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This approach involves treating each tap as an interval [i-ranges[i], i+ranges[i]], and seeking the minimal number of such intervals required to cover the entire garden range [0, n]. The key is to employ a greedy algorithm which iteratively selects the interval that extends the coverage furthest at each point. Specifically, from each starting point, choose the interval that reaches farthest within the current range, then jump to the furthest point reached by the selected intervals and repeat the process.
Time Complexity: O(n), as the algorithm processes the tap ranges in a single pass.
Space Complexity: O(n), due to the storage for the maxRange array.
1class Solution:
2 def minTaps(self, n: int, ranges: List[int]) -> int:
3 max_range = [0] * (n + 1)
4 for i in range(n + 1):
5 left = max(0, i - ranges[i])
6 right = min(n, i + ranges[i])
7 max_range[left] = max(max_range[left], right)
8
9 taps = 0
10 curr_end = 0
11 next_end = 0
12 for i in range(n + 1):
13 if i > next_end:
14 return -1
15 if i > curr_end:
16 taps += 1
17 curr_end = next_end
18 next_end = max(next_end, max_range[i])
19
20 return taps
21
22# Example usage
23sol = Solution()
24print(sol.minTaps(5, [3, 4, 1, 1, 0, 0])) # Output: 1In this Python solution, the aim is to compute the maximum rightward extension possible starting from each point in the garden. The function then proceeds to select minimal and optimal tap combinations that cover the entire garden without leaking.
This approach takes on the problem by formulating it into a dynamic programming challenge. Here, the idea is to maintain an array dp[], where dp[i] denotes the minimum number of taps required to water up to position i. Start by initializing dp[i] with infinity, except dp[0]=0, then update each dp[i] by considering if current tap can extend water coverage from any previous dp positions.
Time Complexity: O(n^2), due to the potential iteration across all tap ranges for each garden position.
Space Complexity: O(n), as a dp array of size n+1 is maintained.
This C implementation establishes an array, dp[], with each index representing the minimal taps needed to reach that point. By updating the array through possible coverage ranges of each tap, it identifies the least number capable to span the whole garden.