The input consists of a graph G and a special node s. If you are already familiar with Dijkstra's algorithm, you can skip to the code snippets. The initial sections are mostly background. We provide implementations in Python and C++. Linear-search Dijkstra: unordered array.Textbook Dijkstra: indexed binary heap.Here is summary of the resulting runtime and space complexities: Throughout the post, let n be the number of nodes and m the number of edges. Roughly, each of the 5 versions corresponds to a different data structure used to implement the priority queue. This is actually impractical due to the complexity and high constant factors of the Fibonacci heap. Theoretical Dijkstra: version that uses a Fibonacci heap for the priority queue in order to achieve the fastest possible runtime in terms of big-O notation.BST Dijkstra: version which uses a self-balancing binary search tree to implement the priority queue functionality, including the "decrease key" operation.Lazy Dijkstra: practical version which does not use the "decrease key" operation at all, at the cost of using some extra space.Linear-search Dijkstra: the most naive implementation, but which is actually optimal for dense graphs.As we said, this often does not hold true in reality. Textbook Dijkstra: the version commonly taught in textbooks where we assume that we have a priority queue with the "decrease key" operation.All in all, we consider 5 versions of Dijkstra (names mostly made up by me): In this blog, we go over the different ways to implement Dijkstra's algorithm with and without this operation, and the implications of using each. While most programming languages offer a priority queue data structure as part of their standard library, this operation is generally not supported (e.g., in C++, Java or Python). However, going from the pseudocode to an actual implementation is made difficult by the fact that it relies on a priority queue with a "decrease key" operation. Actually Implementing Dijkstra's Algorithm IntroductionÄijkstra's algorithm for the shortest-path problem is one of the most important graph algorithms, so it is often covered in algorithm classes.
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