How to use the maximum-flow minimum-cut theorem to find the flow capacity of a network (example 2)
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The Maximum-Flow Minimum-Cut Theorem is a fundamental result in graph theory and network flow problems. It establishes a relationship between the maximum flow that can be sent through a network and the minimum cut that can be made in that network. The theorem can be stated as follows:
"For any network (directed graph with capacities on its edges) with a source node 's' and a sink node 't,' the maximum amount of flow that can be sent from the source to the sink is equal to the minimum capacity of any cut in the network."
Here are some key terms used in this theorem:
1. Network: A network is typically represented as a directed graph with nodes and edges. Each edge has a capacity, which is the maximum amount of flow that can pass through it.
2. Source and Sink: The source node 's' represents the point where the flow enters the network, while the sink node 't' represents the point where the flow exits the network.
3. Cut: A cut in a network is a partition of the nodes into two sets, such that the source node 's' is in one set, and the sink node 't' is in the other set.
The theorem essentially states that to find the maximum flow from the source to the sink in a network, you can find the minimum cut in the network and the flow will be equal to the capacity of this cut. In other words, the maximum flow is bottlenecked by the minimum capacity of the edges crossing the minimum cut.
This theorem has important practical applications in various fields, such as transportation, communication networks, and resource allocation, where you need to find the most efficient way to send flow or data from one point to another through a network with limited capacities. Algorithms like the Ford-Fulkerson method and the Edmonds-Karp algorithm are commonly used to find the maximum flow in a network based on this theorem.
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