Nearest neighbor and cheapest link algorithm

[SamTsui86]

Homework Statement

I need to calculate the relative error of the nearest neighbor algorithm. The book says it is the difference between the worst and nearest neighbor solution to the nearest neighbor solution. Does the mean that for example I have CITY A, find the fastest way to back to city A while reaching all the vertex and find the longest way back to city A while reaching all the vertex. Then I take the difference and divide it by the shortest way?

Can anyone explain how to do the cheapest link algorithm:
My professor told me is
1) Pick the link with the smallest weight
2) Pick the next cheapest link
3) Contine picking the cheapest link availabe
4) Close the circuit

I am very confuse with his step, how do I close the link? How is that different from the nearest neighbor algorithm??

Thank You

Homework Equations

The Attempt at a Solution

 

[Aaron Bex]

The relative error of the nearest neighbor algorithm can be calculated by taking the difference between the longest and shortest paths through the graph and dividing it by the length of the shortest path. This will give you the relative error as a percentage.

For the cheapest link algorithm, the steps your professor gave are straightforward. The first step is to pick the link with the smallest weight. For example, if you have a graph with five nodes, then you would choose the edge with the smallest weight (the smallest number) connecting two of those nodes. The second step is to pick the next cheapest link, which would be the next edge with the smallest weight connecting two nodes. You would continue this until you have connected all of the nodes in the graph. Finally, the fourth step is to close the circuit. This means that you need to connect the last node in the graph back to the first node. This will form a circuit, or a loop, which is what you need for the cheapest link algorithm.

The cheapest link algorithm is different from the nearest neighbor algorithm in that the nearest neighbor algorithm finds the shortest path between two nodes, while the cheapest link algorithm finds the least expensive path between all of the nodes in the graph.

 

[Architeuthis Dux]

The nearest neighbor algorithm and the cheapest link algorithm are both methods used in finding optimal solutions for the traveling salesman problem, which involves finding the shortest route that visits all given points and returns to the starting point. However, they use different approaches to achieve this goal.

The nearest neighbor algorithm works by starting at a given point and then choosing the nearest unvisited point as the next destination. This process is repeated until all points have been visited and the salesman returns to the starting point. The relative error of this algorithm is calculated by finding the difference between the longest and shortest possible routes and dividing it by the shortest route.

On the other hand, the cheapest link algorithm works by finding the link with the smallest weight (distance) between two points and adding it to the route. This process is repeated until all points have been visited and the circuit is closed by connecting the last point to the starting point. This algorithm guarantees the shortest possible route, but it may not be the most efficient in terms of time. The closing of the circuit simply means that the last point in the route is connected back to the starting point to complete the circuit.

The main difference between the two algorithms is that the nearest neighbor algorithm only considers the distance between two consecutive points, while the cheapest link algorithm takes into account the overall distance of the entire route. The cheapest link algorithm may result in a longer route than the nearest neighbor algorithm, but it ensures that it is the shortest possible route.

In terms of calculating the relative error for the cheapest link algorithm, the same approach can be used as in the nearest neighbor algorithm. The difference between the longest and shortest possible routes can be divided by the shortest route to obtain the relative error.

In summary, both algorithms have their own advantages and disadvantages and can be used depending on the specific needs and constraints of the problem at hand. It is important to understand the differences between them and choose the appropriate algorithm for a given situation.