A) Travelling salesman problem: The travelling salesman problem (TSP) is a computational issue in which the goal is to find the quickest distance between a set of points and places. The points in the...

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Extracted text: A) Travelling salesman problem: The travelling salesman problem (TSP) is a computational issue in which the goal is to find the quickest distance between a set of points and places. The points in the problem statement represent cities that a salesperson could visit. The objective of the distributor is to keep all travel costs and distance travelled to a minimum. TSP is always more concerned with finding the cheapest alternative than with finding the most efficient path. The large number of variables in TSP makes finding the quickest distance challenging, which makes estimated, quick, and cheap solutions all the more appealing. Application of TSP: In computer science, TSP is often used to determine the most effective path for data to travel between nodes. • Identifying network or hardware optimization methods is one of the applications. Algorithm to solve TSP: The algorithm Minimum Spanning Tree is used to solve the TSP. There are approximate algorithms for solving TBS the problem. Only if the problem instance satisfies Triangle- Inequality do the approximate algorithms work. According to Triangle-Inequality for any given triangle, the sum of two sides of a triangle is always greater than the third side. We can build an approximation algorithm for TSP problem that returns a route that is never more than twice the cost of an acceptable tour when the cost function satisfies the triangle inequality. The concept is to use a Minimum Spanning Tree (MST). The MST-based algorithm is shown below. Algorithm: 1) Assume 1 is the beginning and ending point for the salesman. 2) Using Prim's Algorithm, construct MST from 1 as the root. 3) Add 1 to the list of vertices visited in the constructed MST's preorder walk.


Extracted text: Let's take a look at the following scenario. The provided graph is the first diagram. The second diagram depicts MST with 1 as the root. MST has a preorder traversal of 1-2-4-3. The performance of this algorithm is 1-2-4-3-1, which is obtained by adding 1 at the end. The cost of the above algorithm's performance is never more than twice the cost of the best possible output. Let's look at how the above algorithm ensures this. To better understand this, let's describe the word "absolute walk." A complete walk lists all vertices when they are visited for the first time in preorder, as well as when they are returned after a subtree is visited for the first time in preorder. The above tree's complete walk will be 1-2-1-4-1-3-1. The following are some important facts to prove the approximations. The best possible Traveling Salesman tour will never be less pricey than MST. (According to the definition, MST is a minimum cost tree that links all edges.) • A complete walk would cost at least twice more than MST (Every edge of MST is visited at-most twice) • The above algorithm produces a result that is less expensive than a complete walk. We print preorder walk as output in the algorithm above. Two or more edges of a complete walk are replaced with a single edge in preorder walk. 2-1 and 1-4, for example, are replaced by 1 edge 2-4. This is always valid if the graph follows triangle inequality. from the above instruction, that we can conclude that the cost of output produced by the approximate algorithm is never more than twice the cost of best possible solution. The output of this algorithm is 1-2-4-3-1. B) Time complexity of the algorithm: Time complexity of minimum spanning tree is o(v).