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The Euler tour technique (ETT), named after Leonhard Euler, is a method in graph theory for representing trees. The tree is viewed as a directed graph that contains two directed edges for each edge in the tree. The tree can then be represented as a Eulerian circuit of the directed graph, known as the Euler tour representation (ETR) of the tree
[2] [4] This solution is based on the Euler tour technique for processing trees. The main observation is that LA(v,d) is the first node of depth d that appears in the Euler tour after the last appearance of v. Thus, by constructing the Euler tour and associated information on depth, the problem is reduced to a query on arrays, named find ...
In this tree, the lowest common ancestor of the nodes x and y is marked in dark green. Other common ancestors are shown in light green. In graph theory and computer science, the lowest common ancestor (LCA) (also called least common ancestor) of two nodes v and w in a tree or directed acyclic graph (DAG) T is the lowest (i.e. deepest) node that has both v and w as descendants, where we define ...
Euler's proof of the degree sum formula uses the technique of double counting: he counts the number of incident pairs (,) where is an edge and vertex is one of its endpoints, in two different ways. Vertex v {\displaystyle v} belongs to deg ( v ) {\displaystyle \deg(v)} pairs, where deg ( v ) {\displaystyle \deg(v)} (the degree of v ...
An Eulerian trail, [note 1] or Euler walk, in an undirected graph is a walk that uses each edge exactly once. If such a walk exists, the graph is called traversable or semi-eulerian. [3] An Eulerian cycle, [note 1] also called an Eulerian circuit or Euler tour, in an undirected graph is a cycle that uses each edge exactly once
Doubling the edges of a T-join causes the given graph to become an Eulerian multigraph (a connected graph in which every vertex has even degree), from which it follows that it has an Euler tour, a tour that visits each edge of the multigraph exactly once. This tour will be an optimal solution to the route inspection problem. [7] [2]
Learn how to download and install or uninstall the Desktop Gold software and if your computer meets the system requirements.
Another data structure that can be used for the same purpose is Euler tour tree. In solving the maximum flow problem , link/cut trees can be used to improve the running time of Dinic's algorithm from O ( V 2 E ) {\displaystyle O(V^{2}E)} to O ( V E log V ) {\displaystyle O(VE\log V)} .