math,tree,minimum-spanning-tree,proof

The sets must have the same weight. Here's a simple proof: suppose they don't. Let's let T1 and T2 be MSTs for some graph G with different multisets of edge weights. Sort those edges into ascending order of weight. Since the two multisets of weights aren't the same, look at...

algorithm,graph-theory,minimum-spanning-tree

It is possible to solve this problem efficiently(in O(n log n) time), but it is not that easy. Just using the Prim's algorithm with a heap does not help(it actually makes it even slower), because its time complexity is O(E log V), which is O(n^2 * log n) in this...

algorithm,complexity-theory,minimum-spanning-tree

log* is the number of times you needed to take log in order to nullify the value. It is NOT infinite time log. Taking infinite time logs will just result in negative number at the end, and then it will be undefined, which does not make sense here. For example:...

python,graph,networkx,minimum-spanning-tree,subgraph

You can get the edges in G that appear in the MST T with a simple comprehension: E = set(T.edges()) # optimization [e for e in G.edges() if e in E or reversed(e) in E] You can then build a new graph from this....

c++,algorithm,graph,minimum-spanning-tree

This problem is easier than the problem of sensitivity analysis of minimum spanning trees, which is to determine how much each tree/nontree edge can increase/decrease in weight before the minimum spanning tree changes. The best known algorithm for MST sensitivity analysis appears to be due to Seth Pettie (2005, arXived...

java,algorithm,minimum-spanning-tree,prims-algorithm

Prims algorithm uses the weight of the edge to determine the best path. Posting the vertex to the PriorityQueue will not work. I am guessing your Edge class will implement Comparable by comparing their lengths....

algorithm,graph,minimum-spanning-tree

Consider a four-node graph, connected in a square, with the left edge having cost 10 and all other edges having cost 1. If you divide the graph into left and right for your recursive step, you will end up with a spanning tree of cost 12, instead of cost 3....

algorithm,greedy,minimum-spanning-tree,kruskals-algorithm

What we do in Kruskal ? Firstly sort the edges according to their weight. Then we choose that edge which has minimal weight. We add that edge if it makes no cycle. Thus we go forward greedily. So it is greedy approach. :) The greedy approach is called greedy because,...

algorithm,graph,graph-theory,minimum-spanning-tree

Note that |E| >= |V|. Choose any vertex, mark it as component1, iterate each connected vertex (along MST edges) and mark component1 as well. That's O(|V|). Find a vertex from the other component by scanning until not marked. That's O(|V|) again. Iterate each vertex in 2nd component (along MST edges),...

java,minimum-spanning-tree,prims-algorithm

The PriorityVertex from this example is just a class that the asker created. It is a simple implementation of a vertex in a graph. This class implements the Comparable interface. The reason for that is that PriorityQueue must compare its elements. This can be achieved in two ways: The elements...

algorithm,graph,minimum-spanning-tree

Let the new edge added be between node i and j.There will be exactly one cycle containing all nodes between node i and j, including them. Also as before it was a tree only one path is there from node i to j. So you can use DFS/BFS to traverse...

java,algorithm,heap,minimum-spanning-tree,prims-algorithm

Thanks friends for investing time to my question, but I figured that I had few mistakes in my implementation due to which I was getting wrong answer. I corrected the state of the vertex when it is added to the MinHeap I corrected the logic of outputting the edge of...

algorithm,graph,graph-theory,discrete-mathematics,minimum-spanning-tree

Yes, the algorithm always finds a spanning tree (if such exist, i.e. the graph has a source). Not sure if it finds a minimal spanning tree though. You have already proven there are no cycles in the resulting graph, so we only need to show connectivity. Proof that the resulting...

parallel-processing,minimum-spanning-tree

I think the parallelization you can use decreases as the algorithm runs. You may try to parallelize the search for the lightest node of each component and leave to a single machine the union part of the algorithm. The single machine will distribute the components to each sub-machine...

algorithm,graph,tree,minimum-spanning-tree

For any connected graph, the spanning tree always contains n-1 edges where n is the number of nodes in the graph. So you will have to remove all the remaining edges. (If I have understood your question correctly) Even for disconnected graphs, the number of edges in a spanning tree...

algorithm,graph,graph-algorithm,minimum-spanning-tree,degrees

To summarize the suggested algorithm [with tightened requirements on epsilon (which you called x)]: Pick a tiny epsilon (such that epsilon * deg(u) is less than d, the smallest non-zero weight difference between any pair of subgraphs). In the case all the original weights are natural numbers, epsilon = 1/(deg(u)+1)...

python,numpy,scipy,minimum-spanning-tree

I had the same problem and managed to figure out a solution using only scipy. All this does is take the MST, locate the maximum weighted edge, delete it (i.e. zero it out), and then use the connected_components method to figure out which nodes remain connected. Here is the complete...

algorithm,graph,minimum-spanning-tree

Answering my own question Here's an efficient answer I've come up with, following also the feedback from @sasha . Suppose we would like to calculate the total weight of the complete graph G, i.e; Let T = (V, E) be a tree of |V| vertices and |E| = |V|-1 edges,...

algorithm,graph,graph-algorithm,minimum-spanning-tree

I would suggest you to do something like the second answer in the given question: This is prim's algorithm : Start by picking any vertex to be the root of the tree. While the tree does not contain all vertices in the graph find shortest edge leaving the tree and...

algorithm,tree,minimum-spanning-tree

You want to check that for each edge (u, v) not in the MST, the path from u to v in the MST has no edge with weight larger than that of (u, v). For a single vertex in the tree, you can use a single BFS or DFS to...

java,tree,minimum-spanning-tree,spanning-tree

In cauda venenum /* Helper Methods */ // search a list of Nodes for a value public static boolean listContainsNode(List<Node<Integer>> list, Integer data){ for(Node<Integer> n : list) if(n.getData() == data) // <-- Can you spot the bug? return true; return false; } The problem is that you compare Integer's with...