Bellman-Ford algorithm is a single-source shortest path algorithm, so when you have negative edge weight then it can detect negative cycles in a graph. The only difference between the two is that Bellman-Ford is also capable of handling negative weights whereas Dijkstra Algorithm can only handle positives. From wiki However, Dijkstra’s algorithm greedily selects the … Read more
Everyone else comparing this to the Travelling Salesman Problem probably hasn’t read your question carefully. In TSP, the objective is to find the shortest cycle that visits all the vertices (a Hamiltonian cycle) — it corresponds to having every node labelled ‘mustpass’. In your case, given that you have only about a dozen labelled ‘mustpass’, … Read more
The reason for using decrease-key rather than reinserting nodes is to keep the number of nodes in the priority queue small, thus keeping the total number of priority queue dequeues small and the cost of each priority queue balance low. In an implementation of Dijkstra’s algorithm that reinserts nodes into the priority queue with their … Read more
Dijkstra’s shortest path algorithm is O(ElogV) where: V is the number of vertices E is the total number of edges Your analysis is correct, but your symbols have different meanings! You say the algorithm is O(VElogV) where: V is the number of vertices E is the maximum number of edges attached to a single node. … Read more
Prim’s algorithm constructs a minimum spanning tree for the graph, which is a tree that connects all nodes in the graph and has the least total cost among all trees that connect all the nodes. However, the length of a path between any two nodes in the MST might not be the shortest path between … Read more
The algorithm you have suggested will indeed find the shortest path in this graph, but not all graphs in general. For example, consider this graph: Let’s trace through the execution of your algorithm. First, you set d(A) to 0 and the other distances to ∞. You then expand out node A, setting d(B) to 1, … Read more
Dijkstra is a special case for A* (when the heuristics is zero).
Dijkstra allows assigning distances other than 1 for each step. For example, in routing the distances (or weights) could be assigned by speed, cost, preference, etc. The algorithm then gives you the shortest path from your source to every node in the traversed graph. Meanwhile BFS basically just expands the search by one “step” (link, … Read more
Recall that in Dijkstra’s algorithm, once a vertex is marked as “closed” (and out of the open set) – the algorithm found the shortest path to it, and will never have to develop this node again – it assumes the path developed to this path is the shortest. But with negative weights – it might … Read more