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The time complexity of Dijkstra's algorithm is $O(|E| + |V|\log|V|)$ when implemented using a Fibonacci heap for the priority queue. Here, $|V|$ is the number of vertices and $|E|$ is the number of edges.

Conditions: The implementation uses a Fibonacci heap data structure for the priority queue.; $|V|$ represents the number of vertices.; $|E|$ represents the number of edges.

The time complexity of Dijkstra's algorithm is $O(|E| + |V|\log|V|)$ when implemented using a Fibonacci heap for the priority queue. Here, $|V|$ is the number of vertices and $|E|$ is the number of edges.

Conditions: The implementation uses a Fibonacci heap data structure for the priority queue.; $|V|$ represents the number of vertices.; $|E|$ represents the number of edges.

The pseudocode initializes distances to infinity and predecessors to nil, sets the source distance to 0, and uses a priority queue H. It repeatedly extracts the vertex u with the minimum distance, relaxes all outgoing edges (u,v), updating dist(v) and prev(v) if a shorter path is found, and decreases the key of v in H.

Conditions: Input: Graph $G = (V, E)$, directed or undirected.; Positive edge lengths {$l_e$ : $e \in E$}.; Source vertex $s \in V$.; Output: dist(u) is the shortest distance from s to u for all reachable u.

The pseudocode initializes distances to infinity and predecessors to nil, sets the source distance to 0, and uses a priority queue H. It repeatedly extracts the vertex u with the minimum distance, relaxes all outgoing edges (u,v), updating dist(v) and prev(v) if a shorter path is found, and decreases the key of v in H.

Conditions: Input: Graph $G = (V, E)$, directed or undirected.; Positive edge lengths {$l_e$ : $e \in E$}.; Source vertex $s \in V$.; Output: dist(u) is the shortest distance from s to u for all reachable u.

Dijkstra's algorithm is a method for finding the shortest path from a single starting node to all other nodes in a weighted directed graph. It is distinct from minimum spanning tree algorithms like Prim's and Kruskal's, which connect all nodes with the minimum total edge weight rather than finding paths from a specific source.

Conditions: The graph must be weighted.; The graph can be directed or undirected, though the video primarily demonstrates it on a directed graph.

Dijkstra's algorithm is a method for finding the shortest path from a single starting node to all other nodes in a weighted directed graph. It is distinct from minimum spanning tree algorithms like Prim's and Kruskal's, which connect all nodes with the minimum total edge weight rather than finding paths from a specific source.

Conditions: The graph must be weighted.; The graph can be directed or undirected, though the video primarily demonstrates it on a directed graph.