Add Tarjan's algorithm

chapter18.tex

@@ -939,7 +939,7 @@ processes each query before receiving the next query.
 However, in many problems, the online
 property is not necessary.
 In this section, we focus on \emph{offline} algorithms
-that are given a collection of queries that can be
+that are given a set of queries that can be
 processed in any order.
 It is often easier to design an offline algorithm
 compared to an online algorithm.
@@ -1138,9 +1138,119 @@ when $a$ and $b$ are C++ standard library data structures.
 
 \subsubsection{Lowest common ancestors}
 
-It turns out that we can also process a collection of
+There is also an offline algorithm
-lowest common ancestor queries using an offline algorithm\footnote{This
+for processing a set of
-algorithm was discovered by R. E. Tarjan in 1979 \cite{tar79}.}.
+lowest common ancestor queries\footnote{This
-This algorithm is based on the union-find data structure
+algorithm was published by R. E. Tarjan in 1979 \cite{tar79}.}.
-(see 15.2), and it is easier to implement than the
+The algorithm is based on the union-find data structure
-previous algorithms presented in this chapter.
+(see Chapter 15.2), and the benefit of the algorithm is
+that it is easier to implement than the
+algorithms discussed earlier in this chapter.
+
+The algorithm is given as input a set of pairs of nodes,
+and it determines for each such pair the
+lowest common ancestor.
+The algorithm performs a depth-first tree traversal
+and maintains disjoint sets of nodes.
+Initially, each node belongs to a separate set.
+For each set, we also maintain the highest node in the
+tree that belongs to the set.
+
+When the algorithm visits a node $x$,
+it goes through all nodes $y$ such that
+the lowest common ancestor of $x$ and $y$
+has to be found.
+If $y$ has already been visited,
+the algorithm reports that the 
+lowest common ancestor of $x$ and $y$
+is the highest node in the set of $y$.
+Then, after processing the subtree of $x$,
+the algorithm combines the sets of $x$ and its parent.
+
+For example, assume that we would like the lowest
+common ancestor of node pairs $(5,8)$
+and $(2,7)$ in the following tree:
+\begin{center}
+\begin{tikzpicture}[scale=0.85]
+\node[draw, circle] (1) at (0,3) {$1$};
+\node[draw, circle] (2) at (2,1) {$4$};
+\node[draw, circle] (3) at (-2,1) {$2$};
+\node[draw, circle] (4) at (0,1) {$3$};
+\node[draw, circle] (5) at (2,-1) {$7$};
+\node[draw, circle] (6) at (-3,-1) {$5$};
+\node[draw, circle] (7) at (-1,-1) {$6$};
+\node[draw, circle] (8) at (-1,-3) {$8$};
+\path[draw,thick,-] (1) -- (2);
+\path[draw,thick,-] (1) -- (3);
+\path[draw,thick,-] (1) -- (4);
+\path[draw,thick,-] (2) -- (5);
+\path[draw,thick,-] (3) -- (6);
+\path[draw,thick,-] (3) -- (7);
+\path[draw,thick,-] (7) -- (8);
+\end{tikzpicture}
+\end{center}
+
+In the following pictures, gray nodes denote visited nodes
+and dashed groups of nodes belong to the same set.
+When the algorithm visits node 8, it notices that
+node 5 has been visited and the highest node
+in its set is 2. Thus, the lowest common ancestor
+of nodes 5 and 8 is 2:
+\begin{center}
+\begin{tikzpicture}[scale=0.85]
+\node[draw, circle, fill=lightgray] (1) at (0,3) {$1$};
+\node[draw, circle] (2) at (2,1) {$4$};
+\node[draw, circle, fill=lightgray] (3) at (-2,1) {$2$};
+\node[draw, circle] (4) at (0,1) {$3$};
+\node[draw, circle] (5) at (2,-1) {$7$};
+\node[draw, circle, fill=lightgray] (6) at (-3,-1) {$5$};
+\node[draw, circle, fill=lightgray] (7) at (-1,-1) {$6$};
+\node[draw, circle, fill=gray] (8) at (-1,-3) {$8$};
+\path[draw,thick,-] (1) -- (2);
+\path[draw,thick,-] (1) -- (3);
+\path[draw,thick,-] (1) -- (4);
+\path[draw,thick,-] (2) -- (5);
+\path[draw,thick,-] (3) -- (6);
+\path[draw,thick,-] (3) -- (7);
+\path[draw,thick,-] (7) -- (8);
+
+\draw [red,thick,dashed,line width=2pt,rotate around={-28:(-2,0)}] (-2.9,1.5) rectangle (-1.9,-2);
+
+
+\draw [red,thick,dashed,line width=2pt] (-1.5,-0.5) rectangle (-0.5,-1.5);
+\draw [red,thick,dashed,line width=2pt] (-1.5,-2.5) rectangle (-0.5,-3.5);
+
+\draw [red,thick,dashed,line width=2pt] (0.5,3.5) rectangle (-0.5,2.5);
+\draw [red,thick,dashed,line width=2pt] (0.5,1.5) rectangle (-0.5,0.5);
+\draw [red,thick,dashed,line width=2pt] (2.5,1.5) rectangle (1.5,0.5);
+\draw [red,thick,dashed,line width=2pt] (2.5,-0.5) rectangle (1.5,-1.5);
+\end{tikzpicture}
+\end{center}
+
+Later, when visiting node 7,
+the algorithm determines that
+the lowest common ancestor of nodes 2 and 7 is 1:
+\begin{center}
+\begin{tikzpicture}[scale=0.85]
+\node[draw, circle, fill=lightgray] (1) at (0,3) {$1$};
+\node[draw, circle, fill=lightgray] (2) at (2,1) {$4$};
+\node[draw, circle, fill=lightgray] (3) at (-2,1) {$2$};
+\node[draw, circle, fill=lightgray] (4) at (0,1) {$3$};
+\node[draw, circle, fill=gray] (5) at (2,-1) {$7$};
+\node[draw, circle, fill=lightgray] (6) at (-3,-1) {$5$};
+\node[draw, circle, fill=lightgray] (7) at (-1,-1) {$6$};
+\node[draw, circle, fill=lightgray] (8) at (-1,-3) {$8$};
+\path[draw,thick,-] (1) -- (2);
+\path[draw,thick,-] (1) -- (3);
+\path[draw,thick,-] (1) -- (4);
+\path[draw,thick,-] (2) -- (5);
+\path[draw,thick,-] (3) -- (6);
+\path[draw,thick,-] (3) -- (7);
+\path[draw,thick,-] (7) -- (8);
+
+\draw [red,thick,dashed,line width=2pt] (0.5,3.5) rectangle (-3.5,-3.5);
+\draw [red,thick,dashed,line width=2pt] (2.5,1.5) rectangle (1.5,0.5);
+\draw [red,thick,dashed,line width=2pt] (2.5,-0.5) rectangle (1.5,-1.5);
+
+\end{tikzpicture}
+\end{center}

list.tex

@@ -352,6 +352,11 @@
   Efficiency of a good but not linear set union algorithm.
   \emph{Journal of the ACM}, 22(2):215--225, 1975.
 
+\bibitem{tar79}
+  R. E. Tarjan.
+  Applications of path compression on balanced trees.
+  \emph{Journal of the ACM}, 26(4):690--715, 1979.
+
 \bibitem{tar84}
   R. E. Tarjan and U. Vishkin.
   Finding biconnected componemts and computing tree functions in logarithmic parallel time.