (va) -- (vh) coordinate [midway] (eah);
\draw (vd) -- (vh) coordinate [midway] (edh);
\draw (vb) -- (vh) coordinate [midway] (ebh);
\draw (vd) -- (vi) coordinate [midway] (edi);
\draw (ve) -- (vi) coordinate [midway] (eei);
\draw (vc) -- (vj) coordinate [midway] (ecj);
\draw (vd) -- (vj) coordinate [midway] (edj);
\draw (ve) -- (vj) coordinate [midway] (eej);
\draw (vf) -- (vk) coordinate [midway] (efk);
\draw (vi) -- (vk) coordinate [midway] (eik);
\draw (ve) -- (vk) coordinate [midway] (eek);
\only<4-4>{
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eab) {1};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebc) {1};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebd) {1};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ecd) {1};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ede) {1};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eef) {1};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eag) {1};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebg) {1};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eah) {1};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (edh) {1};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebh) {1};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (edi) {1};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eei) {1};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ecj) {1};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (edj) {1};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eej) {1};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (efk) {1};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eik) {1};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eek) {1};
}
% motion planning
\only<7-9>{
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eab) {5};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebc) {5};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebd) {6};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ecd) {7};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ede) {9};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eef) {7};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eag) {3};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebg) {6};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eah) {6};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (edh) {7};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebh) {5};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (edi) {8};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eei) {8};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ecj) {8};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (edj) {6};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eej) {4};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (efk) {3};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eik) {6};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eek) {6};
}
\only<10-12>{
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eab) {5};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebc) {5};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebd) {$\infty$};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ecd) {$\infty$};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ede) {$\infty$};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eef) {$\infty$};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eag) {3};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebg) {6};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eah) {6};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (edh) {$\infty$};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebh) {5};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (edi) {$\infty$};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eei) {8};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ecj) {8};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (edj) {$\infty$};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eej) {4};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (efk) {3};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eik) {6};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eek) {$\infty$};
}
% convoy
\only<14->{
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eab) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebc) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebd) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ecd) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ede) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eef) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eag) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebg) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eah) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (edh) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebh) {?};
%\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (edi) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eei) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ecj) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (edj) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eej) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (efk) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eik) {?};
% first uncover
\only<14-15>{\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eek) {?};}
\only<16->{ \node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] (sateek) at (eek) {7.8};}
\only<16>{\draw[ultra thick,red,->] (satnode.south west) -- (sateek.east);}
% second uncover
\only<14-16>{\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (edi) {?};}
\only<17->{ \node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] (satedi) at (edi) {$\infty$};}
\only<17>{\draw[ultra thick,red,->] (satnode.south west) -- (satedi.east);}
}
%\node[left=0.1cm of va] {$v_s$};
%\node[right=0.1cm of vf] {$v_t$};
\only<9>{
\draw (vb) -- (vd) node[pos=0.1,circle,fill=red,inner sep=0.05cm] {};
\draw (vb) -- (vd) node[pos=0.2,circle,fill=red,inner sep=0.05cm] {};
\draw (vb) -- (vd) node[pos=0.3,circle,fill=red,inner sep=0.05cm] {};
\draw (vb) -- (vd) node[pos=0.4,circle,fill=red,inner sep=0.05cm] {};
\draw (vb) -- (vd) node[pos=0.5,circle,fill=red,inner sep=0.05cm] {};
\draw (vb) -- (vd) node[pos=0.6,circle,fill=red,inner sep=0.05cm] {};
\draw (vb) -- (vd) node[pos=0.7,circle,fill=red,inner sep=0.05cm] {};
\draw (vb) -- (vd) node[pos=0.8,circle,fill=red,inner sep=0.05cm] {};
\draw (vb) -- (vd) node[pos=0.9,circle,fill=red,inner sep=0.05cm] {};
}
%\fill[white,opacity=0.8] (0,3) rectangle (12,6.5);
% graph end
\node[draw,thick,minimum height=1.5cm,minimum width=5.5cm,anchor=north] at (3,2.9) {};
\node [anchor=north] at (3,2.9) {\begin{minipage}{5cm}
\begin{center}
Graph $G=(V,E)$:
\end{center}
\vspace{-0.4cm}
% puzzles
\only<3-4>{
$v:$ puzzle state; $|V| = 10^{19}$
$e:$ puzzle move
}
% highlight motion planning
\only<6-10>{
$v:$ arm configuration; $|V| \approx 10^{5}$
$e:$ arm motion
}
\end{minipage}};
\node[draw,thick,minimum height=1.5cm,minimum width=5.5cm,anchor=north] at (9,2.9) {};
\node [anchor=north] at (9,2.9) {\begin{minipage}{5cm}
\begin{center}
Weight Function $w:E \rightarrow \mathbb{R}$:
\end{center}
\vspace{-0.4cm}
% puzzles
\only<4-4>{
$w(e) = 1 \; \forall e \in E$
}
% highlight motion planning
\only<7-7>{%
$w(e) = \left\{ \arraycolsep=2pt \begin{array}{cl}
||e|| & \mbox{if $e$ collision-free} \\
& \\
\end{array} \right.$
}%
\only<8-10>{%
$w(e) = \left\{ \arraycolsep=2pt \begin{array}{cl}
||e|| & \mbox{if $e$ collision-free} \\
\infty & \mbox{otherwise} \\
\end{array} \right.$
}%
\only<12-12>{%
\vspace{0.07cm}
$w(e) = \mbox{BVP}(q_a, q_b)$
{\footnotesize (solve a boundary-value problem)}
}%
\only<15->{%
\vspace{0.2cm}
$w(e) = \mbox{Reconnaissance}(e)$
}%
\end{minipage}};
\node at (1,7) {\includegraphics[width=1.5cm]{figs/rubik.png}};
\node at (3,7) {\includegraphics[width=1.5cm]{figs/15puzzle.png}};
\node at (5,7) {\includegraphics[width=1.5cm]{figs/internet-routers.jpg}};
\node at (7,7) {\includegraphics[width=1.5cm]{figs/lazysp-herbarm/herbarm-path33.png}};
\node at (9,7) {\includegraphics[width=1.5cm]{figs/quad.png}};
\node at (11,7) {\includegraphics[width=1.5cm]{figs/military-truck.jpg}};
% highlight nothing
\only<1>{
\fill[white,opacity=0.8] (0,6) rectangle (12,8);
}
% highlight puzzles
\only<2-4>{
\draw[ultra thick,rounded corners] (0.1,6.1) rectangle (3.9,7.9);
\fill[white,opacity=0.8] (4,6) rectangle (12,8);
}
% highlight motion planning
\only<5-10>{
\draw[ultra thick,rounded corners] (6.1,6.1) rectangle (7.9,7.9);
\fill[white,opacity=0.8] (0,6) rectangle (6,8);
\fill[white,opacity=0.8] (8,6) rectangle (12,8);
\only<6->{
\node[align=center,font=\scriptsize] at (11.0,5.0) {Configuration\\Space};
}
}
% highlight kinodynamic planning
\only<11-12>{
\draw[ultra thick,rounded corners] (8.1,6.1) rectangle (9.9,7.9);
\fill[white,opacity=0.8] (0,6) rectangle (8,8);
\fill[white,opacity=0.8] (10,6) rectangle (12,8);
\node[align=center,font=\scriptsize] at (11.0,5.0) {Kinodynamic\\State Space};
}
% highlight convoy
\only<13->{
\draw[ultra thick,rounded corners] (10.1,6.1) rectangle (11.9,7.9);
\fill[white,opacity=0.8] (0,6) rectangle (10,8);
\only<15->{
\node[draw,align=center,font=\scriptsize] (satnode) at (11.0,5.5) {Satellite};
}
}
\node[draw,align=center,minimum height=1.0cm,minimum width=4cm,thick]
(alg) at (6,0.55) {Shortest Path\\Algorithm};
\draw[->] (4.5,1.40) -- (4.5,1.05);
\draw[->] (7.5,1.40) -- (7.5,1.05);
\node[draw,align=center,shape=document,minimum width=3.0cm,ultra thin]
(query) at (2,0.55) {Query $v_s$, $v_t$\;\;};
\draw[->] (query.east) -- (alg.west);
\node[draw,align=center,shape=document,minimum width=2.0cm,ultra thin]
(path) at (10,0.55) {Path $p^*$\;};
\draw[->] (alg.east) -- (path.west);
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Pathfinding with Expensive Edge Evaluations}
In some domains, determining edge weights\\
is a computationally expensive operation.
\pause
\vspace{0.4cm}
\emph{Question: How can we solve shortest-path problems\\
\quad \quad \quad \quad \, while reducing the number of edges evaluated?}
\pause
\vspace{0.4cm}
Outline of talk:
\begin{itemize}
\pause
\item Review of lazy search approaches
\pause
\item LazySP algorithm outline (using \emph{edge selectors})
\pause
\item Equivalence to existing algorithms
\pause
\item Novel edge selectors
\pause
\item Experimental results
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Review of Prior Approaches}
\begin{tikzpicture}[font=\small]
\tikzset{>=latex} % arrow heads
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
% graph start
\coordinate (va) at ( 2.0,5.6);
\coordinate (vb) at ( 3.5,6.3);
\coordinate (vc) at ( 4.0,7.5);
\coordinate (vd) at ( 5.5,6.1);
\coordinate (ve) at ( 8.0,6.6);
\coordinate (vf) at (10.0,5.6);
\coordinate (vg) at ( 1.5,6.6);
\coordinate (vh) at ( 4.0,5.1);
\coordinate (vi) at ( 7.0,4.8);
\coordinate (vj) at ( 6.5,7.1);
\coordinate (vk) at ( 9.0,5.1);
% start/goal highlighting
\node[circle,fill=black!20,inner sep=0.1cm] at (va) {};
\node[circle,fill=black!20,inner sep=0.1cm] at (vf) {};
% edges, grey
\only<1-2,4,10>{
\draw[black!30] (va) -- (vb) coordinate [midway] (eab);
\draw[black!30] (vb) -- (vc) coordinate [midway] (ebc);
\draw[black!30] (vb) -- (vd) coordinate [midway] (ebd);
\draw[black!30] (vc) -- (vd) coordinate [midway] (ecd);
\draw[black!30] (vd) -- (ve) coordinate [midway] (ede);
\draw[black!30] (ve) -- (vf) coordinate [midway] (eef);
\draw[black!30] (va) -- (vg) coordinate [midway] (eag);
\draw[black!30] (vb) -- (vg) coordinate [midway] (ebg);
\draw[black!30] (va) -- (vh) coordinate [midway] (eah);
\draw[black!30] (vd) -- (vh) coordinate [midway] (edh);
\draw[black!30] (vb) -- (vh) coordinate [midway] (ebh);
\draw[black!30] (vd) -- (vi) coordinate [midway] (edi);
\draw[black!30] (ve) -- (vi) coordinate [midway] (eei);
\draw[black!30] (vc) -- (vj) coordinate [midway] (ecj);
\draw[black!30] (vd) -- (vj) coordinate [midway] (edj);
\draw[black!30] (ve) -- (vj) coordinate [midway] (eej);
\draw[black!30] (vf) -- (vk) coordinate [midway] (efk);
\draw[black!30] (vi) -- (vk) coordinate [midway] (eik);
\draw[black!30] (ve) -- (vk) coordinate [midway] (eek);
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eab) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebc) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebd) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ecd) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ede) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eef) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eag) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebg) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eah) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (edh) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebh) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (edi) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eei) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ecj) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (edj) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eej) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (efk) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eik) {?};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eek) {?};
}
% edges, black
\only<3>{
\draw[ultra thick] (va) -- (vb) coordinate [midway] (eab);
\draw[ultra thick] (vb) -- (vc) coordinate [midway] (ebc);
\draw[ultra thick] (vb) -- (vd) coordinate [midway] (ebd);
\draw[ultra thick] (vc) -- (vd) coordinate [midway] (ecd);
\draw[ultra thick] (vd) -- (ve) coordinate [midway] (ede);
\draw[ultra thick] (ve) -- (vf) coordinate [midway] (eef);
\draw[ultra thick] (va) -- (vg) coordinate [midway] (eag);
\draw[ultra thick] (vb) -- (vg) coordinate [midway] (ebg);
\draw[ultra thick] (va) -- (vh) coordinate [midway] (eah);
\draw[ultra thick] (vd) -- (vh) coordinate [midway] (edh);
\draw[ultra thick] (vb) -- (vh) coordinate [midway] (ebh);
\draw[ultra thick] (vd) -- (vi) coordinate [midway] (edi);
\draw[ultra thick] (ve) -- (vi) coordinate [midway] (eei);
\draw[ultra thick] (vc) -- (vj) coordinate [midway] (ecj);
\draw[ultra thick] (vd) -- (vj) coordinate [midway] (edj);
\draw[ultra thick] (ve) -- (vj) coordinate [midway] (eej);
\draw[ultra thick] (vf) -- (vk) coordinate [midway] (efk);
\draw[ultra thick] (vi) -- (vk) coordinate [midway] (eik);
\draw[ultra thick] (ve) -- (vk) coordinate [midway] (eek);
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eab) {5};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebc) {5};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebd) {6};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ecd) {7};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ede) {9};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eef) {7};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eag) {3};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebg) {6};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eah) {6};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (edh) {7};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ebh) {5};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (edi) {8};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eei) {8};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (ecj) {8};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (edj) {6};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eej) {4};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (efk) {3};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eik) {6};
\node[circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] at (eek) {6};
}
% NEXT, A* stuff
% OPEN list highlighting (A*)
\only<5>{\node[circle,fill=blue!70,inner sep=0.15cm] at (va) {};}
\only<6-9>{\node[circle,fill=black!40,inner sep=0.15cm] at (va) {};}
\only<6>{\node[circle,fill=blue!70,inner sep=0.15cm] at (vb) {};}
\only<7-9>{\node[circle,fill=black!40,inner sep=0.15cm] at (vb) {};}
\only<6-9>{\node[circle,fill=blue!40,inner sep=0.15cm] at (vg) {};}
\only<6>{\node[circle,fill=blue!40,inner sep=0.15cm] at (vh) {};}
\only<7>{\node[circle,fill=blue!70,inner sep=0.15cm] at (vh) {};}
\only<8-9>{\node[circle,fill=black!40,inner sep=0.15cm] at (vh) {};}
\only<7>{\node[circle,fill=blue!40,inner sep=0.15cm] at (vc) {};}
\only<8>{\node[circle,fill=blue!70,inner sep=0.15cm] at (vc) {};}
\only<9>{\node[circle,fill=black!40,inner sep=0.15cm] at (vc) {};}
\only<7-9>{\node[circle,fill=blue!40,inner sep=0.15cm] at (vd) {};}
\only<9>{\node[circle,fill=blue!70,inner sep=0.15cm] at (vj) {};}
% first expansion
\only<5>{
\draw[black!30] (va) -- (vb) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {?};
\draw[black!30] (va) -- (vg) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {?};
\draw[black!30] (va) -- (vh) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {?};
}
\only<6-9>{
\draw[black,ultra thick] (va) -- (vb) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {5};
\draw[black,ultra thick] (va) -- (vg) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {3};
\draw[black,ultra thick] (va) -- (vh) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {6};
}
% second expansion
\only<5-6>{
\draw[black!30] (vb) -- (vc) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {?};
\draw[black!30] (vb) -- (vd) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {?};
\draw[black!30] (vb) -- (vg) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {?};
\draw[black!30] (vb) -- (vh) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {?};
}
\only<7-9>{
\draw[black,ultra thick] (vb) -- (vc) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {5};
\draw[black,ultra thick] (vb) -- (vd) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {$\infty$};
\draw[black,ultra thick] (vb) -- (vg) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {6};
\draw[black,ultra thick] (vb) -- (vh) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {5};
}
% third expansion
\only<5-7>{
\draw[black!30] (vd) -- (vh) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {?};
}
\only<8-9>{
\draw[black,ultra thick] (vd) -- (vh) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {$\infty$};
}
% fourth expansion
\only<5-8>{
\draw[black!30] (vc) -- (vd) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {?};
\draw[black!30] (vc) -- (vj) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {?};
}
\only<9>{
\draw[black,ultra thick] (vc) -- (vd) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {$\infty$};
\draw[black,ultra thick] (vc) -- (vj) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {8};
}
% edges, grey
\only<5-9>{
\draw[black!30] (vd) -- (ve) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {?};
\draw[black!30] (ve) -- (vf) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {?};
\draw[black!30] (vd) -- (vi) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {?};
\draw[black!30] (ve) -- (vi) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {?};
\draw[black!30] (vd) -- (vj) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {?};
\draw[black!30] (ve) -- (vj) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {?};
\draw[black!30] (vf) -- (vk) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {?};
\draw[black!30] (vi) -- (vk) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {?};
\draw[black!30] (ve) -- (vk) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {?};
}
% END A* stuff
% NEXT, LWA* stuff
% OPEN vertex list highlighting (LWA*)
\only<11>{\node[circle,fill=blue!70,inner sep=0.15cm] at (va) {};}
\only<12->{\node[circle,fill=black!40,inner sep=0.15cm] at (va) {};}
\only<13>{\node[circle,fill=blue!70,inner sep=0.15cm] at (vb) {};}
\only<14->{\node[circle,fill=black!40,inner sep=0.15cm] at (vb) {};}
\only<15->{\node[circle,fill=blue!40,inner sep=0.15cm] at (vd) {};}
\only<16>{\node[circle,fill=blue!70,inner sep=0.15cm] at (vh) {};}
\only<17->{\node[circle,fill=black!40,inner sep=0.15cm] at (vh) {};}
\only<19>{\node[circle,fill=blue!70,inner sep=0.15cm] at (vc) {};}
\only<20->{\node[circle,fill=black!40,inner sep=0.15cm] at (vc) {};}
\only<21>{\node[circle,fill=blue!70,inner sep=0.15cm] at (vj) {};}
% edge queue
\only<12->{\draw[blue!40,line width=0.10cm,line cap=round] (va) -- (vg);}
\only<12-14>{\draw[blue!40,line width=0.10cm,line cap=round] (va) -- (vh);}
\only<12>{\draw[blue!70,line width=0.10cm,line cap=round] (va) -- (vb);}
\only<14-17>{\draw[blue!40,line width=0.10cm,line cap=round] (vb) -- (vc);}
\only<14->{\draw[blue!40,line width=0.10cm,line cap=round] (vb) -- (vg);}
\only<14->{\draw[blue!40,line width=0.10cm,line cap=round] (vb) -- (vh);}
\only<14>{\draw[blue!70,line width=0.10cm,line cap=round] (vb) -- (vd);}
\only<15>{\draw[blue!70,line width=0.10cm,line cap=round] (va) -- (vh);}
\only<17>{\draw[blue!70,line width=0.10cm,line cap=round] (vh) -- (vd);}
\only<18>{\draw[blue!70,line width=0.10cm,line cap=round] (vb) -- (vc);}
\only<20->{\draw[blue!40,line width=0.10cm,line cap=round] (vc) -- (vd);}
\only<20>{\draw[blue!70,line width=0.10cm,line cap=round] (vc) -- (vj);}
\only<11-12>{
\draw[black!30] (va) -- (vb) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {5?};
}
\only<13->{
\draw[black,ultra thick] (va) -- (vb) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {5};
}
\only<11-14>{
\draw[black!30] (vb) -- (vd) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {6?};
}
\only<15->{
\draw[black,ultra thick] (vb) -- (vd) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {$\infty$};
}
\only<11-15>{
\draw[black!30] (va) -- (vh) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {6?};
}
\only<16->{
\draw[black,ultra thick] (va) -- (vh) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {6};
}
\only<11-17>{
\draw[black!30] (vh) -- (vd) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {7?};
}
\only<18->{
\draw[black,ultra thick] (vh) -- (vd) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {$\infty$};
}
\only<11-18>{
\draw[black!30] (vb) -- (vc) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {5?};
}
\only<19->{
\draw[black,ultra thick] (vb) -- (vc) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {5};
}
\only<11-20>{
\draw[black!30] (vc) -- (vj) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {8?};
}
\only<21->{
\draw[black,ultra thick] (vc) -- (vj) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {8};
}
% edges, grey
\only<11->{
\draw[black!30] (va) -- (vg) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {3?};
\draw[black!30] (vb) -- (vg) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {6?};
\draw[black!30] (vb) -- (vh) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {5?};
\draw[black!30] (vc) -- (vd) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {7?};
\draw[black!30] (vd) -- (ve) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {9?};
\draw[black!30] (ve) -- (vf) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {7?};
\draw[black!30] (vd) -- (vi) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {8?};
\draw[black!30] (ve) -- (vi) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {8?};
\draw[black!30] (vd) -- (vj) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {6?};
\draw[black!30] (ve) -- (vj) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {4?};
\draw[black!30] (vf) -- (vk) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {3?};
\draw[black!30] (vi) -- (vk) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {6?};
\draw[black!30] (ve) -- (vk) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {6?};
}
% END LWA* stuff
\node[circle,fill=black,inner sep=0.05cm] at (va) {};
\node[circle,fill=black,inner sep=0.05cm] at (vb) {};
\node[circle,fill=black,inner sep=0.05cm] at (vc) {};
\node[circle,fill=black,inner sep=0.05cm] at (vd) {};
\node[circle,fill=black,inner sep=0.05cm] at (ve) {};
\node[circle,fill=black,inner sep=0.05cm] at (vf) {};
\node[circle,fill=black,inner sep=0.05cm] at (vg) {};
\node[circle,fill=black,inner sep=0.05cm] at (vh) {};
\node[circle,fill=black,inner sep=0.05cm] at (vi) {};
\node[circle,fill=black,inner sep=0.05cm] at (vj) {};
\node[circle,fill=black,inner sep=0.05cm] at (vk) {};
\node[left=0.1cm of va] {$v_s$};
\node[right=0.1cm of vf] {$v_t$};
% graph end
\only<2-3>{
\node[fill=blue!20,anchor=north,rounded corners] at (6,4.4) {\begin{minipage}{11cm}
\hangindent=0.35cm
Approach 0: Pre-evaluate all edges, run SP algorithm.
\end{minipage}};
}
\only<4->{
\node[fill=blue!10,anchor=north,rounded corners] at (6,4.4) {\begin{minipage}{11cm}
\hangindent=0.35cm
Approach 0: Pre-evaluate all edges, run SP algorithm.
\end{minipage}};
}
\only<4-9>{
\node[fill=blue!20,anchor=north,rounded corners] at (6,3.57) {\begin{minipage}{11cm}
\hangindent=0.35cm
Approach 1: Run SP (e.g. A*) with JIT and memoized evaluations.\\
(analogue to implicit graph representation)
\end{minipage}};
}
\only<10->{
\node[fill=blue!10,anchor=north,rounded corners] at (6,3.57) {\begin{minipage}{11cm}
\hangindent=0.35cm
Approach 1: Run SP (e.g. A*) with JIT and memoized evaluations.\\
(analogue to implicit graph representation)
\end{minipage}};
}
\only<10->{
\node[fill=blue!20,anchor=north,rounded corners] at (6,2.3) {\begin{minipage}{11cm}
\hangindent=0.35cm
Approach 2: Lazy Weighted A* (LWA*). Uses weight estimator $w_{\ms{est}}$.\\
Maintain independent vertex and edge queues.
\end{minipage}};
}
\only<10->{
%\begin{scope}
%\clip (0,0) rectangle (12,8);
\node[inner sep=0cm] at (6,0.55) {\begin{minipage}{11.0cm}\scriptsize
\hangindent=0.35cm \raggedright
\PaperPortrait\; Cohen, Phillips, and Likhachev,
``Planning Single-arm Manipulations with n-Arm Robots,''
RSS 2014.
\hangindent=0.35cm \raggedright
\PaperPortrait\; Gammell, Srinivasa, and Barfoot,
``Batch Informed Trees (BIT*),''
ICRA 2015.
\end{minipage}};
%\end{scope}
}
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Lazy Shortest Path: Best-First Search over Paths}
\begin{tikzpicture}[font=\small]
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
% graph start
\coordinate (va) at ( 2.0,5.6);
\coordinate (vb) at ( 3.5,6.3);
\coordinate (vc) at ( 4.0,7.5);
\coordinate (vd) at ( 5.5,6.1);
\coordinate (ve) at ( 8.0,6.6);
\coordinate (vf) at (10.0,5.6);
\coordinate (vg) at ( 1.5,6.6);
\coordinate (vh) at ( 4.0,5.1);
\coordinate (vi) at ( 7.0,4.8);