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zJL$aYqauXgj>C6yOI47Mk{x$mwUadgCcQDwc$`F}YuM7sx@qclKvV=esCKf{sB}$- kT|LO!+#G-OafV6%A4{v(70EN|+5i9m07*qoM6N<$g80wra{vGU literal 0 HcmV?d00001 diff --git a/slides/slides.tex b/slides/slides.tex index 9819207..b0bacc8 100644 --- a/slides/slides.tex +++ b/slides/slides.tex @@ -10,7 +10,7 @@ \setbeamertemplate{navigation symbols}{} \setbeamerfont{title page}{family=\rmfamily} -\def\[#1]{~{\color{violet} [#1]}} +\def\[#1]{\hskip0.3em{\color{violet} [#1]}} \def\O{{\cal O}} \begin{frame} @@ -80,6 +80,7 @@ As we approach linear time, we must specify the model. \end{itemize} ~ +\pause Many variants exist, we will use the {\bf Word-RAM:} @@ -103,11 +104,12 @@ Many variants exist, we will use the {\bf Word-RAM:} \end{itemize} ~ +\pause \begin{beamerboxesrounded}[upper=block title example,shadow=true]{Key differences} \begin{itemize} -\item PM has no arrays, we can emulate them in $\O(\log n)$ -\item PM has no arithmetics +\item PM has no arrays, we can emulate them in $\O(\log n)$ time. +\item PM has no arithmetics. \end{itemize} \end{beamerboxesrounded} @@ -142,7 +144,7 @@ We can use RAM as a vector machine: ~ -\begin{example} +\begin{example}[parallel search] \def\sep{\;{\color{brown}0}} \def\seq{\;{\color{brown}1}} \def\sez{\;{\color{red}0}} @@ -153,12 +155,12 @@ We can encode the vector $(1,5,3,0)$ with 3-bit fields as: And then search for 3 by: \begin{center} -\begin{tabular}{rc} - &\seq001\seq101\seq011\seq000 \\ -{\sc xor} &\sep011\sep011\sep011\sep011 \\ +\begin{tabular}{rcl} + &\seq001\seq101\seq011\seq000 & $(1,5,3,0)$ \\ +{\sc xor} &\sep011\sep011\sep011\sep011 & $(3,3,3,3)$ \\ \hline &\seq010\seq110\seq000\seq011 \\ -$-$ &\sep001\sep001\sep001\sep001 \\ +$-$ &\sep001\sep001\sep001\sep001 & $(1,1,1,1)$ \\ \hline &\seq001\seq101\sez111\seq010 \\ {\sc and} &\seq000\seq000\seq000\seq000 \\ @@ -196,25 +198,38 @@ Algorithms for Minimum Spanning Trees: \begin{itemize} \item Classical algorithms \[Borùvka, Jarník-Prim, Kruskal] \item Contractive: $\O(m\log n)$ using flattening on the PM \\ - (lower bound: \[M.]) + (lower bound \[M.]) \item Iterated: $\O(m\,\beta(m,n))$ \[Fredman \& Tarjan~1987] \\ where $\beta(m,n) = \min\{ k: \log_2^{(k)} n \le m/n \}$ -\item Even better: $\O(m\,\alpha(m,n))$ using {\it soft heaps}\\ - \[Chazelle 1998, Pettie 1999] +\item Even better: $\O(m\,\alpha(m,n))$ using {\it soft heaps}\hfil\break\[Chazelle 1998, Pettie 1999] \item MST verification: $\O(m)$ on RAM \[King 1997, M. 2008] -\item Randomized: $\O(m)$ w.h.p. \[Karger et al.~1995] +\item Randomized: $\O(m)$ expected on RAM \[Karger et al.~1995] \end{itemize} \end{frame} \begin{frame}{MST -- Special cases} -We have $\O(m)$ time algorithms for: +Cases for which we have an $\O(m)$ algorithm: + +~ + +Special graph structure: \begin{itemize} \item Planar graphs \[Tarjan 1976, Matsui 1995, M. 2004] (PM) \item Minor-closed classes \[Tarjan 1983, M. 2004] (PM) +\item Dense graphs (by many of the general PM algorithms) +\end{itemize} + +~ +\pause + +Or we can assume more about weights: + +\begin{itemize} \item $\O(1)$ different weights \[folklore] (PM) \item Integer weights \[Fredman \& Willard 1990] (RAM) +\item Sorted weights (RAM) \end{itemize} \end{frame} @@ -225,12 +240,12 @@ There is a~provably optimal comparison-based algorithm \\\[Pettie \& Ramachandra ~ -However, there is a catch: \pause Nobody knows its complexity. +However, there is a catch\alt<2->{:}{ \dots} \pause Nobody knows its complexity. ~ We know that it is $\O({\cal T}(m,n))$ where ${\cal T}(m,n)$ is the depth -of the optimum MST decision tree. +of the optimum MST decision tree. Any other algorithm provides an upper bound. \pause ~ @@ -250,22 +265,24 @@ We insert/delete edges, the structure responds with $\O(1)$ modifications of the MST. \begin{itemize} -\item Unweighted cases similar to dynamic connectivity: +\item Unweighted cases, similar to dynamic connectivity: \begin{itemize} \item Incremental: $\O(\alpha(n))$ \[Tarjan 1975] \item Fully dynamic: $\O(\log^2 n)$ \[Holm et al.~2001] \end{itemize} +\pause \item Weighted cases are harder: \begin{itemize} \item Decremental: $\O(\log^2 n)$ \[Holm et al.~2001] \item Fully dynamic: $\O(\log^4 n)$ \[Holm et al.~2001] - \item Only~$W$ weights: $\O(W\log^2 n)$ \[M. 2008] + \item Only~$C$ weights: $\O(C\log^2 n)$ \[M. 2008] \end{itemize} -\item $K$ best spanning trees: +\pause +\item $K$ smallest spanning trees: \begin{itemize} \item Simple: $\O(T_{MST} + Km)$ \[Katoh et al.~1981, M.~2008] - \item Reduced: $\O(T_{MST} + \min(K^2, Km + K\log K))$ \[Eppst.~1992] - \item Large~$K$: $\O(T_{MST} + \min(K^{3/2}, Km^{1/2}))$ \[Frederickson 1997] + \item Small~$K$: $\O(T_{MST} + \min(K^2, Km + K\log K))$ \[Eppst.~1992] + \item Faster: $\O(T_{MST} + \min(K^{3/2}, Km^{1/2}))$ \[Frederickson 1997] \end{itemize} \end{itemize} @@ -276,8 +293,8 @@ modifications of the MST. Ranking of permutations on the RAM: \[M. \& Straka 2007] \begin{itemize} -\item Traditional algorithms represent a~subset of $\{1,\ldots,n\}$ -\item The result can be $n!$ $\Rightarrow$ word size $=\Omega(n\log n)$ bits +\item We need a DS for the subsets of $\{1,\ldots,n\}$ with ranking +\item The result can be $n!$ $\Rightarrow$ word size is $\Omega(n\log n)$ bits \item We can represent the subsets as RAM vectors \item This gives us an~$\O(n)$ time algorithm for (un)ranking \end{itemize} @@ -293,8 +310,9 @@ Easily extendable to $k$-permutations, also in $\O(n)$ For restricted permutations (e.g., derangements): \[M. 2008] \begin{itemize} +\item Describe restrictions by a~bipartite graph \item Existence of permutation reduces to network flows -\item The ranking problem can be used to calculate permanents,\\ +\item The ranking function can be used to calculate permanents,\\ so it is $\#\rm P$-complete \item However, this is the only obstacle. Calculating $\O(n)$ sub-permanents is sufficient. @@ -304,16 +322,43 @@ For restricted permutations (e.g., derangements): \[M. 2008] \end{frame} -\begin{frame}{My contributions} +\begin{frame}{Summary} + +Summary:\\ \begin{itemize} -\item The lower bound for the Contractive Borùvka's algorithm -\item A~simpler linear-time tree isomorphism algorithm. -\item A~linear-time algorithm for MST on minor-closed classes. -\item Corrected and simplified MST verification. -\item Ranking and unranking algorithms. +\item Low-level algorithmic techniques on RAM and PM +\item Generalized pointer-based sorting and RAM vectors +\item Applied to a~variety of problems: + \begin{itemize} + \item A~short linear-time tree isomorphism algorithm + \item A~linear-time algorithm for MST on minor-closed classes + \item Corrected and simplified MST verification + \item Dynamic MST with small weights + \item {\it Ranking and unranking of permutations} + \end{itemize} +\item Also: + \begin{itemize} + \item A~lower bound for the Contractive Borùvka's algorithm + \item Simplified soft-heaps + \end{itemize} \end{itemize} \end{frame} +\begin{frame}{Good Bye} + +\bigskip + +\centerline{\sc\huge The End} + +\bigskip + +\begin{figure} +\pgfdeclareimage[width=0.3\hsize]{brum}{brum2.png} +\pgfuseimage{brum} +\end{figure} + +\end{frame} + \end{document} -- 2.39.2