Lowest Common Ancestor with Static Trees (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 collection of trees is static but the queries are given on-line. That is, each query must be answered before the next one is known.

Related Problems

Generalizations: Lowest Common Ancestor

Related: Off-Line Lowest Common Ancestor, Lowest Common Ancestor with Linking Roots, Lowest Common Ancestor with Linking, 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
Schieber; Vishkin 1988 $O(n+m)$ $O(n)$ Exact Deterministic Time & Space
Berkman; Vishkin 1993 $O(n+m)$ ? $O(n)$ Exact Deterministic Time
[[Bender; Colton (LCA <=> RMQ) (Lowest Common Ancestor with Static Trees Lowest Common Ancestor)|Bender; Colton (LCA <=> RMQ)]] 2000 $O(n+m)$ $O(n)$ Exact Deterministic Time
Stephen Alstrup, Cyril Gavoille, Haim Kaplan & Theis Rauhe 2004 $O(n+m)$ $O(n)$ Exact Deterministic Time
Aho, Hopcroft, and Ullman (Static Trees) 1976 $O((m+n)$*log(log(n))) $O(n*log(log(n)$)) Exact Deterministic Time & Space
Modified van Leeuwen (Static Trees) 1976 $O(n+m*log(log(n)$)) $O(n)$ Exact Deterministic Space
Harel, Tarjan (Static Trees) 1984 $O(n+m)$ $O(n)$ Exact Deterministic Time & Space
Schieber; Vishkin (Parallel) 1988 $O(m+log(n)$) $O(n)$ total (auxiliary?) Exact Parallel Time & Space
Fischer, Heun 2006 $O(m+n)$ $O(n)$ Exact Parallel Time & Space
Kmett 2015 $O(m*log(h)$) Exact Parallel Time

Time Complexity Graph

Lowest Common Ancestor - Lowest Common Ancestor with Static Trees - Time.png