lowest common ancestor

However, there exist several algorithms for processing trees so that lowest common ancestors may be found more quickly.

It is easy to see that a range minimum query is the lowest common ancestor in of and .

The lowest common ancestor (LCA) is a concept in graph theory and computer science.

If the suffix tree is prepared for constant time lowest common ancestor retrieval, it can be solved in time.

Since the lowest common ancestor is solvable in constant time using a preprocessing of time and space thus so does the range minimum query problem.

Determine, for each edge uv that is not part of the tree, the distance between the root and the lowest common ancestor of u and v.

In a Cartesian tree, this minimum value may be found at the lowest common ancestor of the leftmost and rightmost values in the subsequence.

Probably the simplest solution to sketch here is based on the equivalence between this problem and the Lowest Common Ancestor Problem.

Their algorithm processes any tree in linear time, so that subsequent lowest common ancestor queries may be answered in constant time per query.

RMQs can be used to solve the lowest common ancestor problem, and is used as a tool for many tasks in exact and approximate string matching.

Other contributions by Uzi Vishkin and various co-authors include parallel algorithms for list ranking, lowest common ancestor, spanning trees, and biconnected components.

As had been previously observed by , the range minimum problem can in turn be transformed back into a lowest common ancestor problem using the technique of Cartesian trees.

The lowest common ancestor problem was defined by , but were the first to develop an optimally efficient lowest common ancestor data structure.

Tarjan also found a simpler but less efficient algorithm, based on the union-find data structure, for computing lowest common ancestors of an offline batch of pairs of nodes.

In such a tree, there are n pairs of adjacent leaves, that may be weakly ordered by the height of their lowest common ancestor; this weak ordering determines the tree.

The LCP array is also an essential part of compressed suffix trees which provide full suffix tree functionality like suffix links and lowest common ancestor queries.

In a tree data structure where each node points to its parent, the lowest common ancestor can be easily determined by finding the first intersection of the paths from v and w to the root.

The pseudocode below determines the lowest 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.

A data structure constructed from the minimum spanning tree allows the minimax distance between any pair of vertices to be computed in constant time per distance, using lowest common ancestor queries in a Cartesian tree.

An equivalent way of describing the cograph formed from a cotree is that two vertices are connected by an edge if and only if the lowest common ancestor of the corresponding leaves is labeled by 1.

The leaves of the Cartesian tree represent the vertices of the input graph, and the minimax distance between two vertices equals the weight of the Cartesian tree node that is their lowest common ancestor.

The lowest common ancestor between two nodes v and w is defined as the lowest node in T that has both v and w as descendants (where we allow a node to be a descendant of itself).

Because lowest common ancestors may be found in constant time per query, using a data structure that takes linear space to store and that may be constructed in linear time, the same bounds hold for the range minimization problem.

The same construction, of lowest common ancestors in a Cartesian tree, makes it possible to construct a data structure with linear space that allows the distances between pairs of points in any ultrametric space to be queried in constant time per query.

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.