Q1: Suppose You Are Given A Flow Network N And A Maximum Flo

Q1: Suppose You Are Given A Flow Network N And A Maximum Flow F For N

Suppose you are given a flow network N and a maximum flow f for N. Suppose d, a positive integer, is added to the capacity of one edge of N. 1. Give an efficient algorithm to compute a maximum flow for the new network. 2. What is the worst case time complexity of your algorithm?

Paper For Above instruction

In the realm of network flow algorithms, modifying existing flows efficiently after small changes to the network is a crucial problem. When a new capacity is added to an edge within a flow network, recalculating the maximum flow from scratch can be computationally expensive. Therefore, designing an efficient algorithm to update the maximum flow by leveraging the existing flow and only adjusting as necessary can greatly improve performance.

The problem setting involves a flow network N = (G, c), where G = (V, E) represents the graph with vertices V and edges E, and c denotes capacities on edges. Given a maximum flow f initially computed for N, suppose a positive integer d is added to the capacity of a single edge e = (u, v). The goal is to update the maximum flow to reflect this capacity change with minimal computation.

Algorithm for Updating Maximum Flow

The proposed approach relies on the residual network concept and the pre-existing maximum flow. The core idea is to check whether the added capacity can be utilized to increase the flow, and if so, to adjust the flow accordingly by searching for augmenting paths in the residual network.

The steps of the algorithm are as follows:

  1. Update the capacity: Increase the capacity of edge e = (u, v) by d units, updating c(u, v) to c(u, v) + d.
  2. Recompute the residual network: Construct the residual network based on the current flow f and the updated capacities.
  3. Attempt to augment: Use a breadth-first search (BFS) or depth-first search (DFS) to find an augmenting path from the source s to the sink t in the residual network.
  4. Update flow accordingly: If an augmenting path is found, push additional flow along this path up to the minimum residual capacity on the path, and update the flow in the original network accordingly.
  5. Repeat if necessary: Continue searching for augmenting paths until no more augmenting paths can be found. The resulting flow will be the new maximum flow for the modified network.

This process is efficient because it only involves incremental updates to the residual network based on the capacity change, rather than recomputing the maximum flow from scratch. Since a single edge's capacity is increased, the number of augmenting paths needed to reach the new maximum flow is typically small, especially if the added capacity is significant in comparison to existing bottlenecks.

Analysis of Time Complexity

The initial step of updating the capacity is a constant-time operation. Constructing the residual network involves examining all edges, which can be done in O(E) time. The main component is searching for augmenting paths using BFS or DFS. Each augmenting path search takes O(E) time in the worst case.

In the worst case, the number of augmenting paths needed could be proportional to the capacity increase d, especially if the network is highly constrained. However, in many practical scenarios, the number of augmentations is small because the residual capacities increase significantly after the initial augmentation.

Therefore, the overall worst-case time complexity is O(E min(d, number of augmentations)), which simplifies to O(E) per augmentation step. When considering the total operations, the complexity can be approximated as O(E k), where k is the number of augmentations necessary, which in the worst case could be proportional to the maximum increase d.

Conclusion

Thus, when a capacity on a single edge is increased by a positive integer d, efficiently updating the maximum flow involves augmenting the residual network starting from the existing maximum flow. The process typically requires only a few augmenting path searches, resulting in a time complexity proportional to the number of such augmentations, generally bounded by O(E * d) in the worst case. This approach leverages the residual network structure to avoid recomputing maximum flows from scratch, leading to significant computational savings in practical scenarios.

References

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