Question 3 a) Describe the greedy algorithm approach to solving an optimisation problem. Explain, and provide one example. b) Describe the nearest neighbour algorithm for solving the Travelling Salesman Problem, and explain why it does not yield an optimal solution. c) Describe what kind of algorithm the following is and what it does. S+ empty set WHILE G has at least one edge FIND a vertex v of the highest degree in G ADD v to S REMOVE vertex v from G PRINT S
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- Please help with the following ONLY problems 2, 4, and question 5 (if you can estimate time complexity of straightforward algorithm). (will leave a like) Given a graph G, find the smallest integer k such that the vertex set V of G contains a set A consisting of k elements satisfying the condition: for each edge of G at least one of its ends is in A. The size of the problem is the number n of vertices in G. 1. Suggest a natural greedy algorithm for the problem. 2. Make sure (by estimating the time complexity) that the suggested greedy algorithm is polynomial (its running time is estimated from above by some power of the size of the problem). 3. Find an instance for which the suggested greedy algorithm gives an erroneous answer. 4. Suggest a (straightforward) algorithm which solves the problem correctly. 5. Estimate the time complexity of the straightforward algorithm. Greedy algorithm posted below:Question 3 a) Describe the greedy algorithm approach to solving an optimisation problem. Explain, and provide one example. b) Describe the nearest neighbour algorithm for solving the Travelling Salesman Problem, and explain why it does not yield an optimal solution. c) Describe what kind of algorithm the following is and what it does. S+ empty set WHILE G has at least one edge FIND a vertex v of the highest degree in G ADD v to S REMOVE vertex v from G PRINT S3) The graph k-coloring problem is stated as follows: Given an undirected graph G = (V,E) with N vertices and M edges and an integer k. Assign to each vertex v in Va color c(v) such that 1< c(v)(b) One example of a greedy algorithm is the Dijkstra algorithm for finding the lowest cost path through a weighted graph. The diagram below shows two weighted graphs that a student wants to investigate using Dijkstra's algorithm. In each case the task it to find the lowest cost of reaching every node from v₁. Each graph has a single negative weight in it. ● V₁ ● 10 ● 12 V₂ V3 10 V4 V₁ 15 Graph (a) Graph (b) One of the graphs will yield a correct analysis of the lowest cost for all vertices, and the other will produce an incorrect analysis. Which of the two graphs will produce the incorrect analysis, and explain why the greedy nature of Dijkstra's algorithm is responsible for the incorrect analysis. Your answer should include the key concept of an invariant. V₂ (c) The priority queue is a widely used data structure. Priority queues may be implemented using binary heaps and simple linear arrays. For the basic priority queue operations of: 30 Building an initial queue Taking the highest…From the given information please provide a greedy algorithm and estimate it's time complexity. (will thumbs up) Given: a graph G, find the smallest integer k such that the vertex set V of G contains a set A consisting of k elements satisfying the condition: for each edge of G at least one of its ends is in A. The size of the problem is the number n of vertices in G.Computer Science Consider the d-Independent Set problem: Input: an undirected graph G = (V,E) such that every vertex has degree less or equal than d. Output: The largest Independent Set. Describe a polynomial time algorithm Athat approximates the optimal solution by a factor α(d). Your must write the explicit value of α, which may depend on d. Describe your algorithm in words (no pseudocode) and prove the approximation ratio α you are obtaining. Briefly explain why your algorithm runs in polytime.5. Fleury's algorithm is an optimisation solution for finding a Euler Circuit of Euler Path in a graph, if they exist. Describe how this algorithm will always find a path or circuit if it exists. Describe how you calculate if the graph is connected at each edge removal. Fleury's Algorithm: The algorithm starts at a vertex of v odd degree, or, if the graph has none, it starts with an arbitrarily chosen vertex. At each step it chooses the next edge in the path to be one whose deletion would not disconnect the graph, unless there is no such edge, in which case it picks the remaining edge (a bridge) left at the current vertex. It then moves to the other endpoint of that edge and adds the edge to the path or circuit. At the end of the algorithm there are no edges left ( or all your bridges are burnt). (NOTE: Please elaborate on the answer and explain. Please do not copy-paste the answer from the internet or from Chegg.)Problem 4: Consider the connected simple graph G given below. k Graph G 1. Use depth-first search to produce a spanning tree for the simple graph G. Choose vertex a as the root of this spanning tree. You must show how the spanning tree is constructed step by step as you add vertices and edges. Otherwise, your answer is wrong. Show your work step by step. 2. Use breadth-first search to produce a spanning tree for the simple graph G. Choose vertex a as the root of this spanning tree and assume that all vertices are ordered alphabetically. You must show how the spanning tree is constructed step by step as you add vertices and edges. Otherwise, your answer is wrong. Show your work step by step. 3. Is a spanning tree of a simple connected graph unique? Explain your answer clearly.3) The graph k-coloring problem is stated as follows: Given an undirected graph G= (V,E) with N vertices and M edges and an integer k. Assign to each vertex v in V a color c(v) such that 1Question 8. A connected graph G has 4 vertices and 4 edges of costs 1, 2, 3 and respectively 4. (a) Show that G is not a tree. (b) Show that G has a single minimum spanning tree. (Hint: For example, think how Prim's algorithm constructs the minimum spanning tree). (c) Draw a connected graph G with 4 nodes and 4 edges of costs 1, 2, 3, 4, respectively, so that the minimum spanning tree contains the edge of cost 4. The 4 nodes are below, you need to draw the 4 edges and indicate their cost. a O b dThe following solution designed from a problem-solving strategy has been proposed for finding a minimum spanning tree (MST) in a connected weighted graph G: Randomly divide the vertices in the graph into two subsets to form two connected weighted subgraphs with equal number of vertices or differing by at most Each subgraph contains all the edges whose vertices both belong to the subgraph’s vertex set. Find a MST for each subgraph using Kruskal’s Connect the two MSTs by choosing an edge with minimum wight amongst those edges connecting Is the final minimum spanning tree found a MST for G? Justify your answer.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.SEE MORE QUESTIONSRecommended textbooks for youDatabase System ConceptsComputer ScienceISBN:9780078022159Author:Abraham Silberschatz Professor, Henry F. Korth, S. 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