Minimum Time to Break Locks I

Minimum Time to Break Locks I

What is this problem about?

The Minimum Time to Break Locks I problem presents a set of locks, each requiring a specific strength to break. You gain strength incrementally over time. You must break all locks, and breaking one in a specific order affects how quickly you gain strength for subsequent ones. The goal is to find the minimum total time to break all locks. This Minimum Time to Break Locks I coding problem uses bitmask DP to model the state of which locks have been broken.

Why is this asked in interviews?

IVP asks this medium problem to test bitmask dynamic programming — a pattern for problems where a small set of items must all be processed, with order mattering. The combination of bitmask, DFS, backtracking, bit manipulation, and DP interview pattern is fully exercised here. It's a signal that a candidate can model "which subset has been done" compactly using integer bitmasks.

Algorithmic pattern used

Bitmask DP with DFS or bottom-up DP. State: dp[mask] = minimum time to break exactly the locks indicated by mask. For each state, try breaking each unbroken lock next, computing the time based on current accumulated strength and the lock's requirement. Transition: dp[mask | (1 << i)] = min(dp[mask | (1 << i)], dp[mask] + time_to_break(lock_i, current_strength(mask))). Final answer: dp[(1 << n) - 1].

Example explanation

Locks: [3, 1, 2]. Breaking lock in order 1→0→2 (breaking the easiest first might grant more strength faster).

• Break lock 1 first (strength=0): time = some formula.
• Break lock 0 next (strength increased).
• Break lock 2 last. Compare all orderings via bitmask DP and return the minimum total time.

Common mistakes candidates make

• Trying all permutations explicitly (O(n!)) instead of bitmask DP (O(2^n * n)).
• Incorrectly computing the current accumulated strength from the mask (depends on which locks were broken).
• Not memoizing intermediate states — pure backtracking without DP is too slow.
• Off-by-one in bit indexing when building the mask.

Interview preparation tip

Bitmask DP is the go-to technique when n ≤ 20 and you need to track which subset of items has been processed with optimal ordering. The classic template: define dp[mask] = optimal value for the subset represented by mask, iterate over all masks in increasing order, try each unused item as the next one, and update. Practice the Traveling Salesman Problem bitmask DP as the canonical reference — Minimum Time to Break Locks follows the same structural template.