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To find the shortest disjoint paths in a network, we can use the Edmonds-Karp algorithm.
The Edmonds-Karp algorithm is a variation of the Ford-Fulkerson algorithm that uses Breadth-First Search (BFS) to find the shortest augmenting path. An augmenting path is a path from the source to the sink that has available capacity on all edges. The algorithm repeatedly finds the shortest augmenting path and increases the flow along that path until no more augmenting paths can be found.
To find the shortest disjoint paths, we can modify the Edmonds-Karp algorithm to find k disjoint paths instead of a single path. We start by finding the shortest augmenting path using BFS. We then mark all the edges in that path as visited and decrease their capacity by the flow that was sent along the path. We repeat this process k times, each time ignoring the edges that have already been visited in previous iterations.
If we cannot find k disjoint paths, then we can decrease k and try again until we find the maximum number of disjoint paths possible. The total flow along these paths will be the maximum flow in the network.
Overall, the Edmonds-Karp algorithm is an efficient way to find the shortest disjoint paths in a network. It has a time complexity of O(E^2 * V), where E is the number of edges and V is the number of vertices in the network.
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