How to find the flow capacity of a cut in a network (example 3)
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To find the flow capacity of a cut in a network, you can follow these steps:
1. Identify the Cut:
First, you need to identify a cut in the network. A cut is a partition of the nodes into two sets: one containing the source node 's' and the other containing the sink node 't.' These two sets effectively divide the network into two parts, with some edges crossing from one set to the other.
2. Calculate the Capacity of the Cut:
Once you have identified the cut, calculate the capacity of the cut. The capacity of the cut is the sum of the capacities of all the edges that cross from the source side to the sink side of the cut. In other words, it is the sum of the capacities of the edges that go from the source side to the sink side, or from the set containing 's' to the set containing 't.'
3. Interpret the Capacity:
The capacity of the cut represents the maximum amount of flow that can be sent from the source 's' to the sink 't' while respecting the capacities of the edges. This value is also the minimum capacity of any cut in the network, as per the Maximum Flow Minimum Cut Theorem.
Here's a simple example to illustrate this process:
Consider a network with nodes A, B, C, D, E, F, and edges with the following capacities:
(A, B) with capacity 3
(A, C) with capacity 2
(B, C) with capacity 2
(B, D) with capacity 3
(C, E) with capacity 1
(D, E) with capacity 2
(D, F) with capacity 3
(E, F) with capacity 2
Assume that A is the source node 's,' and F is the sink node 't.' Now, let's find the capacity of a cut that separates the source side from the sink side.
One possible cut includes the nodes {A, B, C} on one side and {D, E, F} on the other side. The edges crossing this cut are (A, B), (A, C), (B, C), (D, E), and (E, F). The sum of their capacities is 3 + 2 + 2 + 2 + 2 = 11. So, the capacity of this cut is 11, which is the maximum flow that can be sent from A to F while respecting the capacities of the edges.
In this example, the cut capacity represents the maximum flow capacity in the network, as stated by the Maximum Flow Minimum Cut Theorem.
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