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Now consider whether more flow can be sent to the sink with Figure-4. Note that the same graph is drawn differently in Figure-3 and Figure-4.

Figure 4 shows that it is possible to send 1 more flow in the path A->C->D->B->E->F! In this path D->B is the opposite edge. That means by canceling 1 flow of D->B edge we can reduce the total flow in the graph by 1. Now let’s see what the residual graph looks like:

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If you have understood so far, then you can take the original graph from Figure-5. 1 flow is sent through C->D edge in Fig-4. So in Figure-5 the residual capacity of C->D edge has decreased to 1-1=0 and the residual capacity of D->C edge has increased to 2+1=3. All the edges are drawn like this.

In this graph, there is no other path to send water from the source to the sink. We get a total flow of 2+2+1=5 in 3 paths, i.e. the water flow rate is 5 liters per second. This is the maximum flow for Grafta.

When we find a path in the residual graph, that path has a special name, that is Augmented path. Then our algorithm is very simple, as long as possible we will find an augmented path in the residual graph and send the flow to that path! It is called Ford-Fulkerson algorithm. Its pseudocode is:

Simply put, a 2-D array Cf[u][v] will have a residual capacity of every edge, edges that are not in the graph have a residual capacity of 0. Now you find a path from the source to the sink such that the residual capacity of each edge in the path is greater than 0. Finds the minimum residual capacity value between the edges of the path. After sending the flow, the residual capacity of each edge will decrease and the residual capacity of the opposite edge will increase. The minimum residual capacity will be added to the total flow value. Finally, you can calculate the amount of flow at an edge by subtracting the residual capacity from the original capacity.

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There is no requirement that the graph be directed. May also have bidirectional edges. If the capacity of A-B bidirectional edge is 10 then Cf[A][B] = Cf[B][A]=10. Now you can find the maximum flow by finding the augmented path as before. But in this case, to find out how much flow is happening in a node, you have to be a little more intelligent.

Algorithms such as BFS/DFS/Bellman-Ford etc. can be used to find the path in step 2 of the algorithm. Depending on the type of problem, each algorithm in this step will provide different benefits, which you will learn to solve the problem. And why not use BFS or whatever to find out the path, while going to each node, the parent of the node should be remembered in an array.

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If you solve using BFS then it is called Edmund Karp algorithm. Ford-Fulkerson Algorithm only says to find the path, it can be found any way, and Edmond-Karp BFS is asked to do the job i.e. one way to implement Ford-Fulkerson is Edmond Karp Algorithm whose complexity is O(VE^2 ).

Now some questions for you to think about:

Question 1: In our problem we had one source and one sink. But how will the algorithm change if water enters the graph through multiple nodes and water exits through multiple nodes?

Question 2: If each node has a certain capacity, that is, the maximum flow that can be sent by a node is specified, then how to solve it?

Question 3: What if there are multiple edges between two nodes?

Question 4: Two friends start from the same node and want to reach the same destination but both of them want to use different roads, that means 2 people can never use the same edge. Given the graph, can you tell if there are 2 such paths? Such a path is called an edge disjoint path.

These questions are discussed in the next section.

Figuring out how to make a graph is much more difficult than coding a maximum flow problem in a programming contest, so you have to practice by solving many problems. Here are some problems to get you started:

http://lightoj.com/volume_showproblem.php?problem=1153

http://lightoj.com/volume_showproblem.php?problem=1155

http://uva.onlinejudge.org/external/110/11045.html

Happy Coding!

**Quantum Computers – 2 (Strengths and Limitations)**

What can a quantum computer do that a normal computer can’t? Can quantum computers solve problems at lightning speed? Richard Feynman took quantum mechanics a step further with the idea of a quantum computer. Although it is not yet possible to build a quantum computer in the lab, we already know a lot in theory, maybe it is not too late for those theories to become reality. Today’s article is about why the quantum computer is strong, and where are its weaknesses.

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Quantum computers are a relatively new subject. I will not write in detail about what quantum computer is in this article, if you want to know, you can read my previous article and Tanvirul Islam’s article on quantum theory. All calculations in a quantum computer are done with “qubits”, a qubit can be a photon particle or an electron or any other particle. These particles can be used to store information, perform various calculations, “quantum information” is information stored in qubits. When scientists saw that it was possible to use quantum information to do things that were not possible with ordinary computers, they naturally became very interested in this matter. Richard Feynman is credited with starting the research field of quantum computers.

Many writings about quantum computers claim that quantum computers will work a thousand or so times faster than ordinary computers, or that it will be possible to solve problems that ordinary computers cannot solve. How much truth and how much imagination of these claims? In this article we will know about these. A large part of the text is a summary of an article by Scott Aaronson of MIT published in Scientific American 2008 [1].

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