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A linear-time algorithm for finding a longest path in a tree was proposed by Edsger Dijkstra around 1960, while a formal proof of this algorithm was published in 2002. [15] Furthermore, a longest path can be computed in polynomial time on weighted trees, on block graphs, on cacti, [16] on bipartite permutation graphs, [17] and on Ptolemaic ...
Dijkstra's algorithm finds the shortest path from a given source node to every other node. [7]: 196–206 It can be used to find the shortest path to a specific destination node, by terminating the algorithm after determining the shortest path to the destination node. For example, if the nodes of the graph represent cities, and the costs of ...
Edmonds' algorithm (also known as Chu–Liu/Edmonds' algorithm): find maximum or minimum branchings; Euclidean minimum spanning tree: algorithms for computing the minimum spanning tree of a set of points in the plane; Longest path problem: find a simple path of maximum length in a given graph; Minimum spanning tree. Borůvka's algorithm ...
The simplest way to find a level ancestor of a node is to climb up the tree towards the root of the tree. On the path to the root of the tree, every ancestor of a node can be visited and therefore reported. In this case, the tree does not need to be preprocessed and the time to answer a query is O(h), where "h" is the height of the tree. This ...
For infinite trees, simple algorithms often fail this. For example, given a binary tree of infinite depth, a depth-first search will go down one side (by convention the left side) of the tree, never visiting the rest, and indeed an in-order or post-order traversal will never visit any nodes, as it has not reached a leaf (and in fact never will ...
Two primary problems of pathfinding are (1) to find a path between two nodes in a graph; and (2) the shortest path problem—to find the optimal shortest path. Basic algorithms such as breadth-first and depth-first search address the first problem by exhausting all possibilities; starting from the given node, they iterate over all potential ...
In computer science, Tarjan's off-line lowest 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 lowest 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 ...
A Fenwick tree or binary indexed tree (BIT) is a data structure that stores an array of values and can efficiently compute prefix sums of the values and update the values. It also supports an efficient rank-search operation for finding the longest prefix whose sum is no more than a specified value.