Lowest Common Ancestor with Linking (Lowest Common Ancestor)

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Given a collection of rooted trees, answer queries of the form, "What is the nearest common ancestor of vertices $x$ and $y$?" In this version of the problem, the queries are on-line. Interspersed with the queries are on-line commands $link(x, y)$ such that $y$, but not necessarily $x$, is a tree root. The effect of a command $link(x, y)$ is to combine the trees containing $x$ and $y$ by making $x$ the parent of $y$.

Related Problems

Generalizations: Lowest Common Ancestor

Related: Off-Line Lowest Common Ancestor, Lowest Common Ancestor with Static Trees, Lowest Common Ancestor with Linking Roots, Lowest Common Ancestors with Linking and Cutting


$n$: number of vertices

$m$: number of total number of operations (queries, links, and cuts)

Table of Algorithms

Name Year Time Space Approximation Factor Model Reference
Aho, Hopcroft, and Ullman (Linking) 1976 $O((m+n)$*log(n)) $O(n*log(n)$) Exact Deterministic Time & Space
Modified van Leeuwen (Linking Roots) 1976 $O(n+m*log(log(n)$)) $O(n)$ Exact Deterministic Space
Sleator and Tarjan (Linking) 1983 $O(n+m*log(n)$) $O(n)$ Exact Deterministic Time & Space
Sleator and Tarjan (Linking and Cutting) 1983 $O(n+m*log(n)$) $O(n)$ Exact Deterministic Time & Space
Harel, Tarjan (Linking Roots) 1984 $O(n+ m*alpha(m + n, n)$) where alpha is the inverse Ackermann function $O(n)$ Exact Deterministic Time & Space