The time complexity of Prim's algorithm depends on the data structures used. Using an adjacency matrix with searching gives O(V2), while using a binary heap with an adjacency list gives O(VlogV+ElogV), where V is the number of vertices and E is the number of edges.
Conditions: The graph is represented using specific data structures.; V denotes the number of vertices.; E denotes the number of edges.
The time complexity of Prim's algorithm depends on the data structures used. Using an adjacency matrix with searching gives O(V2), while using a binary heap with an adjacency list gives O(VlogV+ElogV), where V is the number of vertices and E is the number of edges.
Conditions: The graph is represented using specific data structures.; V denotes the number of vertices.; E denotes the number of edges.
Edge B-E is skipped because both of its endpoints (B and E) are already in the visited set (MST). Prim's algorithm only considers edges that connect a visited vertex to an unvisited vertex to avoid creating cycles.
Conditions: Both endpoints of the edge are already marked as visited.; The algorithm strictly follows the rule of adding only crossing edges.
Edge B-E is skipped because both of its endpoints (B and E) are already in the visited set (MST). Prim's algorithm only considers edges that connect a visited vertex to an unvisited vertex to avoid creating cycles.
Conditions: Both endpoints of the edge are already marked as visited.; The algorithm strictly follows the rule of adding only crossing edges.
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.
To compute the total weight of the MST, sum the weights of all edges that were selected and added to the tree during the execution of Prim's algorithm.
Conditions: The MST edge set has been fully determined.; You have access to the weights of the selected edges.
To compute the total weight of the MST, sum the weights of all edges that were selected and added to the tree during the execution of Prim's algorithm.
Conditions: The MST edge set has been fully determined.; You have access to the weights of the selected edges.
A minimum spanning tree (MST) is a connected, acyclic subgraph that spans all vertices of a weighted graph and has the minimum possible total edge weight.
Conditions: The underlying graph must be weighted.; The graph must be connected for a spanning tree to exist.; The MST minimizes the sum of selected edge weights.; The MST contains no cycles.
A minimum spanning tree (MST) is a connected, acyclic subgraph that spans all vertices of a weighted graph and has the minimum possible total edge weight.
Conditions: The underlying graph must be weighted.; The graph must be connected for a spanning tree to exist.; The MST minimizes the sum of selected edge weights.; The MST contains no cycles.