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A central problem in algorithmic graph theory is the shortest path problem. Hereby, the problem of finding the shortest path between every pair of nodes is known as all-pair-shortest-paths (APSP) problem. As sequential algorithms for this problem often yield long runtimes, parallelization has shown to be beneficial in this field. In this ...
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 ...
Find the Shortest Path: 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 node is marked "closed", and all nodes adjacent to it are added to the open set if they have not already been examined. This process repeats until a path to the destination has been found. Since the lowest distance nodes are examined first, the first time the destination is found, the path to it will be the shortest path. [4]
All these models had one thing in common: they all predicted very short average path length. [1] The average path length depends on the system size but does not change drastically with it. Small world network theory predicts that the average path length changes proportionally to log n, where n is the number of nodes in the network.
The maximum shortest path weight for the source node is defined as ():= { (,): (,) <}, abbreviated . [1] Also, the size of a path is defined to be the number of edges on the path. We distinguish light edges from heavy edges, where light edges have weight at most Δ {\displaystyle \Delta } and heavy edges have weight bigger than Δ ...
The number of shortest paths between and every vertex is calculated using breadth-first search. The breadth-first search starts at s {\displaystyle s} , and the shortest distance d ( v ) {\displaystyle d(v)} of each vertex from s {\displaystyle s} is recorded, dividing the graph into discrete layers.
The algorithm continues until a removed node (thus the node with the lowest f value out of all fringe nodes) is a goal node. [b] The f value of that goal is then also the cost of the shortest path, since h at the goal is zero in an admissible heuristic. The algorithm described so far only gives the length of the shortest path.