Optimize Water Distribution In A Village - Leetcode Solution

Problem description: Given a village with N houses, each house is connected to one or more other houses by pipes. Each pipe has a capacity (in liters per hour) denoted as C[i][j] where i and j are the source and destination of the pipe. The population of the village is P[i], which represents the number of people living in house i. The village has a central water treatment plant with unlimited supply, but the pipes have a limited capacity. The task is to design a water distribution system that will maximize the amount of water supplied to each house such that every house receives water at a minimum rate of one liter per hour.

Solution: This problem is a typical minimum spanning tree (MST) problem, where the edges represent the pipes and the weights represent the capacities. The MST algorithm can be used to find the minimum capacity required to connect all the houses in the village. Then, the flow of water can be optimized using the Ford-Fulkerson algorithm.

Step 1: Build the graph Build a graph with N nodes, where each node represents a house and each edge represents a pipe. The initial graph will have a weight matrix C[N][N] with the capacity of each edge.

Step 2: Find the minimum capacity spanning tree Use Kruskal's algorithm to calculate the minimum capacity spanning tree. During this process, calculate the maximum capacity of the pipes that connect the houses.

Step 3: Implement Ford-Fulkerson algorithm Use the Ford-Fulkerson algorithm to optimize the flow of water from the central water treatment plant to each house. The idea behind Ford-Fulkerson algorithm is to increase the flow from the starting node to the ending node by finding the bottle neck capacity between them, then increase the flow of the bottleneck capacity and update the residual capacity of network, and iterate until no more augmenting paths are found. In this problem, the starting node is the central water treatment plant and the ending node is every house in the village. The residual capacity of the network is calculated as the difference between the total capacity and the current flow.

Step 4: Calculate the maximum flow Calculate the maximum flow of water that can be distributed to each house while maintaining a minimum flow rate of one liter per hour.

Time complexity of the solution: The time complexity of Kruskal's algorithm is O(ElogE), where E is the number of edges in the graph. The time complexity of Ford-Fulkerson algorithm is O(Ef), where f is the maximum flow. Therefore, the time complexity of the solution is O(ElogE + Ef).

Code: The code for the solution of this problem can be found on the LeetCode website.

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