From 479a6bc2f90e1455172faf7cfe33bd0ce7d507c1 Mon Sep 17 00:00:00 2001 From: Antti H S Laaksonen Date: Sat, 6 May 2017 13:50:08 +0300 Subject: [PATCH] Some fixes --- chapter18.tex | 26 +++++++++++++++++++++----- 1 file changed, 21 insertions(+), 5 deletions(-) diff --git a/chapter18.tex b/chapter18.tex index 4e537f1..859fe8e 100644 --- a/chapter18.tex +++ b/chapter18.tex @@ -1047,14 +1047,23 @@ its children are as follows: \end{tikzpicture} \end{center} -It would be too slow to create +If we create such a data structure for each node, +we can easily process all given queries, +because we can handle all queries related +to a node immediately after creating its +data structure. For example, the above +map structure for node 4 +tells us that its subtree +contains two nodes whose value is 3. + +However, it would be too slow to create all data structures from scratch. Instead, at each node $s$, -we create a data structure $\texttt{d}[s]$ -that initially only contains the value of $s$. +we create an initial data structure $\texttt{d}[s]$ +that only contains the value of $s$. After this, we go through the children of $s$ and \emph{merge} $\texttt{d}[s]$ and -all structures of the form +all data structures $\texttt{d}[u]$ where $u$ is a child of $s$. For example, in the above tree, the map @@ -1127,4 +1136,11 @@ swap(a,b); It is guaranteed that the above code works in constant time when $a$ and $b$ are C++ standard library data structures. -\subsubsection{Lowest common ancestors} \ No newline at end of file +\subsubsection{Lowest common ancestors} + +It turns out that we can also process a collection of +lowest common ancestor queries using an offline algorithm\footnote{This +algorithm was discovered by R. E. Tarjan in 1979 \cite{tar79}.}. +This algorithm is based on the union-find data structure +(see 15.2), and it is easier to implement than the +previous algorithms presented in this chapter.