Give an algorithm that determines whether or not a given undirected graph G (V.E) contains a cycle. Your algorithm should run in O(V) time, independent of |E|. %3D
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- 21.2-4 Suppose that all edge weights in a graph are integers in the range from 1 to [V]. How fast can you make Kruskal's algorithm run? What if the edge weights are integers in the range from 1 to W for some constant W? Kruskal's algorithm Kruskal's algorithm finds a safe edge to add to the growing forest by finding, of all the edges that connect any two trees in the forest, an edge (u, v) with the lowest weight. Let C₁ and C₂ denote the two trees that are connected by (u, v). Since (u, v) must be a light edge connecting C₁ to some other tree, Corollary 21.2 implies that (u, v) is a safe edge for C₁. Kruskal's algorithm qualifies as a greedy algorithm because at each step it adds to the forest an edge with the lowest possible weight. 9 (1) 11 14 11 14 7 8 10 10 (K) a 11 14 (1) 11 7. 7. 6 10 (0 4 (m) a 11 7. 6 8 Figure 21.4, continued Further steps in the execution of Kruskal's algorithm. b 14 11 1 141 7 6 10 10 2 1lert dont submit AI generated answer. Write a program that reads a weighted graph.The program must print on the screen the distance matrix and the "PI" matrix of the shortest paths obtained by the Floyd-Warshall algorithm. Input: Receives n and m; n is the total number of vertices and m the total number of arcs.Next, m lines, each line with a trio of integers, corresponding to the beginning and end of the arc, followed by the weight of the arc.(Vertices are identified from 0 to n-1.) Output: Prints the distances and shortest paths obtained by the Floyd-Warshall algorithm. Exemple: Input: 5 9 0 1 3 0 2 8 0 4 -4 1 3 1 1 4 7 2 1 4 3 0 2 3 2 -5 4 3 6 Output: [[ 0 1 -3 2 -4] [ 3 0 -4 1 -1] [ 7 4 0 5 3] [ 2 -1 -5 0 -2] [ 8 5 1 6 0]] [[-1 2 3 4 0] [ 3 -1 3 1 0] [ 3 2 -1 1 0] [ 3 2 3 -1 0] [ 3 2 3 4 -1]]C++ Given a directed graph. The task is to find a shortest path from vertex 0to a target vertex v. You may adapt Breadth First Traversal of thisgraph starting from 0 to achieve this goal.Note: One can move from node u to node v only if there's an edge from u to v and find the BFS traversal of the graph starting from the 0th vertex, from left to right according to the graph. Also, you should only take nodes directly or indirectly connected from Node 0 in consideration. ExampleInput:6 8 20 10 41 20 33 54 55 23 1Output:0 1 2 Use the driver code typed out below. // { Driver Code Starts #include <bits/stdc++.h>using namespace std; // } Driver Code Endsclass Solution { public: // Function to return a path vector consisting of vertex ids from vertex 0 to target vector shortestPath(int V, vector adj[], int target) { // Enter code here! }}; // { Driver Code Starts.int main() { int tc; cin >> tc; while (tc--) { int V, E, target; cin >> V >> E >> target; vector…
- 5. 6. Given an undirected graph with n vertices and m edges, find an O(n+m) time algorithm that determines whether it is possible to color all the vertices red and blue such that every edge is between a red vertex and blue vertex. If such a coloring exists, your algorithm should produce one. Give an algorithm that determines whether or not a given undirected graph contains a cycle. Your algorithm should run in time O(n) and independent of m.A graph G with vertices labelled 0 to 9 is shown below. 8 17 3 Tick all the statements that are correct about the graph. You may need to compare the above graph to the Peterson graph, which is provided as follows: O Gis a Hamiltonian graph O G contains more than five spanning trees O Gis an Eulerian graph O G is a simple graph O Gis bipartite O The complement of G is connected There are two distinct vertices in G that are joined by an Eulerian trail O The diameter of G is 2 O There are two distinct vertices in G that are joined by a Hamiltonian path O Gis isomorphic to the Peterson graph G is a regular graph 6,• An undirected graph G = (V, E) has Eulerian Tour if following two conditions are true: 1. All vertices with non-zero degree are connected. We don't care about vertices with zero degree because they don't belong to Eulerian Cycle or Path. 2. All vertices have even degree. • Write the Python code that finds an Euler Tour of given graph G (if one exists) following the Fleury's Algorithm: 1. Make sure the graph has either 0 or 2 odd vertices. 2. If there are 0 odd vertices, start anywhere. If there are 2 odd vertices, start at one of them. 3. Follow edges one at a time. If you have a choice between a bridge and a non-bridge, always choose the non-bridge. 4. Stop when you run out of edges. Input: Adjacency list representation of the graph G Output: The graph has an Euler Tour / The graph has no Euler Tour [ ] # test 1: Grap with Euler Tour g1 = Graph (5) g1.addEdge(1, 0) g1.addEdge(0, 2) g1.addEdge (2, 1) g1.addEdge(0, 3) g1.addEdge (3, 4) g1.addEdge(4, 0) g1.Euler Tour() # test 2: Grap…
- Exercise 11.11 An undirected graph G = (V, E) is called bipartite if it contains no odd eycle, i.e., no cycle has an odd mumber of edges. Note that a graph with no cycles is also bipartite. We are going to show an important property of bipartite graphs: its verter set can be partitioned into two disjoint sets A and B, i.e., AUB=V and An B = 6, such that all the edges are between vertices in A and vertices in B – in other words, there are no edges between any two vertices in A and any two vertices in B. %3D The goal of this problem is that, given a connected bipartite graph G = (V, E), to find a partition of the vertex set into A and B such that all edges are between some verter in A and some vertex in B. (a) Do Breadth-First Search (BFS) on G from some starting vertex, say a, and compute the distance (i.e., the level in the BFS tree) of every vertex v to a. Show that if G is bipartite then there are no edges between vertices in the same level. (Hint: Use the odd cycle property of…This task is solved in Python. 3. Write a function build_my_graph2 () that: a) creates the following Graph. b) runs Depth First Search (DFS) algorithm starting from node 'a' and prints all the visited nodes. What is printed in the output when you run the function? Note: you can use the implementation of Graph class and the DFS algorithm (provided in Lecture notes). 5.0 g h e5. * Show that there is an algorithm to decide if a graph has a vertex cover of size k, with running time O(kn +20(k)). Hint: use the FPT algorithm discussed in class and the results of the previous problem.
- Use C++ knowledge for the question: Consider the following graph as input and answer the following. 8. 9. 1 7 6. 4 a. What will be the output of the following graph after applying BFS and DFS? The searching strategy following descending order and starting vertex is 1. b. You are given an adjacency matrix of a graph having n number of vertexes as input. Write a function isStronglyConnected() to check whether the given graph is strongly connected (every vertex is reachable from every other vertex) or not. 3.36. Let G be a simple graph on n vertices and has k components. Then the number m of edges of G satisfies n-k ≤m if G is a null graph. This statement is A. sometimes true B. always true C. never true D. Neither true nor falseJAVA: Write a program that takes a graph representing a map between cities and use Dijkstra's Algorithm to find the shortest path given a start and an end vertex. in java 1- Please comment the codes.