Open the Minimum Number of Sprinklers to Cover a Garden

Question

This question asks the candidate to determine the minimum number of sprinklers needed to cover an entire garden based on their coverage ranges.

Codemia · Accepted Answer

To solve the problem, we need to transform the coverage ranges of each sprinkler into intervals and then find the minimum number of intervals that can cover the entire garden. Each sprinkler at index `i` can cover the range `[i - ranges[i], i + ranges[i]]`. We will first create a list of these intervals and then apply a greedy algorithm to select the minimum number of sprinklers.

1. Initialize an empty list for intervals.
2. For each index `i`, calculate the left and right coverage as `max(0, i - ranges[i])` and `min(n, i + ranges[i])`, where `n` is the length of the garden.
3. Sort the intervals based on the left endpoint, and then by the right endpoint in descending order.
4. Use a greedy approach to select the minimum number of sprinklers that cover the entire range from `0` to `n`.

The time complexity of this algorithm is O(n log n) due to the sorting step, and the space complexity is O(n) for storing the intervals. This is efficient enough for typical input sizes.

```python
def minSprinklers(ranges):
    n = len(ranges)
    intervals = []
    for i in range(n):
        left = max(0, i - ranges[i])
        right = min(n, i + ranges[i])
        intervals.append((left, right))
    intervals.sort()

count = 0
    current_end = 0
    farthest = 0
    i = 0
    while current_end < n:
        while i < n and intervals[i][0] <= current_end:
            farthest = max(farthest, intervals[i][1])
            i += 1
        if current_end == farthest:
            return -1  # Not possible to cover the garden
        current_end = farthest
        count += 1
    return count
```

Interviewers look for clarity in your thought process and how well you can articulate your approach. They also assess your coding skills, particularly your ability to write efficient and clean code. Additionally, they are interested in your understanding of the problem's complexity and how you handle edge cases.

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