Solution: Optimize Water Distribution in a Village

Explore how to minimize the total cost of supplying water to houses by implementing the Union-Find data structure. Understand how to model the problem as a graph with wells and pipes, use a virtual node concept, and apply a greedy approach ensuring each house receives water at the lowest possible cost. This lesson helps you grasp effective graph connectivity techniques relevant for coding interviews and real-world optimization.

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Statement
Solution
- Time complexity
- Space complexity

Statement

There are $n$ houses in a village, and we want to supply water to every house. To do this, we have two options:

Build a well directly at a house: The costs for building wells are given in the wells array, where wells[i - 1] represents the cost to build a well at house i (houses are numbered from $1$ to $n$ , but the array is 0-indexed).
Lay water pipes to connect two houses: Allowing one to receive water from the other. The costs for building these connections are given in the pipes array. Each pipe entry is written as [house1, house2, cost], meaning it costs to construct a pipe connecting house1 and house2. The connection is two-way (water can flow in both directions).

In the given data, multiple pipe entries may connect the same pair of houses, but with different costs. Each entry represents an available option; you may choose the most cost-effective one.

Your task is to determine the minimum total cost to ensure every house gets access to water, either by building a well directly at it or connecting it via pipes to another house with access to water.

Constraints:

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