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The algorithm uses a min-priority queue data structure for selecting the shortest paths known so far. Before more advanced priority queue structures were discovered, Dijkstra's original algorithm ran in (| |) time, where | | is the number of nodes.
By removing several elements at once a considerable speedup can be reached. But not all algorithms can use this kind of priority queue. Dijkstra's algorithm for example can not work on several nodes at once. The algorithm takes the node with the smallest distance from the priority queue and calculates new distances for all its neighbor nodes.
In Dijkstra's algorithm for the shortest path problem, vertices of a given weighted graph are extracted in increasing order by their distance from the starting vertex, and a priority queue is used to determine the closest remaining vertex to the starting vertex. Therefore, in this application, the priority queue operations are monotonic.
Use a shortest path algorithm (e.g., Dijkstra's algorithm, Bellman-Ford algorithm) to find the shortest path from the source node to the sink node in the residual graph. Augment the Flow: Find the minimum capacity along the shortest path. Increase the flow on the edges of the shortest path by this minimum capacity.
The key idea behind the speedup over a conventional version of Dijkstra's algorithm is that the sequence of bottleneck distances to each vertex, in the order that the vertices are considered by this algorithm, is a monotonic subsequence of the sorted sequence of edge weights; therefore, the priority queue of Dijkstra's algorithm can be ...
In many applications of priority queues such as Dijkstra's algorithm, the minimum priorities form a monotonic sequence, allowing a monotone priority queue to be used. In these applications, for both the lazy and eager variations of the optimized structure, the sequential searches for non-empty buckets cover disjoint ranges of buckets.
Prim's algorithm has many applications, such as in the generation of this maze, which applies Prim's algorithm to a randomly weighted grid graph. The time complexity of Prim's algorithm depends on the data structures used for the graph and for ordering the edges by weight, which can be done using a priority queue. The following table shows the ...
However, Dijkstra's algorithm uses a priority queue to greedily select the closest vertex that has not yet been processed, and performs this relaxation process on all of its outgoing edges; by contrast, the Bellman–Ford algorithm simply relaxes all the edges, and does this | | times, where | | is the number of vertices in the graph.