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Improve language

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Antti H S Laaksonen 2017-05-29 20:22:28 +03:00
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chapter15.tex
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@ -29,7 +29,7 @@ For example, consider the following graph:
\path[draw,thick,-] (3) -- node[font=\small,label=left:3] {} (6);
\end{tikzpicture}
\end{center}
A possible spanning tree for the graph is as follows:
One spanning tree for the graph is as follows:
\begin{center}
\begin{tikzpicture}[scale=0.9]
\node[draw, circle] (1) at (1.5,2) {$1$};
@ -67,12 +67,9 @@ is 20, and such a tree can be constructed as follows:
\node[draw, circle] (6) at (5,1) {$6$};
\path[draw,thick,-] (1) -- node[font=\small,label=above:3] {} (2);
%\path[draw,thick,-] (2) -- node[font=\small,label=above:5] {} (3);
%\path[draw,thick,-] (3) -- node[font=\small,label=above:9] {} (4);
\path[draw,thick,-] (1) -- node[font=\small,label=below:5] {} (5);
\path[draw,thick,-] (5) -- node[font=\small,label=below:2] {} (6);
\path[draw,thick,-] (6) -- node[font=\small,label=below:7] {} (4);
%\path[draw,thick,-] (2) -- node[font=\small,label=left:6] {} (5);
\path[draw,thick,-] (3) -- node[font=\small,label=left:3] {} (6);
\end{tikzpicture}
\end{center}
@ -92,14 +89,11 @@ example graph is 32:
\node[draw, circle] (4) at (6.5,2) {$4$};
\node[draw, circle] (5) at (3,1) {$5$};
\node[draw, circle] (6) at (5,1) {$6$};
%\path[draw,thick,-] (1) -- node[font=\small,label=above:3] {} (2);
\path[draw,thick,-] (2) -- node[font=\small,label=above:5] {} (3);
\path[draw,thick,-] (3) -- node[font=\small,label=above:9] {} (4);
\path[draw,thick,-] (1) -- node[font=\small,label=below:5] {} (5);
%\path[draw,thick,-] (5) -- node[font=\small,label=below:2] {} (6);
\path[draw,thick,-] (6) -- node[font=\small,label=below:7] {} (4);
\path[draw,thick,-] (2) -- node[font=\small,label=left:6] {} (5);
%\path[draw,thick,-] (3) -- node[font=\small,label=left:3] {} (6);
\end{tikzpicture}
\end{center}
@ -107,15 +101,14 @@ Note that a graph may have several
minimum and maximum spanning trees,
so the trees are not unique.
This chapter discusses algorithms
for constructing spanning trees.
It turns out that it is easy to find
minimum and maximum spanning trees,
because many greedy methods produce optimals solutions.
We will learn two algorithms that both process
It turns out that several greedy methods
can be used to construct minimum and maximum
spanning trees.
In this chapter, we discuss two algorithms
that process
the edges of the graph ordered by their weights.
We will focus on finding minimum spanning trees,
but similar algorithms can be used for finding
We focus on finding minimum spanning trees,
but the same algorithms can find
maximum spanning trees by processing the edges in reverse order.
\section{Kruskal's algorithm}