Routing Total Points 10 Java: The Solution Time Complexity

Routingtotal Points 10 JAVA: The solution time complexity for this lab is

This assignment involves designing an algorithm to determine the maximum minimal bandwidth required for files to reach their rightful storage locations within a network. The problem models a network of storage nodes interconnected by links with varying bandwidths, and the objective is to optimize data transfers so that the slowest link used in the process is as fast as possible. The solution must be efficient enough to handle large inputs, with up to 105 nodes and links, and should produce results within 4-5 seconds.

Given a network of N storage nodes and M bidirectional links, each characterized by an integer bandwidth, the task is to find the highest possible minimal bandwidth such that all files can be moved to their respective storages through a series of swaps. Files are initially located at specified storages, with the goal of rearranging them so that each file fi is at storage si. The swap process can only occur between directly connected storages, and multiple swaps are permitted as part of the overall transfer strategy.

The problem guarantees that a solution exists, meaning that it is always possible to rearrange the files to their correct storages. If the files are already in their correct storages at the start, the output should be -1, indicating no transfer is necessary.

This problem is akin to finding a maximum bottleneck path or a maximum spanning tree approach, where the objective is to maximize the minimum bandwidth along the transfer paths for all files. An effective approach involves using a binary search over possible bandwidth values combined with a union-find (disjoint set) data structure or graph algorithms such as maximum spanning trees, to verify for each candidate bandwidth if a feasible transfer plan exists.

Paper For Above instruction

The task of optimizing data transfers within a network by maximizing the minimal bandwidth of the paths used is a classical problem in graph theory and network optimization, often modeled as the "Maximum Bottleneck Path" problem. Efficient solutions to this problem leverage data structures like union-find along with binary search techniques to identify the highest feasible bandwidth threshold.

The core idea is to perform a binary search over the range of link bandwidths because bandwidth values are bounded by 1 and 10^9. For each candidate bandwidth during the binary search, we construct a subgraph containing only links with bandwidths greater than or equal to the candidate value. Within this subgraph, we check whether all files can reach their destination storages, which are predetermined, by verifying connectivity between the source and target nodes for each file.

To implement this efficiently, we employ a union-find data structure (disjoint-set union - DSU) to manage connectivity among nodes. Starting from the highest bandwidths, we progressively include links and update connectivity components. When a component contains both the source and the destination storages for a file, it implies a feasible path exists for that file at the current bandwidth threshold.

The binary search iteratively narrows down the maximum bandwidth for which all files can be allocated correctly by verifying components using union-find. Once the optimal maximum minimal bandwidth is identified, the solution is output. If the files are initially correctly placed, the result is -1, indicating no transfer is needed.

This approach satisfies the problem constraints, including large N and M (up to 10^5), by ensuring a time complexity of approximately O(M log(max bandwidth)), which is efficient enough for the problem's input size.

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