+\documentclass{beamer}
+\usepackage[latin2]{inputenc}
+\usepackage{palatino}
+\usetheme{Warsaw}
+\title[Graph Algorithms]{Graph Algorithms\\Spanning Trees and Ranking}
+\author[Martin Mare¹]{Martin Mare¹\\\texttt{mares@kam.mff.cuni.cz}}
+\institute{Department of Applied Mathematics\\MFF UK Praha}
+\date{2008}
+\begin{document}
+\setbeamertemplate{navigation symbols}{}
+\setbeamerfont{title page}{family=\rmfamily}
+
+\def\[#1]{~{\color{violet} [#1]}}
+\def\O{{\cal O}}
+
+\begin{frame}
+\titlepage
+\end{frame}
+
+\begin{frame}{The Minimum Spanning Tree Problem}
+
+{\bf 1. Minimum Spanning Tree Problem:}
+
+\begin{itemize}
+\item Given a~weighted undirected graph,\\
+ what is its lightest spanning tree?
+\item In fact, a~linear order on edges is sufficient.
+\item Efficient solutions are very old \[Borùvka 1926]
+\item A~long progression of faster and faster algorithms.
+\item Currently very close to linear time, but still not there.
+\end{itemize}
+
+\end{frame}
+
+\begin{frame}{The Ranking Problems}
+
+{\bf 2. Ranking of Combinatorial Structures:}
+
+\begin{itemize}
+\item We are given a~set~$C$ of objects with a~linear order $\prec$.
+\item {\bf Ranking function $R_\prec(x)$:} how many objects precede~$x$?
+\item {\bf Unranking function $R^{-1}_\prec(i)$:} what is the $i$-th object?
+\end{itemize}
+
+\pause
+
+\begin{example}[toy]
+$C=\{0,1\}^n$ with lexicographic order
+
+\pause
+$R$ = conversion from binary\\
+$R^{-1}$ = conversion to binary
+\end{example}
+
+\pause
+\begin{example}[a~real one]
+$C=$ set of all permutations on $\{1,\ldots,n\}$
+\end{example}
+
+\pause
+How to compute the (un)ranking function efficiently?
+
+For permutations, an $\O(n\log n)$ algorithm was known \[folklore].
+
+We will show how to do that in $\O(n)$.
+
+\end{frame}
+
+\begin{frame}{Models of computation: RAM}
+
+As we approach linear time, we must specify the model.
+
+~
+
+{\bf 1. The Random Access Machine (RAM):}
+
+\begin{itemize}
+\item Works with integers
+\item Memory: an array of integers indexed by integers
+\end{itemize}
+
+~
+
+Many variants exist, we will use the {\bf Word-RAM:}
+
+\begin{itemize}
+\item Machine words of $W$ bits
+\item The ``C operations'': arithmetics, bitwise logical op's
+\item Unit cost
+\item We know that $W\ge\log_2 \vert \hbox{input} \vert$
+\end{itemize}
+
+\end{frame}
+
+\begin{frame}{Models of computation: PM}
+
+{\bf 2. The Pointer Machine (PM):}
+
+\begin{itemize}
+\item Memory cells accessed via pointers
+\item Each cell contains $\O(1)$ pointers and $\O(1)$ symbols
+\item Operates only on pointers and symbols
+\end{itemize}
+
+~
+
+\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
+\end{itemize}
+\end{beamerboxesrounded}
+
+~
+
+We can emulate PM on RAM with constant slowdown.
+
+Emulation of RAM on PM is more expensive.
+
+\end{frame}
+
+\begin{frame}{PM Techniques}
+
+{\bf Bucket Sorting} does not need arrays.
+
+~
+
+Interesting consequences:
+
+\begin{itemize}
+\item Flattening of multigraphs in $\O(m+n)$
+\item Unification of sequences in $\O(n+\sum_i\ell_i+\vert\Sigma\vert)$
+\item (Sub)tree isomorphism in $\O(n)$ simplified \[M. 2008]
+\item Batched graph computations \[Buchsbaum et al.~1998]
+\end{itemize}
+
+\end{frame}
+
+\begin{frame}{RAM Techniques}
+
+We can use RAM as a vector machine:
+
+~
+
+\begin{example}
+\def\sep{\;{\color{brown}0}}
+\def\seq{\;{\color{brown}1}}
+\def\sez{\;{\color{red}0}}
+We can encode the vector $(1,5,3,0)$ with 3-bit fields as:
+\begin{center}
+\sep001\sep101\sep011\sep000
+\end{center}
+And then search for 3 by:
+
+\begin{center}
+\begin{tabular}{rc}
+ &\seq001\seq101\seq011\seq000 \\
+{\sc xor} &\sep011\sep011\sep011\sep011 \\
+\hline
+ &\seq010\seq110\seq000\seq011 \\
+$-$ &\sep001\sep001\sep001\sep001 \\
+\hline
+ &\seq001\seq101\sez111\seq010 \\
+{\sc and} &\seq000\seq000\seq000\seq000 \\
+\hline
+ &\seq000\seq000\sez000\seq000 \\
+\end{tabular}
+\end{center}
+\end{example}
+
+\end{frame}
+
+\begin{frame}{RAM Data Structures}
+
+We can translate vector operations to $\O(1)$ RAM instructions
+
+\smallskip
+
+\dots\ as long as the vector fits in $\O(1)$ words.
+
+~
+
+We can build ``small'' data structures operating in $\O(1)$ time:
+
+\begin{itemize}
+\item Sets
+\item Ordered sets with ranking
+\item ``Small'' heaps of ``large'' integers \[Fredman \& Willard 1990]
+\end{itemize}
+
+\end{frame}
+
+\begin{frame}{Minimum Spanning Trees}
+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.])
+\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 MST verification: $\O(m)$ on RAM \[King 1997, M. 2008]
+\item Randomized: $\O(m)$ w.h.p. \[Karger et al.~1995]
+\end{itemize}
+\end{frame}
+
+\begin{frame}{MST -- Special cases}
+
+We have $\O(m)$ time algorithms for:
+
+\begin{itemize}
+\item Planar graphs \[Tarjan 1976, Matsui 1995, M. 2004] (PM)
+\item Minor-closed classes \[Tarjan 1983, M. 2004] (PM)
+\item $\O(1)$ different weights \[folklore] (PM)
+\item Integer weights \[Fredman \& Willard 1990] (RAM)
+\end{itemize}
+
+\end{frame}
+
+\begin{frame}{MST -- Optimality}
+
+There is a~provably optimal comparison-based algorithm \\\[Pettie \& Ramachandran 2002]
+
+~
+
+However, there is a catch: \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.
+
+\pause
+~
+
+\begin{corollary}
+It runs on the PM, so we know that if there is a~linear-time algorithm,
+it does not need any special RAM data structures. (They can however
+help us to find it.)
+\end{corollary}
+
+\end{frame}
+
+\begin{frame}{MST -- Dynamic algorithms}
+
+Sometimes, we need to find the MST of a~changing graph. \\
+We insert/delete edges, the structure responds with $\O(1)$
+modifications of the MST.
+
+\begin{itemize}
+\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}
+\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]
+ \end{itemize}
+\item $K$ best 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]
+ \end{itemize}
+\end{itemize}
+
+\end{frame}
+
+\begin{frame}{Back to Ranking}
+
+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 can represent the subsets as RAM vectors
+\item This gives us an~$\O(n)$ time algorithm for (un)ranking
+\end{itemize}
+
+~
+
+Easily extendable to $k$-permutations, also in $\O(n)$
+
+\end{frame}
+
+\begin{frame}{Restricted permutations}
+
+For restricted permutations (e.g., derangements): \[M. 2008]
+
+\begin{itemize}
+\item Existence of permutation reduces to network flows
+\item The ranking problem 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.
+\item For derangements, we have achieved $\O(n)$ time after $\O(n^2)$ time
+ preprocessing.
+\end{itemize}
+
+\end{frame}
+
+\begin{frame}{My contributions}
+
+\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.
+\end{itemize}
+
+\end{frame}
+
+\end{document}
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