Dijkstra's algorithm (794209), страница 2

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The A* algorithm is a generalization of Dijkstra's algorithm that cuts down on the size of the subgraph that must be explored, if additional information is available that provides a lower bound on the "distance" to the target. This approach can be viewed from the perspective of linear programming: there is a natural linear program for computing shortest paths, and solutions to its dual linear program are feasible if and only if they form a consistent heuristic (speaking roughly, since the sign conventions differ from place to place in the literature). This feasible dual / consistent heuristic defines a nonnegative reduced cost and A* is essentially running Dijkstra's algorithm with these reduced costs. If the dual satisfies the weaker condition of admissibility, then A* is instead more akin to the Bellman-Ford algorithm.

The process that underlies Dijkstra's algorithm is similar to the greedy process used in Prim's algorithm. Prim's purpose is to find a minimum spanning tree for a graph.

Dynamic programming perspective

From a dynamic programming point of view, Dijkstra's algorithm is a successive approximation scheme that solves the dynamic programming functional equation for the shortest path problem by the Reaching method.

In fact, Dijkstra's explanation of the logic behind the algorithm,^] namely

Problem 2. Find the path of minimum total length between two given nodes P and Q.

We use the fact that, if R is a node on the minimal path from P to Q, knowledge of the latter implies the knowledge of the minimal path from P to R.

is a paraphrasing of Bellman's famous Principle of Optimality in the context of the shortest path problem.

Conclusion

I told you about fundamental principles of Dijkstra’s algorithm. I also asked on following question: “why is this algorithm needed?”. In modern world we often have to use different devices that use different types of this algorithm. Of course, it doesn’t allow to work with graphs with negative weights. This problem is solved by another graph algorithms like Bellman-Ford’s algorithm.

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