Lowest Common Ancestor (Lowest Common Ancestor)
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Description
Given a collection of rooted trees, answer queries of the form, "What is the nearest common ancestor of vertices $x$ and $y$?"
Related Problems
Subproblem: Off-Line Lowest Common Ancestor, Lowest Common Ancestor with Static Trees, Lowest Common Ancestor with Linking Roots, Lowest Common Ancestor with Linking, Lowest Common Ancestors with Linking and Cutting
Related: Lowest Common Ancestor with Static Trees, 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 |
---|---|---|---|---|---|---|
Tarjan's off-line lowest common ancestors algorithm | 1984 | $O(n+m)$ | $O(n)$ | Exact | Deterministic | Time & Space |
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 (Offline) | 1976 | $O(n+ m*alpha(m + n, n)$) where alpha is the inverse Ackermann function | $O(n)$ | Exact | Deterministic | Time & Space |
Aho, Hopcroft, and Ullman (Static Trees) | 1976 | $O((m+n)$*log(log(n))) | $O(n*log(log(n)$)) | Exact | Deterministic | Time & Space |
Aho, Hopcroft, and Ullman (Linking) | 1976 | $O((m+n)$*log(n)) | $O(n*log(n)$) | Exact | Deterministic | Time & Space |
Modified van Leeuwen (Static Trees) | 1976 | $O(n+m*log(log(n)$)) | $O(n)$ | Exact | Deterministic | 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 (Static Trees) | 1984 | $O(n+m)$ | $O(n)$ | 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 |
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
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