#3376

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Minimum Time to Break Locks I

Solve the Minimum Time to Break Locks I problem using state transition dynamic programming to minimize the time to break all locks.

array dynamic programming backtracking bit manipulation depth first search

Problem Statement

Bob is trapped in a dungeon and must break a set of locks, each requiring a certain amount of energy to break. The energy for each lock is stored in the array strength where strength[i] represents the energy needed for the i-th lock. Bob has a sword that can break the locks, but each break takes time equal to the energy required for that lock.

Your task is to determine the minimum amount of time in minutes required for Bob to break all the locks and escape the dungeon. You can break the locks in any order, and Bob can break at most k locks at a time. Return the minimum total time required to break all locks.

Examples

Example 1

Input: strength = [3,4,1], k = 1

Output: 4

The locks cannot be broken in less than 4 minutes; thus, the answer is 4.

Example 2

Input: strength = [2,5,4], k = 2

Output: 5

The locks cannot be broken in less than 5 minutes; thus, the answer is 5.

Constraints

n == strength.length
1 <= n <= 8
1 <= K <= 10
1 <= strength[i] <= 106

Solution Approach

State Transition Dynamic Programming

The key approach involves modeling the problem as a dynamic programming problem with state transitions. We use a DP table where each state represents the locks that have been broken, and the minimum time to break them. The transitions depend on choosing locks in different orders, allowing the computation of the minimum time.

Backtracking with Bit Masking

A backtracking solution with bit masking can explore all possible ways to break the locks. Each lock can either be broken or not, and by masking the states, we can efficiently explore all combinations of broken locks and the time taken for each. This helps to avoid brute force and unnecessary recomputations.

Permutations and Time Calculation

Try all n! permutations of the locks, computing the time taken for each permutation. This approach is feasible due to the small constraint of n (maximum of 8). By calculating the time required for each permutation, we can return the minimum time across all possible orders.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity of this solution depends on the method used to explore the lock permutations. The backtracking approach with bit masking can explore all combinations of broken locks efficiently. The space complexity also depends on the implementation, as we store intermediate results in the DP table or through recursion stacks in backtracking.

What Interviewers Usually Probe

Look for the candidate's ability to model the problem using dynamic programming.
Evaluate the candidate's understanding of bit manipulation and backtracking techniques.
Assess if the candidate can effectively handle permutations and manage time calculations.

Common Pitfalls or Variants

Common pitfalls

Overlooking the need for bit masking or efficient state transitions in dynamic programming.
Incorrectly assuming that brute-forcing all permutations is the only solution.
Failing to consider the impact of breaking locks in different orders on the overall time.

Follow-up variants

Try solving with different values for k to see how it impacts the time calculation.
Handle larger values of n and test the performance of the solution.
Explore using different optimization techniques, such as greedy methods or additional pruning.

FAQ

What is the primary approach to solving the Minimum Time to Break Locks I problem?

The primary approach is state transition dynamic programming, which allows efficient tracking of time for each combination of broken locks.

How can bit manipulation help in solving the Minimum Time to Break Locks I problem?

Bit manipulation helps by efficiently representing the state of broken locks and reducing the complexity of exploring all possible combinations.

What is the time complexity of the Minimum Time to Break Locks I problem?

The time complexity depends on the number of permutations of locks, typically O(n!), and the complexity of handling each permutation.

Can the Minimum Time to Break Locks I problem be solved using a greedy approach?

A greedy approach might not work for this problem, as it requires considering all permutations and the specific order of breaking locks.

What is the maximum value for n in the Minimum Time to Break Locks I problem?

The maximum value for n is 8, making the problem feasible for brute-force approaches like backtracking with permutations.

terminal

Solution

Solution 1

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class Solution:
    def findMinimumTime(self, strength: List[int], K: int) -> int:
        @cache
        def dfs(i: int) -> int:
            if i == (1 << len(strength)) - 1:
                return 0
            cnt = i.bit_count()
            x = 1 + cnt * K
            ans = inf
            for j, s in enumerate(strength):
                if i >> j & 1 ^ 1:
                    ans = min(ans, dfs(i | 1 << j) + (s + x - 1) // x)
            return ans

        return dfs(0)

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