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The flip graphs of a pentagon and a hexagon, corresponding to rotations of three-node and four-node binary trees. Given a family of triangulations of some geometric object, a flip is an operation that transforms one triangulation to another by removing an edge between two triangles and adding the opposite diagonal to the resulting quadrilateral.
During execution, the distance of a node N is the length of the shortest path discovered so far between the starting node and N. [18] From the unvisited set, select the current node to be the one with the smallest (finite) distance; initially, this is the starting node (distance zero). If the unvisited set is empty, or contains only nodes with ...
The rotation distance between any two binary trees with the same number of nodes is the minimum number of rotations needed to transform one into the other. With this distance, the set of n-node binary trees becomes a metric space: the distance is symmetric, positive when given two different trees, and satisfies the triangle inequality.
With given nodes, the minimum possible tree height is = (+) with which the tree is a balanced full tree or perfect tree. With a given height h {\displaystyle h} , the number of nodes can't exceed the 2 h + 1 − 1 {\displaystyle 2^{h+1}-1} as the number of nodes in a perfect tree.
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. Once the minimum spanning tree edges have been sorted, this Cartesian tree can be constructed in linear time. [16]
The leaves of the Cartesian tree represent points of the metric space, and the lowest common ancestor of two leaves in the Cartesian tree is the heaviest edge between those two points in the minimum spanning tree, which has weight equal to the distance between the two points. Once the minimum spanning tree has been found and its edge weights ...
Shortest path (A, C, E, D, F), blue, between vertices A and F in the weighted directed graph. In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. [1]
The numbers beside the vertices indicate the distance from the root vertex. In mathematics and computer science, a shortest-path tree rooted at a vertex v of a connected, undirected graph G is a spanning tree T of G, such that the path distance from root v to any other vertex u in T is the shortest path distance from v to u in G.