Homework Answers
Here are the procedure that
the Kruskal algorithm find all the MST.
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4. Suppose that the edge weights in G are not distinct and therefore, there can be more than one ...
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.
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...
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...
Let G = (V. E) be an undirected, connected graph with weight function w : E → R
Problem 4 Let G = (V. E) be an undirected, connected graph with weight function w : E → R. Furthermore, suppose that E 2 |V and that all edge weights are distinct. Prove that the MST of G is unique (that is, that there is only one minimum spanning tree of G).
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.
#4. TSP a) Solve with 2 MST approx. algorithm. Note: you can assume weights of edges:...
#4. TSP a) Solve with 2 MST approx. algorithm. Note: you can assume weights of edges: (CE) = 36 and w(C,A)=33 А B 24 1) Find MST 2) Double MST 3) Find Eulerian cycle 4) Do shortcuts (show steps here) 10 11 С. 30 25 8 E 28 Report the resulting Hamiltonian cycle and its length:
Updating an MST when an edge weight changes. You have a graph G= (V, E) with...
Updating an MST when an edge weight changes. You have a graph G= (V, E) with edge weights given in the graph (whatever they are). In addition, a minimum spanning tree T= (V, E′) of this graph has also been given to you. Now, say we need to increase the weight of one particular edge e. Does the MST change? If so, show how to compute the new MST in linear time. You should consider two cases: 1). when e∈E′and...
Suppose you are given a connected graph G, with edge costs that you may assume are...
Suppose you are given a connected graph G, with edge costs that you may assume are all distinct. G has n vertices and m edges. A particular edge e of G is specified. Give an algorithm with running time O(m + n) to decide whether e is contained in a minimum spanning tree of G.
3. Consider the the following graphs for each of the two subproblems. Each subproblem can be answ...
3. Consider the the following graphs for each of the two subproblems. Each subproblem can be answered (or blank) independently of the other ( subject to the 4 total blank for partial credit rule). s MST algorithm on the graph below and left, starting with vertex all work done so far: al (40 points) You are runing Prim' a. You are about to take vertex g out of the min-ehave not done so yet. Show the order that vertices wer...
a) Solve with 2 MST approx. algorithm. Note: you can assume weights of edges: w(C,E) =...
a) Solve with 2 MST approx. algorithm. Note: you can assume weights of edges: w(C,E) = 36 and w(C,A)=33 A B 24 1) Find MST 2) Double MST 3) Find Eulerian cycle 4) Do shortcuts (show steps here) 9 lol 11 30 25 8 E 28 Report the resulting Hamiltonian cycle and its length:
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.
#4. TSP a) Solve with 2 MST approx. algorithm. Note: you can assume weights of edges: (CE) = 36 and w(C,A)=33 А B 24 1) Find MST 2) Double MST 3) Find Eulerian cycle 4) Do shortcuts (show steps here) 10 11 С. 30 25 8 E 28 Report the resulting Hamiltonian cycle and its length:
3. Consider the the following graphs for each of the two subproblems. Each subproblem can be answered (or blank) independently of the other ( subject to the 4 total blank for partial credit rule). s MST algorithm on the graph below and left, starting with vertex all work done so far: al (40 points) You are runing Prim' a. You are about to take vertex g out of the min-ehave not done so yet. Show the order that vertices wer...
a) Solve with 2 MST approx. algorithm. Note: you can assume weights of edges: w(C,E) = 36 and w(C,A)=33 A B 24 1) Find MST 2) Double MST 3) Find Eulerian cycle 4) Do shortcuts (show steps here) 9 lol 11 30 25 8 E 28 Report the resulting Hamiltonian cycle and its length:
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