Tarjan's off-line least common ancestors algorithm

In computer science, Tarjan's off-line least common ancestors algorithm is an algorithm for computing lowest common ancestors for pairs of nodes in a tree, based on the union-find data structure. The least common ancestor of two nodes "d" and "e" in a rooted tree "T" is the node "g" that is an ancestor of both "d" and "e" and that has the greatest depth in "T". It is named after Robert Tarjan, who discovered the technique in 1979. Tarjan's algorithm is "offline"; that is, unlike other lowest common ancestor algorithms, it requires that all pairs of nodes for which the lowest common ancestor is desired must be specified in advance. The simplest version of the algorithm uses the union find data structure, which unlike other lowest common ancestor data structures can take more than constant time per operation when the number of pairs of nodes is similar in magnitude to the number of nodes. A later refinement by harvtxt|Gabow|Tarjan|1983 speeds the algorithm up to linear time.

Pseudocode

The pseudocode below determines the least common ancestor of each pair in "P", given the root "r" of a tree in which the children of node "n" are in the set "n.children". For this offline algorithm, the set "P" must be specified in advance. It uses the "MakeSet", "Find", and "Union" functions of a disjoint-set forest. "MakeSet(u)" removes "u" to a singleton set, "Find(u)" returns the standard representative of the set containing "u", and "Union(u,v)" merges the set containing "u" with the set containing "v".TarjanOLCA("r") is first called on the root "r".

function TarjanOLCA(u) MakeSet(u); u.ancestor := u; for each v in u.children do TarjanOLCA(v); Union(u,v); Find(u).ancestor := u; u.colour := black; for each v such that {u,v} in P do if v.colour = black print "Tarjan's Least Common Ancestor of " + u + " and " + v + " is " + Find(v).ancestor + ".";

Each node is initially white, and is colored black after it and all its children have been visited. The least common ancestor of the pair "{u,v}" is available as "Find(v).ancestor" immediately (and only immediately) after "u" is colored black, provided "v" is already black. Otherwise, it will be available later as "Find(u).ancestor", immediately after "v" is colored black.

For reference, here are optimized versions of "MakeSet", "Find", and "Union" for a disjoint-set forest:

function MakeSet(x) x.parent := x x.rank := 0 function Union(x, y) xRoot := Find(x) yRoot := Find(y) if xRoot.rank > yRoot.rank yRoot.parent := xRoot else if xRoot.rank < yRoot.rank xRoot.parent := yRoot else if xRoot != yRoot yRoot.parent := xRoot xRoot.rank := xRoot.rank + 1 function Find(x) if x.parent = x return x else x.parent := Find(x.parent) return x.parent

References

*citation
last1 = Gabow | first1 = H. N.
last2 = Tarjan | first2 = R. E.
authorlink2 = Robert Tarjan
contribution = A linear-time algorithm for a special case of disjoint set union
title = Proceedings of the 15th ACM Symposium on Theory of Computing (STOC)
year = 1983 | pages = 246–251 | doi = 10.1145/800061.808753.

*citation
last = Tarjan | first R. E.
authorlink = Robert Tarjan
title = Applications of path compression on balanced trees
journal = Journal of the ACM
volume = 26
issue = 4
year = 1979
pages = 690–715
doi = 10.1145/322154.322161.