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The amortized performance of a Fibonacci heap depends on the degree (number of children) of any tree root being (), where is the size of the heap. Here we show that the size of the (sub)tree rooted at any node x {\displaystyle x} of degree d {\displaystyle d} in the heap must have size at least F d + 2 {\displaystyle F_{d+2}} , where F i ...
Dijkstra's algorithm (/ ˈ d aɪ k s t r ə z / DYKE-strəz) is an algorithm for finding the shortest paths between nodes in a weighted graph, which may represent, for example, a road network. It was conceived by computer scientist Edsger W. Dijkstra in 1956 and published three years later.
A pairing heap is either an empty heap, or a pairing tree consisting of a root element and a possibly empty list of pairing trees. The heap ordering property requires that parent of any node is no greater than the node itself. The following description assumes a purely functional heap that does not support the decrease-key operation.
The Dijkstra algorithm originally was proposed as a solver for the single-source-shortest-paths problem. However, the algorithm can easily be used for solving the All-Pair-Shortest-Paths problem by executing the Single-Source variant with each node in the role of the root node. In pseudocode such an implementation could look as follows:
From a dynamic programming point of view, Dijkstra's algorithm for the shortest path problem is a successive approximation scheme that solves the dynamic programming functional equation for the shortest path problem by the Reaching method. [8] [9] [10] In fact, Dijkstra's explanation of the logic behind the algorithm, [11] namely Problem 2.
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 ...
The time complexity of Yen's algorithm is dependent on the shortest path algorithm used in the computation of the spur paths, so the Dijkstra algorithm is assumed. Dijkstra's algorithm has a worse case time complexity of (), but using a Fibonacci heap it becomes (+ ), [3] where is the number of edges in the graph.
In computer science, smoothsort is a comparison-based sorting algorithm.A variant of heapsort, it was invented and published by Edsger Dijkstra in 1981. [1] Like heapsort, smoothsort is an in-place algorithm with an upper bound of O(n log n) operations (see big O notation), [2] but it is not a stable sort.