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Augmenting Paths Any algorithm for maximum flow that uses the Ford-Fulkerson method has to cope with the problem of picking a "good" augmenting path. If a "bad" augmenting path is chosen, the flow will increase, but only a little. This will lead to numerous iterations and a slow algorithm. What if there aren't any good augmenting paths? what if they just don't exist? The Max Flow Min Cut Theorem won't help us here because if there are still bad augmenting paths, then we have not achieved maximum flow yet. For this problem, you will show that there is always a sequence of at most E augmenting paths (starting from zero flow) that leads to the maximum flow. Assignment: 1. You are given a flow network G. Suppose that you "cheat" and peek at a max flow f. (I.e., you look at the answer before doing the problem.) How would you pick a good augmenting path for the zero flow in G? (Remember that you are trying to minimize the number of iterations in the Ford-Fulkerson method.) 2. How would you update the max flow f so you can pick another good augmenting path? This augmenting path would augment the good path you picked above and still result in a valid flow for G. 3. Argue that you can keep picking good augmenting paths this way. 4. Argue that you will eventually reach max flow if you pick good augmenting paths this way. 5. Argue that you will pick at most E augmenting paths. Note: This observation isn't very useful in designing a max flow algorithm, since you need to know the max flow already before you can pick a good augmenting path. However, it does show that the difficulty isn't that good augmenting paths do not exist. The problem is that they are hard to find.

Respuesta :

Finding the shortest routes from the sources to the sink inside the recurrent neural network would be the first step in selecting a suitable augmenting route again for zero flow in G.

What  is algorithm?
The algorithm is a series of guidelines or procedures used to resolve a conflict or finish a job. It is a series of doable actions that result in a particular result. Numerous disciplines, including mathematics, computer science, construction, or business, use algorithms. The most effective way to solve a problem or finish a task is frequently determined using an algorithm. Additionally, it can be applied to decision-making and process optimization. Algorithms are crucial to computer programming because they give machines a means to process information and finish tasks.

Then, among those paths, we would choose the enhancing path with the maximum capacity.
To pick a good augmenting path for the zero flow in G, we would start by finding the shortest paths from the source to the sink in the residual network. We would then select the augmenting path with the highest capacity among those paths.

To update the max flow f, we would add the capacity of the augmenting path we chose to the max flow f. This would result in a larger max flow, and we would then look for another augmenting path in the residual network with the highest capacity.
We can keep picking good augmenting paths this way because the max flow will increase with each augmenting path. We will eventually reach a point where there are no more augmenting paths with a capacity greater than 0.
We will eventually reach max flow if we pick good augmenting paths this way because the max flow will be continually increasing, and at some point we will reach a max flow that is equal to the maximum capacity of all the edges in the graph.
We will pick at most E augmenting paths because there are only E edges in the graph, and each augmenting path will increase the max flow by at least 1. Therefore, the maximum number of augmenting paths that can be picked is E.

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