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\begin{center} {\Large\bf Notes on Kruskal's Algorithm for Minimal Spanning Tree}
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In Kruskal's algorithm (\S 23.2) the edges are ordered $e_1,\ldots,e_E$ by
of weight and $e_i$ is added to the tree if and only if its addition does not
cause a cycle. The data structure that does this efficiently is covered in
detail in Chapter 21, which we are not covering. Instead, these notes give
a specific implementation of the algorithm. Assume the edges have already been
ordered by weight and $x_i,y_i$ are the vertices of $e_i$. To each vertex $x$
we have functions $\pi(x)$ and $SIZE(x)$, initially all $\pi(x)\leftarrow x$ and
all $SIZE(x)\leftarrow 1$.
\par For $i=1$ to $E$ we set (for notational convenience) $x\leftarrow x_i$,
$y\leftarrow y_i$ and do the following:
\\ WHILE $\pi(x)\neq x$
\\ \hspace*{1cm} $x\leftarrow \pi(x)$
\\ WHILE $\pi(y)\neq y $
\\ \hspace*{1cm} $y\leftarrow \pi(y)$
\\ IF $x\neq y$ then DO
\\ \hspace*{1cm} IF $SIZE(x)\leq SIZE(y)$ then DO
\\ \hspace*{2cm} $\pi(x)\leftarrow y$
\\ \hspace*{2cm} $SIZE(y)\leftarrow SIZE(y)+SIZE(x)$
\\ \hspace*{1cm} OTHERWISE DO
\\ \hspace*{2cm} $\pi(y)\leftarrow x$
\\ \hspace*{2cm} $SIZE(x)\leftarrow SIZE(x)+SIZE(y)$
\\ \hspace*{1cm} Add $e_i$ to Minimal Spanning Tree
At any time the $\pi(x)$ will give a rooted forest with $\pi(x)=x$ exactly
when $x$ is a root. In that case $SIZE(x)$ will be the size of the forest.
Certain edges will have already been put in the Minimal Spanning Tree so that
the structure will be a forest. That forest and the forest given by $\pi(x)$
will have the same components (though they may have different edges).
\par Example. $e_1=(a,c),e_2=(c,b),e_3=(d,e),e_4=(a,d),e_5=(b,d)$. With $i=1$ we
add $e_1$ to tree, $\pi(a)\leftarrow c$ and $SIZE(c)\leftarrow 2$. With $i=2$
we add $e_2$ to tree, as $SIZE(c)>SIZE(b)$ we set $pi(b)\leftarrow c$ and
$SIZE(c)\leftarrow 3$. With $i=3$ we add $e_3$ to tree, $\pi(d)\leftarrow e$
and $SIZE(e)\leftarrow 2$. Now $i=4$ so $x\leftarrow a;y\leftarrow d$. The
WHILE parts trace $x$ down to its root $c$ and $y$ down to its root $e$. As
$SIZE(c)>SIZE(e)$ we set $\pi(e)\leftarrow c$, $SIZE(c)\leftarrow 5$ and add
$e_4$ to the tree. Note that the current state of the Minimal Spanning Tree and
the forest given by $\pi$ have different edges but the same components.
Now with $i=5$, $x\leftarrow b; y\leftarrow e$. Both $x,y$ trace down
with the WHILE loops to the same $c$ so we do nothing and $e_5$ is not added to
the tree.
To analyze the time we note that the process is done $E$ times, so we
analyze the process with a particular $x=x_i,y=y_i$. The key aspect
to the time is we must iterate $\pi(x)\leftarrow x$ until reaching
a root. (Similarly for $y$.) At first blush, this seems like it might
take time $V$. ($V$ is number of vertices.)
{\em However,} here we use the fact that when we earlier
considered an edge $x,y$ and we moved them down to their roots we then
reset $\pi(x)\leftarrow y$ where $SIZE[y]$ had been bigger than $SIZE[x]$.
Now the new $SIZE[y]$ became the old $SIZE[x]+SIZE[y]$. That is, the
new $SIZE[y]$ is at least double the old $SIZE[x]$. As $x$ is no longer
a root its value of $SIZE[x]$ will never change. The value of $SIZE[y]$
may change later, but it can only get larger. Hence we will have
$2\cdot SIZE[x] \leq SIZE[y]$ forevermore. Therefore, as we look
at a path $x,\pi(x),\pi(\pi(x)),\ldots$ the value $SIZE(\cdot)$ at
least doubles each iteration. Therefore the path can only be of length
$\log V$.
This is a big savings over the length $V$ without this aspect.
Now the process with a particular $x,y$ takes time $O(\log V)$ and therefore
the total time is $O(E\log V)$.
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