Consider the following weighted graph G:
Consider the following weighted graph G:
Use Prim's algorithm to find a minimal spanning tree T of this graph starting at the vertex s. You do not need to show every step of the algorithm, but to receive full credit you should:
list the edges of T in the order in which they're added;
redraw G and indicate which edges belong to T;
compute the cost of T.
Homework Answers
Consider the weighted graph below: Demonstrate Prim's algorithm starting from vertex A. Write the edges in...
Consider the weighted graph below: Demonstrate Prim's algorithm starting from vertex A. Write the edges in the order they were added to the minimum spanning tree. Demonstrate Dijkstra's algorithm on the graph, using vertex A as the source. Write the vertices in the order which they are marked and compute all distances at each step.
Consider the following weighted undirected graph.
Consider the following weighted undirected graph. (a) Explain why edge (B, D) is safe. In other words, give a cut where the edge is the cheapest edge crossing the cut. (b) We would like to run the Kruskal's algorithm on this graph. List the edges appearing in the Minimum Spanning Tree (MST) in the order they are added to the MST. For simplicity, you can refer to each edge as its weight. (c) 1We would like to run the Prim's algorithm on this...
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.
Given the following weighted graph G. use Prim's algorithm to determine the Minimum-Cost Spanning Tree (MCST)...
Given the following weighted graph G. use Prim's algorithm to determine the Minimum-Cost Spanning Tree (MCST) with node 1 as the "root". List the vertices in the order in which the algorithm adds them to the solution, along with the edge and its weight used to make the selection, one per line. Each line should look like this: add vertex y: edge = (x,y), weight = 5 When the algorithm ends there are, generally, edges left in the heap. List...
Question 6 Let G be the weighted graph (a) Use Dijkstra's algorithm to find the shortest path from A to F. You can...
Question 6 Let G be the weighted graph (a) Use Dijkstra's algorithm to find the shortest path from A to F. You can do all the work on a single diagram, but, to show that you have used the algorithm correctly, if an annotation needs updating do not erase itjust put a line through it and write the new annotation above that b) In what order are the vertices added to the tree? (c) Notice that the algorithm does not,...
3. In this problem, you will show the execution of the minimum spanning tree algorithms that...
3. In this problem, you will show the execution of the minimum spanning tree algorithms that you studied in class on the following graph: START 10 40 5 20 35 15 6 30 62 12 (a) (5 points) Trace the execution of Prim's algorithm to find the minimum spanning tree for this graph. At each step, you should show the vertex and the edge added to the tree and the resulting values of D after the relaxation operation. Use START...
Let G be the weighted graph
Let G be the weighted graph (a) Find a minimum spanning tree for G using Prim's algorithm, showing all interme- diate steps. What is the cost of this tree? (b) Find a minimum spanning tree for G using Kruskal's algorithm, showing all inter- mediate steps. What is the cost of this tree?
Consider the graph below. Use Prim's algorithm to find a minimal spanning tree of the graph rooted in vertex A.
Consider the graph below. Use Prim's algorithm to find a minimal spanning tree of the graph rooted in vertex A. Note: enter your answer as a set of edges [E1, E2, ...) and write each edge as a pair of nodes between parentheses separate by a comma and one blank space e.g. (A,B)
Run Prim (starting from vertex "f") and Kruskal algorithms on the graph below: 3 2 9...
Run Prim (starting from vertex "f") and Kruskal algorithms on the graph below: 3 2 9 3 . (5 points) Prim's algorithm: draw a table that shows the vertices in the queue at each iteration, similar to example from the notes (2 points) Prim's algorithm: using the table from the first part, list the order in which edges are added to the tree (3 points) Kruskal's algorithm: list the order in which edges are added to the tree
Question II - Graph Traversal and Minimum Spanning Trees [40 Points] Consider the following graph: B...
Question II - Graph Traversal and Minimum Spanning Trees [40 Points] Consider the following graph: B 10 1 4 1 H 9 4 a) Traverse the graph starting from vertex A, and using the Breadth-First Search algorithm. Show the traversal result and the data structure you are using. [10 Points] b) Traverse the graph starting from vertex A, and using the Depth-First Search (Post-order) algorithm. Show the traversal result and the data structure you are using. [10 Points] c) Apply...
Consider the weighted graph below: Demonstrate Prim's algorithm starting from vertex A. Write the edges in the order they were added to the minimum spanning tree. Demonstrate Dijkstra's algorithm on the graph, using vertex A as the source. Write the vertices in the order which they are marked and compute all distances at each step.
Given the following weighted graph G. use Prim's algorithm to determine the Minimum-Cost Spanning Tree (MCST) with node 1 as the "root". List the vertices in the order in which the algorithm adds them to the solution, along with the edge and its weight used to make the selection, one per line. Each line should look like this: add vertex y: edge = (x,y), weight = 5 When the algorithm ends there are, generally, edges left in the heap. List...
Question 6 Let G be the weighted graph (a) Use Dijkstra's algorithm to find the shortest path from A to F. You can do all the work on a single diagram, but, to show that you have used the algorithm correctly, if an annotation needs updating do not erase itjust put a line through it and write the new annotation above that b) In what order are the vertices added to the tree? (c) Notice that the algorithm does not,...
3. In this problem, you will show the execution of the minimum spanning tree algorithms that you studied in class on the following graph: START 10 40 5 20 35 15 6 30 62 12 (a) (5 points) Trace the execution of Prim's algorithm to find the minimum spanning tree for this graph. At each step, you should show the vertex and the edge added to the tree and the resulting values of D after the relaxation operation. Use START...
Run Prim (starting from vertex "f") and Kruskal algorithms on the graph below: 3 2 9 3 . (5 points) Prim's algorithm: draw a table that shows the vertices in the queue at each iteration, similar to example from the notes (2 points) Prim's algorithm: using the table from the first part, list the order in which edges are added to the tree (3 points) Kruskal's algorithm: list the order in which edges are added to the tree
Question II - Graph Traversal and Minimum Spanning Trees [40 Points] Consider the following graph: B 10 1 4 1 H 9 4 a) Traverse the graph starting from vertex A, and using the Breadth-First Search algorithm. Show the traversal result and the data structure you are using. [10 Points] b) Traverse the graph starting from vertex A, and using the Depth-First Search (Post-order) algorithm. Show the traversal result and the data structure you are using. [10 Points] c) Apply...
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