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Minimum Time to Break Locks I

Difficulty: Medium

Problem Description

Bob is stuck in a dungeon and must break n locks, each requiring some amount of energy to break. The required energy for each lock is stored in an array called strength where strength[i] indicates the energy needed to break the i-th lock. Your task is to determine the minimum time in minutes required for Bob to break all n locks and escape the dungeon.

Key Insights

  • The initial energy of Bob's sword is 0, and it increases every minute by a factor that starts at 1 and increases by k after each lock is broken.
  • Bob must reach the required energy for each lock in minimal time, which means calculating the time it takes to reach the strength needed for each lock while considering the increase in the factor x after each lock is broken.
  • The problem can be approached using a backtracking method to explore different sequences of breaking the locks to find the optimal time.

Space and Time Complexity

Time Complexity: O(n * max_strength) where max_strength is the maximum value in the strength array, since we may need to simulate time until we reach the required strength for each lock. Space Complexity: O(n) for the recursion stack in backtracking.

Solution

The solution involves simulating the process of increasing the sword's energy until it meets the requirements to break a lock. We will use a backtracking approach to try breaking the locks in different orders, keeping track of the minimum time required.

Code Solutions

def minTimeToBreakLocks(strength, k):
    n = len(strength)
    
    def backtrack(index, current_time, current_x):
        if index == n:
            return current_time
        
        min_time = float('inf')
        energy = 0
        
        for time in range(current_time, current_time + 100):
            energy = (time - current_time + 1) * current_x
            
            if energy >= strength[index]:
                new_time = backtrack(index + 1, time + 1, current_x + k)
                min_time = min(min_time, new_time)
        
        return min_time
    
    return backtrack(0, 0, 1)

# Example usage
print(minTimeToBreakLocks([3, 4, 1], 1))  # Output: 4
print(minTimeToBreakLocks([2, 5, 4], 2))  # Output: 5
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