Minimum Time to Break Locks II

Minimum Time to Break Locks II

What is this problem about?

The Minimum Time to Break Locks II problem extends its predecessor to a harder variant where the lock-breaking order has more complex interdependencies modeled as a graph. Instead of bitmask DP, the problem requires identifying the optimal traversal order through a directed graph structure where each lock's breaking cost depends on previously broken locks. This Minimum Time to Break Locks II coding problem combines graph traversal with DFS-based optimization.

Why is this asked in interviews?

IVP asks this hard-level problem to test whether candidates can model dependency-based ordering problems as directed graphs and optimize traversal cost. The array, graph, and DFS interview pattern is central. It rewards candidates who can identify the correct graph structure from the problem description and then apply DFS or topological reasoning to minimize total cost.

Algorithmic pattern used

DFS with graph modeling. Build a directed graph where edges represent "must break lock A before lock B" constraints. Use DFS to explore all valid orderings that respect these constraints. At each node, compute the cost of breaking the current lock given the set of already-broken locks (which determines the current strength). Track the minimum total cost across all valid orderings. Memoization on the graph state avoids redundant recomputation.

Example explanation

Locks: [A, B, C]. Constraints: A must be broken before C; B can be broken independently.

• Order B→A→C: time = time(B) + time(A) + time(C with A broken first).
• Order A→B→C: time = time(A) + time(B) + time(C with A broken first) → same for C. DFS explores both valid orderings and returns the one with minimum total time.

Common mistakes candidates make

• Ignoring graph constraints and trying all permutations (doesn't respect dependency order).
• Not memoizing DFS states, leading to exponential redundancy.
• Incorrectly computing the breaking cost when multiple predecessors have been broken.
• Missing the case where a lock has no predecessors (can be broken at any time).

Interview preparation tip

Hard problems that combine graph structure with ordering optimization often require DFS with careful state tracking. Before coding, draw the dependency graph explicitly and identify which orderings are valid. Then design your DFS state to capture exactly what information is needed to compute costs at each step. Practice building DFS solutions on DAGs (directed acyclic graphs) with accumulated costs — this pattern generalizes to many scheduling and dependency-resolution problems seen in senior engineering interviews.