Commit e9ac7079 authored by Ondřej Mička's avatar Ondřej Mička
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Graphs: Minor changes

parent ae8844c7
......@@ -185,14 +185,13 @@ time.
\section[linkcut]{Link-cut trees}
Link-cut trees are dynamic version of the heavy-light decomposition. They allow us to
change structure of the represented forest by either linking two trees or by cutting an
edge inside a tree. Link-cut trees were introduced in a paper by Sleator and Tarjan \TODO
reference. However, we will show later version that uses splay trees instead of original
biased binary trees \TODO reference. Although it achieves the time complexity only in amortized
case, it is significantly easier to analyze.
edge inside a tree. Link-cut trees were introduced in a paper by Sleator and Tarjan in
1982. However, we will show later version from 1985, also by Sleator and Tarjan, that uses splay
trees instead of original biased binary trees. Although it achieves the time complexity
only in amortized case, it is significantly easier to analyze.
Link-cut tree represents a forest $F$ of \em{rooted} trees; each edge is oriented towards the
respective root. It supports following operations:
\TODO proper formatting
\list{o}
\: Structural queries:
\tightlist{-}
......@@ -258,9 +257,8 @@ we jump to the top of the newly created fat path and repeat the whole process.
\figure[expose-idea]{expose-idea.pdf}{}{One step of $\Expose$ in the thin-fat
decomposition.}
\theorem{
\theoremn{Sleator, Tarjan'82}{
$\Expose$ operation performs $\O(\log n)$ steps amortized.
\TODO reference S\&T'82
}
By using a balanced binary tree to represent fat paths, we obtain
......
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