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What is krushkal algorithm?
Kruskal's algorithm is an algorithm in graph theory that finds a minimum spanning tree for a connected weighted graph. This means it finds a subset of the edges that forms a tree that includes every vertex, where the total weight of all the edges in the tree is minimized. If the graph is not connected, then it finds a minimum spanning forest (a minimum spanning tree for each connected component). Kruskal's algorithm is an example of a greedy algorithm.
What is the complexity of kruskal's minimum spanning tree algorithm on a graph with n nodes and e edges?
o(eloge)
Why prims algorithm is better than kruskals algorithm?
"What are difference between Prim's algorithm and Kruskal's algorithm for finding the minimum spanning tree of a graph?" Prim's method starts with one vertex of a graph as your tree, and adds the smallest edge that grows your tree by one more vertex. Kruskal starts with all of the vertices of a graph as a forest, and adds the smallest edge that joins two trees in the forest. Prim's method is better when * You can only concentrate on one tree at a time * You can concentrate on only a few edges at a time Kruskal's method is better when * You can look at all of the edges at once * You can hold all of the vertices at once * You can hold a forest, not just one tree Basically, Kruskal's method is more time-saving (you can order the edges by weight and burn through them fast), while Prim's method is more space-saving (you only hold one tree, and only look at edges that connect to vertices in your tree).
When prim algorithm fail?
When there are directed edges in the graph, as it is impossible to move back from B to A when the edges are directed.
Will either kruskal or prim's algorithm work on negative edge graph?
The correctness of either Prim's or Kruskal's algorithm, is not affected by negative edges in the graph. They both work fine with negative edges. The question boils down to "Does a Priority Queue of numbers work with negative numbers?" because of the fact that both Prim's and Kruskal's algorithm use a priority queue. Of course -- as negative numbers are simply numbers smaller than 0. The "<" sign will still work with negative numbers.
What is krushkal algorithm?
Kruskal's algorithm is an algorithm in graph theory that finds a minimum spanning tree for a connected weighted graph. This means it finds a subset of the edges that forms a tree that includes every vertex, where the total weight of all the edges in the tree is minimized. If the graph is not connected, then it finds a minimum spanning forest (a minimum spanning tree for each connected component). Kruskal's algorithm is an example of a greedy algorithm.
What is the pseudocode for implementing the Kruskal algorithm to find the minimum spanning tree of a graph?
The pseudocode for implementing the Kruskal algorithm to find the minimum spanning tree of a graph involves sorting the edges by weight, then iterating through the sorted edges and adding them to the tree if they do not create a cycle. This process continues until all vertices are connected.
How can you find minimum spanning trees?
Minimum spanning trees can be found using algorithms like Prim's algorithm or Kruskal's algorithm. These algorithms work by starting with an empty spanning tree and iteratively adding edges with the smallest weights until all vertices are connected. The resulting tree will have the minimum total weight possible.
Can you provide the pseudocode for Kruskal's algorithm?
Here is the pseudocode for Kruskal's algorithm: Sort all the edges in non-decreasing order of their weights. Initialize an empty minimum spanning tree. Iterate through all the edges in sorted order: a. If adding the current edge does not create a cycle in the minimum spanning tree, add it to the tree. Repeat step 3 until all vertices are included in the minimum spanning tree. This algorithm helps find the minimum spanning tree of a connected, undirected graph.
What is the complexity of kruskal's minimum spanning tree algorithm on a graph with n nodes and e edges?
o(eloge)
Prove that a graph G is connected and only if it has a spanning tree?
Proving this is simple. First, you prove that G has a spanning tree, it is connected, which is pretty obvious - a spanning tree itself is already a connected graph on the vertex set V(G), thus G which contains it as a spanning sub graph is obviously also connected. Second, you prove that if G is connected, it has a spanning tree. If G is a tree itself, then it must "contain" a spanning tree. If G is connected and not a tree, then it must have at least one cycle. I don't know if you know this or not, but there is a theorem stating that an edge is a cut-edge if and only if it is on no cycle (a cut-edge is an edge such that if you take it out, the graph becomes disconnected). Thus, you can just keep taking out edges from cycles in G until all that is left are cut-gees. Since you did not take out any cut-edges, the graph is still connected; since all that is left are cut-edges, there are no cycles. A connected graph with no cycles is a tree. Thus, G contains a spanning tree. Therefore, a graph G is connected if and only if it has a spanning tree!
Prove that a graph G is connected if and only if it has a spanning tree?
Proving this is simple. First, you prove that G has a spanning tree, it is connected, which is pretty obvious - a spanning tree itself is already a connected graph on the vertex set V(G), thus G which contains it as a spanning sub graph is obviously also connected. Second, you prove that if G is connected, it has a spanning tree. If G is a tree itself, then it must "contain" a spanning tree. If G is connected and not a tree, then it must have at least one cycle. I don't know if you know this or not, but there is a theorem stating that an edge is a cut-edge if and only if it is on no cycle (a cut-edge is an edge such that if you take it out, the graph becomes disconnected). Thus, you can just keep taking out edges from cycles in G until all that is left are cut-gees. Since you did not take out any cut-edges, the graph is still connected; since all that is left are cut-edges, there are no cycles. A connected graph with no cycles is a tree. Thus, G contains a spanning tree. Therefore, a graph G is connected if and only if it has a spanning tree!
What is the runtime complexity of Kruskal's algorithm for finding the minimum spanning tree of a graph?
The runtime complexity of Kruskal's algorithm is O(E log V), where E is the number of edges and V is the number of vertices in the graph.
What is the runtime complexity of the Prim's algorithm for finding the minimum spanning tree of a graph?
The runtime complexity of Prim's algorithm is O(V2) or O(E log V), where V is the number of vertices and E is the number of edges in the graph.
Why prims algorithm is better than kruskals algorithm?
"What are difference between Prim's algorithm and Kruskal's algorithm for finding the minimum spanning tree of a graph?" Prim's method starts with one vertex of a graph as your tree, and adds the smallest edge that grows your tree by one more vertex. Kruskal starts with all of the vertices of a graph as a forest, and adds the smallest edge that joins two trees in the forest. Prim's method is better when * You can only concentrate on one tree at a time * You can concentrate on only a few edges at a time Kruskal's method is better when * You can look at all of the edges at once * You can hold all of the vertices at once * You can hold a forest, not just one tree Basically, Kruskal's method is more time-saving (you can order the edges by weight and burn through them fast), while Prim's method is more space-saving (you only hold one tree, and only look at edges that connect to vertices in your tree).
What are the differences between Dijkstra's algorithm and Breadth-First Search (BFS) when it comes to finding the shortest path in a graph?
Dijkstra's algorithm and Breadth-First Search (BFS) are both used to find the shortest path in a graph, but they have key differences. Dijkstra's algorithm considers the weight of edges, making it suitable for graphs with weighted edges, while BFS treats all edges as having the same weight. Additionally, Dijkstra's algorithm guarantees the shortest path, but BFS may not always find the shortest path in weighted graphs.
What is the time complexity of the Kosaraju algorithm for finding strongly connected components in a directed graph?
The time complexity of the Kosaraju algorithm for finding strongly connected components in a directed graph is O(V E), where V is the number of vertices and E is the number of edges in the graph.
