Prove that in any graph with edge-weights that are distinct (no pair of edges exists with...
Prove that in any graph with edge-weights that are distinct (no pair of edges exists with the same weight), then both Prim and Kruskal must output the same MST, regardless of the starting node.
Homework Answers
Suppose there are two minimum trees, A and B. Let e be the edge in just one of A,B with the smallest cost. Suppose it is in A but not B. Suppose e is the edge PQ. Then B must contain a path from P to Q which is not simply the edge e. So if we add e to B, then we get a cycle. If all the other edges in the cycle were in A, then A would contain a cycle, which it cannot. So the cycle must contain an edge f not in A. Hence, by the definition of e (and the fact that all edge-costs are different) the cost of f must be greater than the cost of e. So if we replace f by e we get a spanning tree with smaller total cost. Contradiction.
Hence, it is proved that in any graph with edge-weights that are distinct (no pair of edges exists with the same weight), then both Prim and Kruskal must output the same MST, regardless of the starting node.
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Prove that in any graph with edge-weights that are distinct (no
pair of edges exists with...
Consider a graph with distinct edge weights such that Prim and Kruskal select the edges of...
Consider a graph with distinct edge weights such that Prim and Kruskal select the edges of the spanning tree T in opposite order. How must T look like?
You are given an undirected graph G = (V, E) with positive weights on the edges....
You are given an undirected graph G = (V, E) with positive weights on the edges. If the edge weights are distinct, then there is only one MST, so both Prim’s and Kruskal’s algorithms will find the same MST. If some of the edge weights are the same, then there can be several MSTs and the two algorithms could find different MSTs. Describe a method that forces Prim’s algorithm to find the same MST of G that Kruskal’s algorithm finds.
This problem is dealing with Discrete Math. Please answer fully and clearly, and show/explain all...
This problem is dealing with Discrete Math. Please answer fully
and clearly, and show/explain all steps or proofs that you state in
the answer.
4. Let (G, w) be a connected graph with weights on edges so that all weights are distinct positive real numbers. Suppose we find a MST (minimum spanning trees ) in G by using Prim's algorithm. Prove that no matter what vertex we begin with in Prim algorithm, the set of all weights on edges in...
Help. I need to write a small program that executes the following graph algorithms in any language: 1. All-Pairs Shortest Path (Floyd-Warshall). It must ask for the vertices and edges for the user to...
Help. I need to write a small program that executes the following graph algorithms in any language: 1. All-Pairs Shortest Path (Floyd-Warshall). It must ask for the vertices and edges for the user to enter them. As an output, deploy the resulting matrix. This will be done only for directed graphs. 2. Kruskal or Prim algorithm whatever you want to do. It must ask for a graph and present it at the end. The minimum coating tree that results from...
Suppose we have a graph G = (V, E) with weights on the edges of E,...
Suppose we have a graph G = (V, E) with weights on the edges of E, and we are interested in computing a Minimum Spanning Tree (MST) of G. Suppose we modify the DFS algorithm so that when at a vertex v, we next visit the unvisited neighbor u such that the weight of (u, v) is minimized. Does this produce a MST of G? prove that it does or provide a counter example.
Problem A: Consider the following graph.
Problem A: Consider the following graph. (a). Find a minimum spanning tree of the graph using Kruskal's algorithm. List the edges in the order they are put into the tree. (b). Apply Prim's algorithm to the same graph starting with node A. List the edges, in order added to the MST. (c). Suppose the cost of every edge touching node A is increased by a constant. Are we guaranteed that the MST remains the MST? Explain.
Algorithm Question： The following questions are on minimum spanning tree. (a) Suppose we have an undirected...
Algorithm Question： The following questions are on minimum spanning tree. (a) Suppose we have an undirected graph with weights that can be either positive or negative. Do Prim’s and Kruskal’s algorithim produce a MST for such a graph? Explain. (b) Prove that for any weighted undirected graph such that the weights are distinct (no two edges have the same weight), the minimal spanning tree is unique.
Prim's and Kruskal's algorithms both create MST. It is possible that both algorithms will create dierent...
Prim's and Kruskal's algorithms both create MST. It is possible that both algorithms will create dierent trees. Show using a contradiction why the graphs created by both algorithms MUST have the same total weight. (Note: speak in general. For example: Assume that you have a graph G that has two different MST. Prim(G) = A Kruskal(G) = B where weight(A) != weight(B) Example why this can't be possible.)
MST For an undirected graph G = (V, E) with weights w(e) > 0 for each edge e ∈ E, you are given a MST T. Unfortunately one of the edges e* = (u, z) which is in the MST T is deleted from the graph G...
MST For an undirected graph G = (V, E) with weights w(e) > 0 for each edge e ∈ E, you are given a MST T. Unfortunately one of the edges e* = (u, z) which is in the MST T is deleted from the graph G (no other edges change). Give an algorithm to build a MST for the new graph. Your algorithm should start from T. Note: G is connected, and G − e* is also connected. Explain...
need help filling in the code def prim(G): Use Prim's algorithm to find a MST for the graph G … # Initialize tree T with a single vertex and no edges v = next(iter( )) # while the vertex set of T...
need help filling in the code
def prim(G): Use Prim's algorithm to find a MST for the graph G … # Initialize tree T with a single vertex and no edges v = next(iter( )) # while the vertex set of T is smaller than the v+tex set of G, # (i.e. while the vertex set of T is a proper subset of the vertex set of G), find the edge e with minimum weight so that # Tte is...
This problem is dealing with Discrete Math. Please answer fully
and clearly, and show/explain all steps or proofs that you state in
the answer.
4. Let (G, w) be a connected graph with weights on edges so that all weights are distinct positive real numbers. Suppose we find a MST (minimum spanning trees ) in G by using Prim's algorithm. Prove that no matter what vertex we begin with in Prim algorithm, the set of all weights on edges in...
Suppose we have a graph G = (V, E) with weights on the edges of E, and we are interested in computing a Minimum Spanning Tree (MST) of G. Suppose we modify the DFS algorithm so that when at a vertex v, we next visit the unvisited neighbor u such that the weight of (u, v) is minimized. Does this produce a MST of G? prove that it does or provide a counter example.
need help filling in the code
def prim(G): Use Prim's algorithm to find a MST for the graph G … # Initialize tree T with a single vertex and no edges v = next(iter( )) # while the vertex set of T is smaller than the v+tex set of G, # (i.e. while the vertex set of T is a proper subset of the vertex set of G), find the edge e with minimum weight so that # Tte is...
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