diff --git a/om-graphs/graphs.tex b/om-graphs/graphs.tex index b71b2880e3badbf67a9ab9313153942d759d1896..947780bd9b47360912fb4271d2e2881891206846 100644 --- a/om-graphs/graphs.tex +++ b/om-graphs/graphs.tex @@ -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