What is the brute force solution for Minimum Number of Seconds to Make Mountain Height Zero?

The brute force approach for Minimum Number of Seconds to Make Mountain Height Zero is Brute Force, with O(n²) time complexity and O(1) space complexity. The brute force approach involves simulating the work of each worker for every possible time until the mountain height is reduced to zero. This is straightforward but inefficient as it checks every second. The optimal Optimal Solution approach improves this to O(n log m).

What algorithm or data structure is used to solve Minimum Number of Seconds to Make Mountain Height Zero?

Minimum Number of Seconds to Make Mountain Height Zero uses the following concepts: Array, Math, Binary Search, Greedy, Heap (Priority Queue). The recommended patterns to study are: Binary Search, Greedy, Heap.

What are the interview tips for Minimum Number of Seconds to Make Mountain Height Zero?

Always clarify if workers can work simultaneously before diving into a solution. Consider edge cases such as when workerTimes has only one worker or when mountainHeight is zero. Practice binary search problems to become comfortable with the approach.

What are common mistakes when solving Minimum Number of Seconds to Make Mountain Height Zero?

Not considering simultaneous work of workers leading to incorrect total time calculations. Failing to optimize the search for the minimum time, leading to inefficient solutions.

#3296

Minimum Number of Seconds to Make Mountain Height Zero

Medium

ArrayMathBinary SearchGreedyHeap (Priority Queue)Binary SearchGreedyHeap

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Approaches

Brute ForceOptimal

Complexity Comparison

	Brute Force	Optimal Solution★
Time	O(n²)	O(n log m)
Space	O(1)	O(1)

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Intuition

Time O(n log m)Space O(1)

The optimal solution uses binary search to efficiently find the minimum time required. We check if a given time is sufficient to reduce the mountain height to zero by calculating the total reduction possible within that time.

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Algorithm

4 steps

1Step 1: Define a function to check if a given time can reduce the mountain height to zero.
2Step 2: Use binary search to find the minimum time by setting a range from 1 to a large enough value.
3Step 3: For each mid-point in the binary search, calculate the total height reduction possible by all workers within that time.
4Step 4: Adjust the search range based on whether the height can be reduced to zero.

solution.py23 lines

1# Full working Python code
2
3def can_reduce_to_zero(time, mountainHeight, workerTimes):
4    total_reduction = 0
5    for time_per_worker in workerTimes:
6        x = 1
7        while time_per_worker * x * (x + 1) // 2 <= time:
8            total_reduction += x
9            x += 1
10            if total_reduction >= mountainHeight:
11                return True
12    return total_reduction >= mountainHeight
13
14
15def min_seconds_optimal(mountainHeight, workerTimes):
16    left, right = 1, 10**9
17    while left < right:
18        mid = (left + right) // 2
19        if can_reduce_to_zero(mid, mountainHeight, workerTimes):
20            right = mid
21        else:
22            left = mid + 1
23    return left

ℹ

Complexity note: The time complexity is O(n log m) where n is the number of workers and m is the maximum time we are searching through, as we check each worker's contribution in logarithmic steps.

1Workers can work simultaneously, so we need to consider the maximum time taken by any worker.
2The contribution of each worker increases quadratically with the amount they reduce the mountain.

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