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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 ...
Pointer jumping or path doubling is a design technique for parallel algorithms that operate on pointer structures, such as linked lists and directed graphs. Pointer jumping allows an algorithm to follow paths with a time complexity that is logarithmic with respect to the length of the longest path.
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 fact in order to answer a level ancestor query, the algorithm needs to jump from a path to another until it reaches the root and there could be Θ(√ n) of such paths on a leaf-to-root path. This leads us to an algorithm that can pre-process the tree in O( n ) time and answers queries in O( √ n ).
Consider finding a shortest path for traveling between two cities by car, as illustrated in Figure 1. Such an example is likely to exhibit optimal substructure. That is, if the shortest route from Seattle to Los Angeles passes through Portland and then Sacramento, then the shortest route from Portland to Los Angeles must pass through Sacramento too.
The first three stages of Johnson's algorithm are depicted in the illustration below. The graph on the left of the illustration has two negative edges, but no negative cycles. The center graph shows the new vertex q, a shortest path tree as computed by the Bellman–Ford algorithm with q as starting vertex, and the values h(v) computed at each other node as the length of the shortest path from ...
Local search is an anytime algorithm; it can return a valid solution even if it's interrupted at any time after finding the first valid solution. Local search is typically an approximation or incomplete algorithm because the search may stop even if the current best solution found is not optimal. This can happen even if termination happens ...
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 ...