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Jul 10, 2016 · Let ST mean spanning tree and MST mean minimum spanning tree. Let me define some less common terms first. An edge is unique-cycle-heaviest if it is the unique heaviest edge in some cycle. An edge is non-cycle-heaviest if it is never a heaviest edge in any cycle. An edge is unique-cut-lightest if it is the unique lightest edge to cross some cut.
If I do that however, the resulting Spanning Tree is not Minimum because there is the edge weight of 1 that can be used to lower the cost of connecting them instead of using edge weight of 2. When the edge with the least weight is used to connect S and S', the end result is a Minimum Spanning Tree because we kept the Minimum property in place.
May 2, 2020 · Minimum Spanning tree: It is a spanning tree where the summation of edge weights is minimum. Now, does this mean, while retrieving an MST, if we come across a path in G which has more edges (as compared to some other path) but least weight on the summation of edge weights(as compared to all other paths possible), we won't be considering it as an MST?
Mar 30, 2012 · As far as I can tell, removal requires O(n^2), because for each edge (assume sorted already in a list), we need to find the smallest edge which connects the two spanning trees. This can take up to O(n^2) if the only edge that connects them is also the edge with the highest value.
May 16, 2019 · More generally, a minimum weight spanning tree must contain every edge that is a bridge (a bridge is an edge whose removal disconnects the graph). Consider a graph which contains two bridges, both having the same weight, and with the edges in the rest of the graph having distinct weights.
May 23, 2016 · There is only one minimum spanning tree in the graph where the weights of vertices are different. I think the best way of finding the number of minimum spanning tree must be something using this property. EDIT: I found a solution to this problem, but I am not sure, why it works. Can anyone please explain it.
Dec 5, 2017 · With Prim's algorithm we find minimum cost spanning tree. The goal is to find minimum cost to cover all nodes. with Dijkstra we find Single Source Shortest Path. The goal is find the shortest path from the source to every other node. Prim’s algorithm works exactly as Dijkstra’s, except. It does not keep track of the distance from the source.
Aug 20, 2017 · Briefly, the answer is no, we cannot construct minimum spanning tree for an un-directed graph with distinct weights using BFS or DFS algorithm. This post provides a counterexample. Computing MST using DFS/BFS would mean it is solved in linear time, but (as Yuval Filmus commented) it is unknown if such algorithm exists.
Minimum spanning tree is a tree in a graph that spans all the vertices and total weight of a tree is minimal. Shortest path is quite obvious, it is a shortest path from one vertex to another. What I don't understand is since minimum spanning tree has a minimal total weight, wouldn't the paths in the tree be the shortest paths?
Jan 17, 2017 · Thank you. Just wanna add that the decisional version of the problem is NP-complete though: given an integer k>=2 representing the maximum number of leaves allowed in a spanning tree, a graph G and a candidate solution tree T, it is trivial to check in polynomial time if T is a spanning tree for G with a number of leaves lesser or equal than k.
