### Educational Content on Dijkstra's Algorithm#### Diagram DescriptionThe diagram represents a graph with vertices labeled A, B, C, D, E, and F. The edges between these vertices are weighted, with specific values indicating the cost of traveling from one vertex to another:- A to B: 8- A to C: 2- B to C: 5- C to D: 5- C to E: 3- D to E: 4- E to F: 9- F to D: 5- D to F: 11Currently, vertices A and C are within the "shortest path cloud," marked with a dotted line.#### Questions and Instructions1. **Vertex Addition in Dijkstra's Algorithm:** Suppose Dijkstra's algorithm is being run starting from vertex A, and vertices A and C are already included in the shortest path cloud. The next step is to determine which vertex should be added next. The answer to this is the vertex with the smallest tentative distance from the current set of vertices in the shortest path cloud. - Select the vertex to be added next: [Dropdown menu with vertex options; suggested selection is D]2. **Computing the Value of d[F]:** After adding the selected vertex and performing edge relaxation, the task is to calculate the new value of d[F], which represents the shortest distance from the starting vertex A to vertex F. - Enter the value of d[F]: [Input box]3. **Runtime Complexity:** Understand the worst-case runtime of Dijkstra’s algorithm on a connected graph with N vertices and M edges. It is essential to know this to gauge the efficiency of the algorithm. - Select the worst-case runtime: [Dropdown menu; common answer: O((N + M) log N) with a priority queue]4. **Auxiliary Data Structure:** It's crucial to identify the auxiliary data structure commonly used in implementing Dijkstra’s algorithm to efficiently find the next vertex to add to the shortest path tree. - Choose the auxiliary data structure: [Dropdown menu; common answer: Priority Queue or Min-Heap]This educational activity is designed to enhance your understanding of Dijkstra's algorithm and its implementation details.

8. В 11 Suppose Dijkstra's algorithm is being run from vertex A and vertices A and C are in the shortest path cloud so far, with the graph and d[] values as shown. What vertex would be added next? D + After adding that vertex and relaxing across the edges coming out of it, what would the value of d[F] be? What is the worst-case runtime of Dijkstra's algorithm on a connected graph with N vertices and M edges? What auxiliary data structure is normally used to find the next vertex to add to the shortest path tree? 3. 2.

8. В 11 Suppose Dijkstra's algorithm is being run from vertex A and vertices A and C are in the shortest path cloud so far, with the graph and d[] values as shown. What vertex would be added next? D + After adding that vertex and relaxing across the edges coming out of it, what would the value of d[F] be? What is the worst-case runtime of Dijkstra's algorithm on a connected graph with N vertices and M edges? What auxiliary data structure is normally used to find the next vertex to add to the shortest path tree? 3. 2.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Attached is the pseudocode for Boruvka's algorithm. What is a counterexample that shows this algorithm does not work? Draw out the graph of this counterexample and explain why the algorithm would be unable to compute an MST with your example.
Given the following graph, A C 3 1 E 4 В F 7 D 5 What is the next edge and vertex(s) to be included in the solution if a. we are using Prim's algorithm to find a minimum spanning tree and we have started with vertex E? Edge: Vertex(s): What is the next edge and vertex(s) to be included in the solution if b. we are using Kruskal's algorithm to find a minimum spanning tree Edge: Vertex(s): What is the next edge and vertex(s) to be included in the solution if c. we are using Dijkstra's shortest path algorithm to find shortest paths and we have started with vertex E, with C and F already included in the shortest path tree? Edge: Vertex(s):
The Floyd-Warshall algorithm is a dynamic algorithm for searching the shortest path in a graph. Each vertex pair has its assigned weight. You are asked to draw the initial directed graph and show the tables for each vertex from Mo to Ms by finding all the shortest paths. Below is the algorithm as a guide. Algorithm 1: Pseudocode of Floyd-Warshall Algorithm Data: A directed weighted graph G(V, E) Result: Shortest path between each pair of vertices in G for each de V do | distance|d][d] «= 0; end for each edge (s, p) € E do | distance[s][p] + weight(s, p); end n = cardinality(V); for k = 1 to n do for i = 1 to n do for j = 1 to n do if distancefi][j] > distance/i][k] + distance/k][j] then | distance i]lj] + distancefi|[k] + distance/k|[j]; end end end end Consider the relation R = {(1,4) =4, (2,1)=3, (2,5)=-3, (3,4)=2, (4,2)=1, (4,3)=1, (5,4)=2 } on A = (1,2,3,4,5) solve the Floyd-Warshall Algorithm.
Implement the dijkstra's algorithm on a directed graph from a given vertex. all edges have non-negative edge weights. Output the edge as they are added to the shortest path trees. Compute and print the weight of the shortest path to every reachable vertex. The source vertex is S. show the following: -program code -Screenshot of the output -Representation of the graph transversal
There are many applications of Shortest Path Algorithm. Consider the problem of solving a jumbled Rubik's Cube in the fewest number of moves. I claim that this problem can be solved using a Shortest Path Algorithm. Determine whether this statement is TRUE or FALSE. NOTE: if you want to check if this statement is TRUE, think about how the Rubik's Cube Problem can be represented as a graph. What are the vertices? Which pairs of vertices are connected with edges? What is your source vertex and what is your destination vertex? How would Dijkstra's Algorithm enable you to find the optimal sequence of moves to solve a jumbled cube in the fewest number of moves?
Consider the following graph G.
a. For the graph below, give the order in which Dijkstra’s Algorithm would visit each vertex, starting from vertex A. 3 2 B F A G 4 ( E 4 D H 4 b. Change one of the weights in the graph so that the shortest paths tree returned by Dijkstra's is not correct. Hint: we showed in class that Dijkstra’s returns the correct SPT as long as all edges are non- negative. Set the weight of the edge connecting vertex . and vertex to c. Suppose we use the following heuristic: h(A) = 2 h(B) = 2 h(C) = 20 h(D) = 2 h(E) = 6 h(F) = 2 h(G) = 0 2 %3D h(H) %3D Recall that A* is just Dijkstra’s algorithm, except that vertices are given a priority equal to the sum of their Dijkstra priority (distance from the source) plus the heuristic distance, and also that we quit when the target is visited. Give the path (not order visited) that A* returns from A to G, you may not need all of the blanks: _A_ 2.
Please give me correct solution.
Please give me correct solution.
Exercise 12.9 Given a weighted (connected) undirected graph G = (V, E, w), with m =n+ 0(1) edges (i.e., the number of edges is at most the number of vertices plus some constant), give an O(n)-time algorithm to compute the minimum spanning tree of G. Prove the correctness and the running time of your algorithm. (Hint: Think of how one can eliminate an edge from being in the MST.)
Suppose G = (V,E) is a DAG. Design a linear-time algorithm that counts the total number of simple paths (i.e., a path that does not visit any node more than once) between two vertices s and t in G. Note that your algorithm only need to return the number of simple paths and does not need to list the actual paths.