Minimum Time to Reach Target With Limited Power - Video Solutions

Problem Overview: You need to reach a target position while consuming a limited amount of power. Each move takes time and may reduce available power, so the goal is to find the path that reaches the target in the minimum time without exceeding the power limit.

Approach 1: Brute Force Path Exploration (Exponential Time)

Enumerate every possible sequence of moves from the starting state while tracking remaining power. Each state stores the current position, elapsed time, and remaining power. Use recursive exploration or a queue to generate all paths and discard those that exceed the power limit. This approach quickly becomes impractical because the number of possible paths grows exponentially with the graph or grid size. Time complexity is O(b^d) where b is branching factor and d is depth; space complexity is also O(b^d) due to stored states.

Approach 2: BFS with State Tracking (O(V * P + E * P))

If every move takes the same amount of time, treat the problem as a layered graph where each node is represented as (position, remainingPower). Perform a BFS starting from the initial state and push valid transitions that do not exceed the power limit. Maintain a visited structure to avoid revisiting the same state with equal or worse power. BFS guarantees the first time you reach the target is the minimum time under uniform edge weights. Time complexity is O((V + E) * P) and space complexity is O(V * P), where P is the maximum power.

Approach 3: Dijkstra's Algorithm with Extended State (Optimal for Weighted Moves)

When moves have different time costs, model the problem as a shortest-path search over a state graph. Each state becomes (node, remainingPower). Use a priority queue ordered by total time. From each state, iterate through possible transitions, update time, and subtract power cost. Push the new state into the heap only if it improves the best known time for that configuration. This technique combines ideas from graph algorithms and priority queues. Time complexity is O((V * P + E * P) log (V * P)) and space complexity is O(V * P). In practice this performs well because many dominated states are pruned.

Approach 4: Pruned Shortest Path with Dominance Optimization

You can reduce the search space by discarding states that are strictly worse than previously seen ones. If you reach the same node with less remaining power and higher time, that state can never lead to an optimal solution. Maintaining a best-time table per node and power level significantly cuts the number of pushes into the priority queue. The complexity remains O((V * P) log (V * P)) in the worst case but performs closer to linear exploration in practice.

Recommended for interviews: Start by explaining the brute force exploration to show you understand the constraints. Then move quickly to the shortest-path formulation where the state includes remaining power. Interviewers typically expect the Dijkstra with extended state approach because it handles weighted transitions and demonstrates familiarity with shortest path algorithms and state compression techniques.