Lowest Common Ancestor with Static Trees

Description

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

Parameters

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

Space Complexity Graph

Pareto Frontier Improvements Graph